# Solutions for Physics for Scientists and Engineers with Modern Physics (2022)

Table of Content

 Physics and Measurement Motion in One Dimension Vectors Motion in Two Dimensions The Laws of Motion Circular Motion and Other Applications of Newtonâ€™s Laws Energy of a System Conservation of Energy Linear Momentum and Collisions Rotation of a Rigid Object About a Fixed Axis Angular Momentum Static Equilibrium and Elasticity Universal Gravitation Fluid Mechanics Oscillatory Motion Wave Motion Sound Waves Superposition and Standing Waves Temperature The First Law of Thermodynamics The Kinetic Theory of Gases Heat Engines, Entropy, and the Second Law of Thermodynamics Electric Fields Gaussâ€™s Law Electric Potential Capacitance and Dielectrics Current and Resistance Direct-Current Circuits Magnetic Fields Sources of the Magnetic Field Faradays Law Inductance Alternating-Current Circuits Electromagnetic Waves The Nature of Light and the Principles of Ray Optics Image Formation Wave Optics Diffraction Patterns and Polarization Relativity Introduction to Quantum Physics Quantum Mechanics Atomic Physics Molecules and Solids Nuclear Structure Applications of Nuclear Physics Particle Physics and Cosmology Particle Physics and Cosmology
• A cylinder that has a 40.0 -cm radius and is 50.0 cmcm deep is filled with air at 20.0∘0∘C and 1.00 atm (Fig. Pl9.62 a). A 20.0 -kg piston is now lowered into the cylinder, compressing the air trapped inside as it takes equilibrium height hi(Fig.P19.62b).hi(Fig.P19.62b). Finally, a 25.0−kg25.0−kg dog stands on the piston, further compressing the air, which remains at 20∘C20∘C.(Fig. P19.62c).P19.62c). (a) How far down (Δh)(Δh) does the piston move when the dog steps onto it? (b) To what temperature should the gas be warmed to raise the piston and dog back to hi?hi?
• Your dog is running around the grass in your back yard. He undergoes successive displacements 3.50 m south, 8.20 m northeast, and 15.0 m west. What is the resultant displacement?
• Consider a bright star in our night sky. Assume its distance from the Earth is 20.0 light-years (ly) and its power output is 4.00×1028W4.00×1028W , about 100 times that of the Sun. (a) Find the intensity of the starlight at the Earth. (b) Find the power of the starlight the Earth intercepts. One light-year is the distance traveled by light through a vacuum in one year.
• What If? The two capacitors of Problem 13 (C1=5.00μF(C1=5.00μF and C2=12.0μF)C2=12.0μF) are now connected in series and to a 9.00−V9.00−V battery. Find (a) the equivalent capacitance of the combination, (b) the potential difference across each capacitor, and (c) the charge on each capacitor.
• A block of mass 2.20 kg is accelerated across a rough surface by a light cord passing over a small pulley as shown in Figure P5.73. The tension TT in the cord is maintained at 10.0 NN , and the
pulley is 0.100 mm above the top of the block. The coefficient of kinetic friction is 0.400 . (a) Determine the acceleration of the block when x=0.400mx=0.400m (b) Describe the general behavior of the acceleration as the block slides from a location where xx is large to x=0.x=0. (c) Find the maximum value of the acceleration and the position xx for which it occurs. (d) Find the value of xx for which the acceleration is zero.
• Consider nn equal positively charged particles each of magnitude Q/nQ/n placed symmetrically around a circle of radius aa (a) Calculate the magnitude of the electric field at a point a distance xx from the center of the circle and on the line passing through the center and perpendicular to the plane of the circle. (b) Explain why this result is identical to the result of the calculation done in Example 23.7 .
• The Earth’s atmosphere consists primarily of oxygen (21%)(21%) and nitrogen (78%).(78%). The rms speed of oxygen molecules (O2)(O2) in the atmosphere at a certain location is 535 m/sm/s . (a) What is the temperature of the atmosphere at this location? (b) Would the rms speed of nitrogen molecules (N2)(N2) at this location be higher, equal to, or lower than 535 m/sm/s ? Explain. (c) Determine the rms speed of N2N2
at his location.
• The Earth reflects approximately 38.0% of the incident sunlight from its clouds and surface. (a) Given that the intensity of solar radiation at the top of the atmosphere is 1370 W/m2W/m2 , find the radiation pressure on the Earth, in pascals, at the location where the Sun is straight overhead. (b) State how this quantity compares with normal atmospheric pressure at the Earth’s surface, which is 101 kPakPa.
• Sodium is a mono valent metal having a density of 0.971 g/cm3g/cm3 and a molar mass of 23.0g/mol.23.0g/mol. Use this information to calculate (a) the density of charge carriers and (b) the Fermi energy of sodium.
• Gliese 581c is the first Earth-like extrasolar terrestrial planet discovered. Its parent star, Gliese 581, is a red dwarf that radiates electromagnetic waves with power 5.00×1024W, which is only 1.30% of the power of the Sun. Assume the emissivity of the planet is equal for infrared and for visible light and the planet has a uniform surface temperature. Identify (a) the projected area over which the planet absorbs light from Gliese 581 and (b) the radiating area of the planet. (c) If an average temperature of 287 K is necessary for life to exist on Gliese 581c, what should the radius of the planet’s orbit be?
• Use the quantum-particle-in-a-box model to calculate the first three energy levels of a neutron trapped
in an atomic nucleus of diameter 20.0 fm. (b) Explain whether the energy-level differences have a realistic order of magnitude.
• Figure P35.67P35.67 shows the path of a light beam through several slabs with different indices of refraction. (a) If θ1=θ1= 30.0∘,30.0∘, what is the angle θ2θ2 of the emerging beam? (b) What must the incident angle θ1θ1 be to have total internal reflection at the surface between the medium with n=1.20n=1.20 and the medium with n=1.00?n=1.00?
• A certain rain cloud at an altitude of 1.75 kmkm contains 3.20×107kg3.20×107kg of water vapor. How long would it take a 2.70−kW2.70−kW pump to raise the same amount of water from the Earth’s surface to the cloud’s position?
• Astronomers observe a 60.0 -MHz radio source both directly and by reflection from the sea as shown in Figure P37.15. If the receiving dish is 20.0 mm above sea level, what is the angle of the radio source above the horizon at first maximum?
• Laser light with a wavelength of 632.8 nm is directed through one slit or two slits and allowed to fall on a screen 2.60 m beyond. Figure P38.48 shows the pattern on the screen, with a centimeter ruler below it. (a) Did the light pass through one slit or two slits? Explain how you can determine the answer. (b) If one slit, find its width. If two slits, find the distance between their centers.
• A truck loaded with cannonball watermelons stops suddenly to avoid running over the edge of a washed-out bridge (Fig. P4.60). The quick stop causes a number of melons to fly off the truck. One melon leaves the hood of the truck with an initial speed vi=10.0m/svi=10.0m/s in the horizontal direction. A cross section of the bank has the shape of the bottom half of a parabola, with its vertex at the initial location of the projected watermelon and with the equation y2=16x,y2=16x, where xx and yy are measured in meters. What are the xx and yy coordinates of the melon when it splatters on the bank?
• Why is the following situation impossible? An experiment is performed on an atom. Measurements of the atom when it is in a particular excited state show five possible values of the zz component of orbital angular momentum, ranging between 3.16×10−34kg⋅m2/s3.16×10−34kg⋅m2/s and −3.16×10−34kg⋅m2/s−3.16×10−34kg⋅m2/s .
• A 20.0 -kg cannonball is fired from a cannon with muzzle speed of 1000 m/sm/s at an angle of 37.0∘0∘ with the horizontal. A second ball is fired at an angle of 90.0∘.90.0∘. Use the isolated system model to find (a) the maximum height reached by each ball and (b) the total mechanical energy of the ball- Earth system at the maximum height for each ball. Let y=0y=0 at the cannon.
• A toroid having a rectangular cross section (a=2.00cm by b=3.00cm) and inner radius R=4.00cm consists of N=500 turns of wire that carry a sinusoidal current I= Imaxsinωt, with Imax=50.0A and a frequency f=ω/2π= 60.0 Hz. A coil that consists of N′=20 turns of wire is wrapped around one section of the toroid as shown in Figure P31.17. Determine the emf induced in the coil as a function of time.
• Two protons in a molecule are 3.80×10−10m3.80×10−10m apart. Find the magnitude of the electric force exerted by one proton on the other. (b) State how the magnitude of this force compares with the magnitude of the gravitational force exerted by one proton on the other. (c) What If? What must be a particle’s charge-to-mass ratio if the magnitude of the gravitational force between two of these particles is equal to the magnitude of electric force between them?
• At saturation, when nearly all the atoms have their magnetic moments aligned, the magnetic field is equal to the permeability constant μ0μ0 multiplied by the magnetic moment per unit volume. In a sample of iron, where the number density of atoms is approximately 8.50×10288.50×1028 atoms /m3/m3 , the magnetic field can reach 2.00 TT . If each electron con- tributes a magnetic moment of 9.27×10−24A⋅27×10−24A⋅m2 (1 Bohr magneton), how many electrons per atom contribute to the saturated field of iron?
• A proton moves through a region containing a uniform electric field given by →E=50.0ˆjV/mE→=50.0j^V/m and a uniform magnetic field →B=(0.200ˆi+0.300j+0.400ˆk)T.B→=(0.200i^+0.300j+0.400k^)T. Determine the acceleration of the proton when it has a velocity →v=200ˆim/sv→=200i^m/s
• A block of mass M, supported by a string, rests on a frictionless incline making an angle θθ with the horizontal (Fig. P16.53). The length of the string is L,L, and its mass is m<<Mm<<M . Derive an expression for the time interval required for a transverse wave to travel from one end of the string to the other.
• With particular experimental methods, it is possible to produce and observe in a long, thin rod both a transverse wave whose speed depends primarily on tension in the rod and a longitudinal wave whose speed is determined by Young’s modulus and the density of the material according to the expression v=√Y/ρ.v=Y/ρ−−−−√. The transverse wave can be modeled as a wave in a stretched string. A particular metal rod is 150 cmcm long and has a radius of 0.200 cmcm and a mass of 50.9 gg . Young’s modulus for the material is 6.80×1010N/m26.80×1010N/m2 . What must the tension in the rod be if the ratio of the speed of longitudinal waves to the speed of transverse waves is 8.00??
• A student proposes to study the gravitational force by suspending two 100.0-kg spherical objects at the lower ends of cables from the ceiling of a tall cathedral and measuring the deflection of the cables from the vertical. The 45.00-m-long cables are attached to the ceiling 1.000 m apart. The first object is suspended, and its position is carefully measured. The second object is suspended, and the two objects attract each other gravitationally. By what distance has the first object moved horizontally from its initial position due to the gravitational attraction to the other object? Suggestion: Keep in mind that this distance will be very small and make appropriate approximations.
• What are the wavelengths of electromagnetic waves in free space that have frequencies of (a) 5.00×1019Hz5.00×1019Hz and (b) 4.00×109Hz4.00×109Hz ?
• A disk of radius RR (Fig.P25.73) has a nonuniform surface charge density σ=σ= Cr, where CC is a constant and rr is measured from the center of the disk to a point on the surface of the disk. Find (by direct integration) the electric potential at P.P.
• Two forces →F1F→1 and →F2F→2 act along the two sides of an equilateral triangle as shown in Figure P 11.9. Point OO is the intersection of the altitudes of the triangle. (a) Find a third force →F3F→3 to be applied at BB and along BCBC that will make the total torque zero about the point O.O. (b) What If? Will the total torque change if →F3F→3 is applied not at BB but at any other point along BC?BC?
• Calculate the absolute pressure at an ocean depth of 1000 $\mathrm{m}$ . Assume the density of seawater is 1030 $\mathrm{kg} / \mathrm{m}^{3}$ and the air above exerts a pressure of 101.3 $\mathrm{kPa}$ . (b) At this depth, what is the buoyant force on a spherical submarine having a diameter of 5.00 $\mathrm{m}$ ?
• Determine the equilibrium charge on the capaci- tor in the circuit of Figure P28.62 as a function of R.
(b) Evaluate the charge when R 5 10.0 V. (c) Can the charge on the capacitor be zero? If so, for what value of R? (d) What is the maximum possible magnitude of thecharge on the capacitor? For what value of R is it achieved?
(e) Is it experimentally meaningful to take R 5 `? Explain your answer. If so, what charge magnitude does it imply?
• A possible means of space flight is to place a perfectly reflecting aluminized sheet into orbit around the Earth and then use the light from the Sun to push this “solar sail.” Suppose a sail of area A=6.00×105m2A=6.00×105m2 and mass m=6.00×103kgm=6.00×103kg is placed in orbit facing the Sun. Ignore all gravitational effects and assume a solar intensity of 1370 W/m2W/m2 . (a) What force is exerted on the sail? ( b) What is the sail’s acceleration? (c) Assuming the acceleration calculated in part (b) remains constant, find the time interval required for the sail to reach the Moon, 3.84×108m3.84×108m away, starting from rest at the Earth.
• A quartz watch contains a crystal oscillator in the form of a block of quartz that vibrates by contracting and expanding. An electric circuit feeds in energy to maintain the oscillation and also counts the voltage pulses to keep time. Two opposite faces of the block, 7.05 mm apart, are antinodes, moving alternately toward each other and away from each other. The plane halfway between these two faces is a node of the vibration. The speed of sound in quartz is equal to 3.70×103m/s3.70×103m/s . Find the frequency of the vibration.
• Two long, straight wires cross each other perpendicularly as shown in Figure P30.61. The wires do not touch. Find (a) the magnitude and (b) the direction of the magnetic field at point P,P, which is in the same plane as the two wires. (c) Find the magnetic field at a point 30.0 cmcm above the point of intersection of the wires along the zz axis; that is, 30.0 cmcm out of the page, toward you.
• A 1.00−mF1.00−mF capacitor is connected to a North American electrical outlet (ΔVrms=120V,f=60.0Hz).(ΔVrms=120V,f=60.0Hz). Assuming the energy stored in the capacitor is zero at t=0,t=0, determine the magnitude of the current in the wires at t=1180st=1180s .
• A material having an index of refraction of 1.30 is used as an antireflective coating on a piece of glass (n=1.50).(n=1.50). What should the minimum thickness of this film be to minimize reflection of 500 -nm light?
• A singly charged ion of mass mm is accelerated from rest by a potential difference ΔVΔV . It is then deflected by a uniform magnetic field (perpendicular to the ion’s velocity) into a semicircle of radius R.R. Now a doubly charged ion of mass m′m′ is accelerated through the same potential difference and deflected by the same magnetic field into a semicircle of radius R′=2R.R′=2R. What is the ratio of the masses of the ions?
• (a) Derive an expression for the buoyant force on a spherical balloon, submerged in water, as a function of the depth hh below the surface, the volume ViVi of the balloon at the surface, the pressure P0P0 at the surface, and the density ρwρw of the water. Assume the water temperature does not change with depth. (b) Does the buogant force increase or decrease as the balloon is submerged? (c) At what depth is the buoyant force one-half the surface value?
• The law of atmospheres states that the number density of molecules in the atmosphere depends on height yy above sea level according to nV(y)=n0e−m0g0/kBTnV(y)=n0e−m0g0/kBT
where n0n0 is the number density at sea level (where y=0).y=0). The average height of a molecule in the Earth’s atmosphere is given by
yavg=∫∞0ynV(y)dy∫∞0nV(y)dy=∫∞0ye−m0g/kBTdy∫∞0e−m0g/kBTdyyavg=∫∞0ynV(y)dy∫∞0nV(y)dy=∫∞0ye−m0g/kBTdy∫∞0e−m0g/kBTdy
(a) Prove that this average height is equal to kBT/m0gkBT/m0g .
(b) Evaluate the average height, assuming the temperature is 10.0∘0∘C and the molecular mass is 28.9u,28.9u, both uniform throughout the atmosphere.
• A rigid, mass less rod has three particles with equal masses attached to it as shown in Figure P 11.49. The rod is free to rotate in a vertical plane about a friction less axle perpendicular to the rod through the point PP and is released from rest in the horizontal position at t=0t=0 . Assuming mm and dd are known, find (a) the moment of inertia of the system of three particles about the pivot, (b) the torque acting on the system at t=0,t=0, (c) the angular acceleration of the system at t=0,t=0, (d) the linear acceleration of the particle labeled 3 at t=0,(e)t=0,(e) the maximum kinetic energy of the system, (f) the maximum angular speed reached by the rod, (g)(g) the maximum angular momentum of the system, and (h)(h) the maximum speed reached by the particle labeled 2 .
• At 20.0∘C,20.0∘C, an aluminum ring has an inner diameter of 5.0000 cmcm and a brass rod has a diameter of 5.0500cm.5.0500cm. (a) If only the ring is warmed, what temperature must it reach so that it will just slip over the rod? (b) What If? If both the ring and the rod are warmed together, what temperature must they both reach so that the ring barely slips over the rod? (c) Would this latter process work? Explain. Hint: Consult Table 20.2 in the next chapter.
• Figure P31.25 shows a bar of mass m=0.200kg that can slide without friction on a pair of rails separated by a distance ℓ=1.20m and located on an inclined plane that makes an angle θ=25.0∘ with respect to the ground. The resistance of the resistor is R=1.00Ω and a uniform magnetic field of magnitude B=0.500T is directed downward, perpendicular to the ground, over the entire region through which the bar moves. With what constant speed v does the bar slide along the rails?
• A concave spherical mirror has a radius of curvature of magnitude $24.0 \mathrm{cm} .$ (a) Determine the object position for which the resulting image is upright and larger than the object by a factor of $3.00 .$ (b) Draw a ray diagram to deter- mine the position of the image. (c) Is the image real or virtual?
• Two 2.00×103−kg2.00×103−kg cars both traveling at 20.0 m/sm/s undergo a head-on collision and stick together. Find the change in entropy of the surrounding air resulting from the collision if the air temperature is 23.0∘0∘C . Ignore the energy carried away from the collision by sound.
• A light spring of constant $k=90.0 \mathrm{N} / \mathrm{m}$ is attached vertically to a table (Fig. Pl4.6 la). A 2.00 -g balloon is filled with helium (density $=0.179 \mathrm{kg} / \mathrm{m}^{3} )$ to a volume of 5.00 $\mathrm{m}^{3}$ and is then connected with a light cord to the spring, causing the spring to stretch as shown in Figure Pl4.61b. Determine the extension distance $L$ when the balloon is in
equilibrium.
• A beam of monochromatic light is incident on a single slit of width 0.600 mm. A diffraction pattern forms on a wall 1.30 m beyond the slit. The distance between the positions of zero intensity on both sides of the central maximum is 2.00 mm. Calculate the wavelength of the light.
• An aluminum block of mass m1=2.00kgm1=2.00kg and a copper block of mass m2=6.00kgm2=6.00kg are connected by a light string over a frictionless pulley. They sit on a steel surface as shown in Figure P5.62, where θ=30.0∘.θ=30.0∘. (a) When they are released from rest, will they start to move? If they do, determine (b) their acceleration and (c) the tension in the string. If they do not move, determine (d) the sum of the magnitudes of the forces of friction acting on the blocks.
• Why is the following situation impossible? A 10.0−μF10.0−μF capacitor has plates with vacuum between them. The capacitor is charged so that it stores 0.0500 JJ of energy. A particle with
charge −3.00μC−3.00μC is fired from the positive plate toward the negative plate with an initial kinetic energy equal to 1.00×10−4J.1.00×10−4J. The particle arrives at the negative plate with a reduced kinetic energy.
• Two resistors R1 and R2 are in parallel with each other. Together they carry total current I. (a) Determine the current in each resistor. (b) Prove that this division of the total current I between the two resistors results in less power delivered to the combination than any other division. It is a general principle that current in a direct current circuit distributes itself so that the total power delivered to the circuit is a
• Calculate the difference in binding energy per nucleon for the nuclei2311Na2311Na and 2312Mg.2312Mg. (b) How do you account for the difference?
• A series RLCRLC circuit contains the following components: R=150Ω,L=0.250H,C=2.00μFR=150Ω,L=0.250H,C=2.00μF , and a source with ΔVmax=210VΔVmax=210V operating at 50.0 HzHz . Our goal is to find the phase angle, the power factor, and the power input for this circuit. (a) Find the inductive reactance in the circuit. (b) Find the capacitive reactance in the circuit. (c) Find the impedance in the circuit. (d) Calculate the maximum current in the circuit. (e) Determine the phase angle between the current and source voltage. (f) Find the power factor for the circuit. (g) Find the power input to the circuit.
• Following a collision in outer space, a copper disk at 850∘C850∘C is rotating about its axis with an angular speed of 25.0 rad/srad/s . As the disk radiates infrared light, its temperature falls to 20.0∘C20.0∘C . No external torque acts on the disk. (a) Does the angular speed change as the disk cools? Explain how it changes or why it does not. (b) What is its angular speed at the lower temperature?
• At an intersection of hospital hallways, a convex spherical mirror is mounted high on a wall to help people avoid collisions. The magnitude of the mirror’s radius of curvature is 0.550 $\mathrm{m}$ (a) Locate the image of a patient 10.0 $\mathrm{m}$ from the mirror. (b) Indicate whether the image is upright or inverted. (c) Determine the magnification of the image.
• A block of mass m=2.50kgm=2.50kg is pushed a distance d=2.20md=2.20m along aa frictionless, horizontal table by a constant applied force of magnitude F=16.0NF=16.0N directed at an angle θ=25.0∘θ=25.0∘ below the horizontal as shown in Figure P7.1. Determine the work done on the block by (a) the applicd force, (b) the normal force exerted by the table, (c) the gravitational force, and (d) the net force on the block.
• Show that the density of an ideal gas occupying a volume VV is given by ρ=PM/RT,ρ=PM/RT, where MM is the molar mass. (b) Determine the density of oxygen gas at atmospheric pressure and 20.0∘0∘C .
• Two waves are described by the wave functions
y1(x,t)=5.00sin(2.00x−10.0t)y1(x,t)=5.00sin⁡(2.00x−10.0t)
y2(x,t)=10.0cos(2.00x−10.0t)y2(x,t)=10.0cos⁡(2.00x−10.0t)
where x,y1,x,y1, and y2y2 are in meters and tt is in seconds. (a) Show that the wave resulting from their superposition can be expressed as a single sine function. (b) Determine the amplitude and phase angle for this sinusoidal wave.
• A Carnot engine has a power output PP . The engine operates between two reservoirs at temperature TcTc and TkTk . (a) How much energy enters the engine by heat in a time interval Δt?Δt? (b) How much energy is exhausted by heat in the time interval ΔtΔt ?
• Two identical parallel-plate capacitors, each with capacitance C,C, are charged to potential difference ΔVΔV and then disconnected from the battery. They are then connected to each other in parallel with plates of like sign connected. Finally, the plate separation in one of the capacitors is doubled. (a) Find the total energy of the system of two capacitors before the plate separation is doubled. (b) Find the potential difference across each capacitor after the plate separation is doubled. (c) Find the total energy of the system after the plate separation is doubled. (d) Reconcile the difference in the answers to parts (a) and (c) with the law of conservation of energy.
• The output voltage of an AC source is given by Δv=Δv= 120sin30.0πt,120sin30.0πt, where ΔvΔv is in volts and tt is in seconds. The source is connected across a 0.500−H0.500−H inductor. Find (a) the frequency of the source, (b) the rms voltage across the inductor, (c) the inductive reactance of the circuit, (d) the rms current in the inductor, and (e) the maximum current in the inductor.
• Why is the following situation impossible? A uniform beam of mass mb=3.00kgmb=3.00kg and length ℓ=1.00mℓ=1.00m supports blocks with masses m1=5.00kgm1=5.00kg and m2=15.0kgm2=15.0kg at two positions as shown in Figure P12.2.P12.2. The beam rests on two triangular blocks, with point PP a distance d=0.300md=0.300m to the right of the center of gravity of the beam. The position of the object of mass m2m2 is adjusted along the length of the beam until the normal force on the beam at OO is zero.
• A metal rod having a mass per unit length λλ carries a current I.I. The rod hangs from two wires in a uniform vertical magnetic field as shown in Figure P 29.69. The wires make an angle θθ with the vertical when in equilibrium. Determine the magnitude of the magnetic field.
• A string fixed at both ends and having a mass of 4.80 g, a length of 2.00 m, and a tension of 48.0 N vibrates in its second (n=2)(n=2) normal mode. (a) Is the wavelength in air of the sound emitted by this vibrating string larger or smaller than the wavelength of the wave on the string? (b) What is the ratio of the wavelength in air of the sound emitted by this vibrating string and the wavelength of the wave on the string?
• The bar of mass m in Figure P31.74 is pulled horizontally across parallel, frictionless rails by a massless string that passes over a light, frictionless pulley and is attached to a suspended object of mass M. The uniform upward magnetic field has a magnitude B , and the distance between the rails is ℓ. The only significant electrical resistance is the load resistor R shown connecting the rails at one end. Assuming the suspended object is released with the bar at rest at t=0, derive an expression that gives the bar’s horizontal speed as a function of time.
• When an object is displaced by an amount x from stable equilibrium, a restoring force acts on it, tending to return the object to its equilibrium position. The magnitude of the restoring force can be a complicated function of x. In such cases, we can generally imagine the force function F(x) to be expressed as a power series in x as F(x)=−(k1x+k2x2+ k3x3+…). The first term here is just Hooke’s law, which describes the force exerted by a simple spring for small displacements. For small excursions from equilibrium, we generally ignore the higher-order terms, but in some cases it may be desirable to keep the second term as well. If we model the restoring force as F=−(k1x+k2x2), how much work is done on an object in displacing it from x=0 to x= xmax by an applied force −F?
• A pitcher throws a 0.142−kg0.142−kg baseball at 47.2 m/sm/s . As it trav-
els 16.8 mm to home plate, the ball slown to 42.5 m/sm/s
because of air resistance. Find the change in temperature of the air through which it passes. To find the greatest possible temperature change, you may make the following assumptions. Air has a molar specific heat of CP=72RCP=72R and an equivalent molar mass of 28.9g/mol.28.9g/mol. The process is so rapid that the cover of the baseball acts as thermal insulation and the temperature of the ball itself does not change. A change in temperature happens initially only for the air in a cylinder 16.8 mm in length and 3.70 cmcm in radius. This air is initially at 20.0∘0∘C .
• An unstable particle at rest spontaneously breaks into two fragments of unequal mass. The mass of the first fragment is 2.50×10−28kg2.50×10−28kg , and that of the other is 1.67×1.67× 10−27kg10−27kg . If the lighter fragment has a speed of 0.893cc after the breakup, what is the speed of the heavier fragment?
• The speed of a one-dimensional compressional wave traveling along a thin copper rod is 3.56 km/s. The rod is given a sharp hammer blow at one end. A listener at the far end of the rod hears the sound twice, transmitted through the metal and through air, with a time interval ΔtΔt between the two pulses. (a) Which sound arrives first? (b) Find the length of the rod as a function of Δt.Δt. (c) Find the length of the rod if Δt=127msΔt=127ms . (d) Imagine that the copper rod is replaced by another material through which the speed of sound is vrvr . What is the length of the rod in terms of tt and v2rv2r (e) Would the answer to part (d) go to a well-defined limit as the speed of sound in the rod goes to infinity? Explain your answer.
• Consider a system of electrons confined to a three-dimensional box. Calculate the ratio of the number of allowed energy levels at 8.50 eV to the number at 7.05 eV. (b) What If? Copper has a Fermi energy of 7.05 eV at 300 K. Calculate the ratio of the number of occupied levels in copper at an energy of 8.50 eV to the number at the Fermi energy. (c) How does your answer to part (b) compare with that obtained in part (a)?
• Prepare a table like Table 22.1 by using the same procedure (a) for the case in which you draw three marbles from your bag rather than four and (b) for the case in which you draw five marbles rather than four.
• Calculate the potential difference between points aa and
bb in Figure P28.53P28.53 and (b)(b) identify which point is at the
higher potential.
• A rectangular loop of area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to B=Bmaxe−t/τ, where Bmax and τ are constants. The field has the constant value Bmax for t<0 . Find the emf induced in the loop as a function of time.
• Determine the magnetic field (in terms of I,a,I,a, and d)d) at the origin due to the current loop in Figure P30.17. The loop extends to infinity above the figure.
• An incompressible, nonviscous fluid is initially at rest in the vertical portion of the pipe shown in Figure Pl4.71a, where $L=2.00 \mathrm{m} .$ When the valve is opened, the fluid flows into the horizontal section of the pipe. What is the fluid’s speed when all the fluid is in the horizontal section as shown in Figure $\mathrm{P} 14.71 \mathrm{b}$ ? Assume the cross-sectional area of the entire pipe is constant.
• Marie Cornu, a physicist at the Polytechnic Institute in Paris, invented phasors in about 1880. This problem helps you see their general utility in representing oscillations. Two mechanical vibrations are represented by the expressions
y1=12.0sin4.50t
and
y2=12.0sin(4.50t+70.0∘)
where y1 and y2 are in centimeters and t is in seconds. Find the amplitude and phase constant of the sum of these functions (a) by using a trigonometric identity (as from Appendix B) and (b) by representing the oscillations as phasors. (c) State the result of comparing the answers to parts (a) and (b). (d) Phasors make it equally easy to add traveling
Find the amplitude and phase constant of the sum of the three waves represented by
y1=12.0sin(15.0x−4.50t+70.0∘)
y2=15.5sin(15.0x−4.50t−80.0∘)
y3=17.0sin(15.0x−4.50t+160∘)
where x,y1,y2, and y3 are in centimeters and t is in seconds.
• Vector →AA→ has a magnitude of 29 units and points in the positive yy direction. When vector →BB→ is added to →A,A→, the resultant vector →A+→BA→+B→ points in the negative yy direction with a magnitude of 14 units. Find the magnitude and direction of →BB→ .
• A long solenoid with 1.00×103 turns per meter and radius 2.00 cm carries an oscillating current I=5.00 sin 100πt, where I is in amperes and t is in seconds. (a) What is the electric field induced at a radius r=1.00cm from the axis of the solenoid? (b) What is the direction of this electric field when the current is increasing counterclockwise in the solenoid?
• A goldfish is swimming at 2.00 $\mathrm{cm} / \mathrm{s}$ toward the front wall of a rectangular aquarium. What is the apparent speed of the fish measured by an observer looking in from outside the front wall of the tank?
• Why is the beta decay p→n+e++ν forbidden for a free proton? (b) What If? Why is the same reaction possible if the proton is bound in a nucleus? For example, the following reaction occurs:
137N→136C+e++ν
(c) How much energy is released in the reaction given in part (b)?
• A child of mass m swings in a swing supported by two chains, each of length R. If the tension in each chain at the lowest point is T, find (a) the child’s speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)
• A block weighing 40.0 NN is suspended from a spring that has a force constant of 200 N/mN/m . The system is undamped (b=0)(b=0) and is subjected to a harmonic driving force of frequency 10.0 HzHz , resulting in a forced-motion amplitude of 2.00cm.2.00cm. Determine the maximum value of the driving force.
• A particle moving along the xx axis in simple harmonic motion starts from its equilibrium position, the origin, at t=0t=0 and moves to the right. The amplitude of its motion is 2.00cm,2.00cm, and the frequency is 1.50 HzHz . (a) Find an expression for the position of the particle as a function of time. Determine (b) the maximum speed of the particle and (c) the earliest time (t>0)(t>0) at which the particle has this speed. Find (d) the maximum positive acceleration of the particle and (e) the earliest time (t>0)(t>0) at which the particle has this acceleration. (f) Find the total distance traveled by the particle between t=0t=0 and t=1.00t=1.00 s.
• In a nuclear power plant, the fuel rods last 3 yr before they are replaced. The plant can transform energy at a maximum possible rate of 1.00 GWGW . Supposing it operates at 80.0%% capacity for 3.00yr,3.00yr, what is the loss of mass of the fuel?
• Radio waves of wavelength 125 mm from a galaxy reach a radio telescope by two separate paths as shown in Figure P37.15.P37.15. One is a direct path to the receiver, which is situated on the edge of a tall cliff by the ocean, and the second is by reflection off the water. As the galaxy rises in
the east over the water, the first minimum of destructive interference occurs when the galaxy is θ=25.0∘θ=25.0∘ above the horizon. Find the height of the radio telescope dish above the water.
• A hard rubber ball, released at chest height, falls to the pavement and bounces back to nearly the same height. When it is in contact with the pavement, the lower side of the ball is temporarily flattened. Suppose the maximum depth of the dent is on the order of 1 cm. Find the order of magnitude of the maximum acceleration of the ball while it is in contact with the pavement. State your assumptions, the quantities you estimate, and the values you estimate for them.
• A radar transmitter contains an LCLC circuit oscillating at 1.00×1010Hz1.00×1010Hz . ( a) For a one-turn loop having an inductance of 400 pHpH to resonate at this frequency, what capacitance is required in series with the loop? (b) The capacitor has square, parallel plates separated by 1.00 mmmm of air. What should the edge length of the plates be? (c) What is the common reactance of the loop and capacitor at resonance?
• A cylinder is closed at both ends and has insulating walls. It is divided into two compartments by an insulating piston that is perpendicular to the axis of the cylinder as shown in Figure P21.71a. Each compartment contains 1.00 mol of oxygen that behaves as an ideal gas with g 5 1.40. Initially,
the two compartments have equal volumes and their temperatures are 550 K and 250 K. The piston is then allowed to move slowly parallel to the axis of the cylinder until it comes to rest at an equilibrium position (Fig. P21.71b). Find the final temperatures in the two compartments.
• Refer to Problem 74 for the statement of Fermat’s principle of least time. Derive the law of reflection (Eq. 35.2) from Fermat’s principle.
• In 1963, astronaut Gordon Cooper orbited the Earth 22 times. The press stated that for each orbit, he aged two-millionths of a second less than he would have had he remained on the Earth. (a) Assuming Cooper was 160 km above the Earth in a circular orbit, determine the difference in elapsed time between someone on the Earth and the orbiting astronaut for the 22 orbits. You may use the approximation
1√1−x≈1+x211−x−−−−−√≈1+x2
for small xx (b) Did the press report accurate information? Explain.
• Protons in an accelerator at the Fermi National Laboratory near Chicago are accelerated to a total energy that is 400 times their rest energy. (a) What is the speed of these protons in terms of c?c? (b) What is their kinetic energy in MeV?
• A battery with ε=6.00Vε=6.00V and no internal resistance supplies current to the circuit shown in Figure P28.11.P28.11. When the double-throw switch SS is open as shown in the figure, the current in the battery is 1.00 mAmA . When the switch is closed in position a,a, the current in the battery is 1.20 mAmA . When the switch is closed in position bb , the current in the battery is 2.00 mAmA . Find the resistances (a) R1,(b)R2,R1,(b)R2, and (c) R3R3 .
• A triangular glass prism with apex angle Φ=60.0∘Φ=60.0∘ has an index of refraction n=1.50n=1.50 (Fig, P35.37). What is the smallest angle of incidence θ1θ1 for which a light ray can emerge from the other side?
• A power plant, having a Carnot efficiency, produces electric power PP from turbines that take in energy from steam at temperature ThTh and discharge energy at temperature TcTc through a heat exchanger into a flowing river. The water downstream is warmer by ΔTΔT due to the output of the power plant. Determine the flow rate of the river.
• On a clear day at a certain location, a 100−V/m100−V/m vertical electric field exists near the Earth’s surface. At the same place, the Earth’s magnetic field has a magnitude of 0.500×0.500×
10−410−4 T. Compute the energy densities of (a) the electric field and (b) the magnetic field.
• Consider a light wave passing through a slit and propagating toward a distant screen. Figure P 38.72 shows the intensity variation for the pattern on the screen. Give a mathematical argument that more than 90% of the transmitted energy is in the central maximum of the diffraction pattern. Suggestion: You are not expected to calculate the precise percentage, but explain the steps of your reasoning. You may use the identification
112+132+152+…=π28112+132+152+…=π28
• A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a frictionless, horizontal track (Fig. P15.75). The force constant of the spring is k, and the equilibrium length is ,. Assume all portions of the spring oscillate in phase and the velocity of a segment of the spring of length dx is proportional to the distance xx from the fixed end; that is, vx=(x/ℓ)vvx=(x/ℓ)v . Also,
notice that the mass of a segment of the spring is dm=dm= (m/ℓ)dx(m/ℓ)dx . Find (a)(a) the kinetic energy of the system when the block has a speed vv and (b)(b) the period of oscillation.
• A 1.00 -kg beaker containing 2.00 $\mathrm{kg}$ of oil (density = 916.0 $\mathrm{kg} / \mathrm{m}^{3}$ ) rests on a scale. A. 2.00 -kg block of iron suspended from a spring scale is completely submerged in the oil as shown in Figure $\mathrm{P} 14.65 .$ Determine the equilibrium readings of both scales.
• Calculate the de Broglie wavelength for a proton moving with a speed of 1.00×106m/s1.00×106m/s .
• The quark composition of the proton is uud, whereas that of the neutron is udd. Show that the charge, baryon number, and strangeness of these particles equal the sums of these numbers for their quark constituents.
• A light spring with spring constant k1 is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant k2 . An object of mass m is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system.
• Consider the system shown in Figure P10.44 with m1=20.0kg,m2=12.5kg,R=0.200m,m1=20.0kg,m2=12.5kg,R=0.200m, and the mass of the pulley M=5.00kg.M=5.00kg. Object m2m2 is resting on the floor, and object m1m1 is 4.00 mm above the floor when it is released from rest. The pulley axis is frictionless. The cord is light, does not stretch, and does not slip on the pulley. (a) Calculate the time interval required for m1m1 to hit the floor. (b) How would your answer change if the pulley were massless?
• The average thermal conductivity of the walls (including the windows) and roof of the house depicted in Figure P20.64P20.64 is 0.480W/m⋅∘C,0.480W/m⋅∘C, and their average thickness is 21.0cm.21.0cm. The house is kept warm with natural gas having a heat of combustion (that is, the energy provided per cubic meter of gas burned) of 9300kcal/m3.9300kcal/m3. How many cubic meters of gas must be burned each day to maintain an inside temperature of 25.0∘0∘C if the outside temperature is 0.0∘C0.0∘C ? Disregard radiation and the energy transferred by heat through the ground.
• A 1.00-mol sample of a monatomic ideal gas is taken through the cycle shown in Figure P22.62. At point A, the pressure, volume, and temperature are Pi,Vi, and Ti , respectively. In terms of R and Ti , find (a) the total energy entering the system by heat per cycle, (b) the total energy leaving the system by heat per cycle, and (c) the cfficiency of an engine operating in this cycle. (d) Explain how the efficiency compares with that of an engine operating in a Carnot cycle between the same temperature extremes.
• In the Bohr model of the hydrogen atom, an electron travels in a circular path. Consider another case in which an electron travels in a circular path: a single electron moving perpendicular to a magnetic field →BB→ . Lev Davidovich Landau (1908−1968)(1908−1968) solved the Schrödinger equation for such an electron. The electron can be considered as a model atom without a nucleus or as the irreducible quantum limit of the cyclotron. Landau proved its energy is quantized in uniform steps of eℏB/me.eℏB/me. In 1999,1999, a single electron was trapped by a Harvard University research team in an evacuated centimeter-size metal can cooled to a temperature of 80 mKmK . In a magnetic field of magnitude 5.26 TT , the electron circulated for hours in its lowest energy level. (a) Evaluate the size of a quantum jump in the electron’s energy. (b) For comparison, evaluate kBTkBT as a measure of the energy available to the electron in blackbody radiation from the walls of its container. Microwave radiation was introduced to excite the electron. Calculate (c) the frequency and (d) the wavelength of the photon the electron absorbed as it jumped to its second energy level. Measurement of the resonant absorption frequency verified the theory and permitted precise determination of properties of the electron.
• A solid sphere is released from height hh from the top of an incline making an angle θθ with the horizontal. Calculate the speed of the sphere when it reaches the bottom of the incline (a) in the case that it rolls without slipping and (b) in the case that it slides frictionlessly without rolling. (c) Compare the time intervals required to reach the bottom in cases (a) and (b).
• Consider an object with any one of the shapes displayed in Table 10.2. What is the percentage increase in the moment of inertia of the object when it is warmed from 0∘C0∘C to 100∘C100∘C if it is composed of (a) copper or (b) aluminum? Assume the average linear expansion coefficients shown in Table 19.1 do not vary between 0∘C0∘C and 100∘C100∘C . (c) Why are the answers for parts (a) and (b) the same for all the shapes?
• At one location on the Earth, the rms value of the magnetic field caused by solar radiation is 1.80μTμT . From this value, calculate (a) the rms electric field due to solar radiation, (b) the average energy density of the solar component of electromagnetic radiation at this location, and (c) the average magnitude of the Poynting vector for the Sun’s radiation.
• A thin, uniform, rectangular signboard hangs vertically above the door of a shop. The sign is hinged to a stationary horizontal rod along its top edge. The mass of the sign is 2.40 kg, and its vertical dimension is 50.0 cm. The sign is swinging without friction, so it is a tempting target for children armed with snowballs. The maximum angular displacement of the sign is 25.0∘0∘ on both sides of the vertical. At a moment when the sign is vertical and moving to the left, a snowball of mass 400 g, traveling horizontally with a velocity of 160 cm/s to the right, strikes perpendicularly at the lower edge of the sign and sticks there. (a) Calculate the angular speed of the sign immediately before the impact. (b) Calculate its angular speed immediately after the impact. (c) The spattered sign will swing up through what maximum angle?
• A method called neutron activation analysis can be used for chemical analysis at the level of isotopes. When a sample is irradiated by neutrons, radioactive atoms are produced continuously and then decay according to their characteristic half-lives. (a) Assume one species of radioactive nuclei is produced at a constant rate RR and its decay is described by the conventional radioactive decay law. Assuming irradiation begins at time t=0t=0 , show that the number of radio- active atoms accumulated at time tt is
N=Rλ(1−e−λt)N=Rλ(1−e−λt)
(b) What is the maximum number of radioactive atoms that can be produced?
• Figure P 29.67 shows a schematic representation of an apparatus that can be used to measure magnetic fields. A rectangular coil of wire contains N turns and has a width w.w. The coil is attached to one arm of a balance and is suspended between the poles of a magnet. The magnetic field is uniform and perpendicular to the plane of the coil. The system is first balanced when the current in the coil is zero. When the switch is closed and the coil carries a current II, a mass mm must be added to the right side to balance the system. (a) Find an expression for the magnitude of the magnetic field. (b) Why is the result independent of the vertical dimensions of the coil? (c) Suppose the coil has 50 turns and a width of 5.00 cm. When the switch is closed, the coil carries a current of 0.300 A, and a mass of 20.0 g must be added to the right side to balance the system. What is the magnitude of the magnetic field?
• A magnetic field directed into the page changes with time according to B=0.0300t2+1.40, where B is in teslas and t is in seconds. The field has a circular cross section of radius R=2.50cm (see Fig. P31.33). When t=3.00 s and r2=0.0200m, what are (a) the magnitude and (b) the direction of the electric field at point P2?
• Extremely low-frequency (ELF) waves that can penetrate the oceans are the only practical means of communicating with distant submarines. (a) Calculate the length of a quarter-wavelength antenna for a transmitter generating ELF waves of frequency 75.0 Hz into air. (b) How practical is this means of communication?
• Consider the following wave function in SI units:
ΔP(r,t)=(25.0r)sin(1.36r−2030t)ΔP(r,t)=(25.0r)sin(1.36r−2030t)
Explain how this wave function can apply to a wave radiating from a small source, with r being the radial distance from the center of the source to any point outside the source. Give the most detailed description of the wave that you can. Include answers to such questions as the following and give representative values for any quantities that can be evaluated. (a) Does the wave move more toward the right or the left? (b) As it moves away from the source, what happens to its amplitude? (c) Its speed? (d) Its frequency? (e) Its wavelength? (f) Its power? (g) Its intensity?
• In 1993, the U.S. government instituted a requirement that all room air conditioners sold in the United States must have an energy efficiency ratio (EER) of 10 or higher. The EER is defined as the ratio of the cooling capacity of the air conditioner, measured in British thermal units per hour, or Btu/h, to its electrical power requirement in watts. (a) Convert the EER of 10.0 to dimensionless form, using the conversion 1Btu=1055J1Btu=1055J . (b) What is the appropriate name for this dimensionless quantity? (c) In the 1970 ss , it was common to find room air conditioners with EERs, of 5 or lower. State how the operating costs compare for 10 000 -Btu/h air conditioners with EERs of 5.00 and 10.0 . Assume each air conditioner operates for 1500 h during the summer in a city where electricity costs 17.0€€ per kWhkWh .
• The circuit in Figure P28.37 has been connected for a long time. (a) What is the potential difference across the capacitor? (b) If the battery is disconnected from the circuit, over what time interval does the capacitor discharge to one-tenth its initial voltage?
• Three 100−Ω100−Ω resistors are connected as shown in Figure P28.7. The maximum power that can safely be delivered to any one resistor is 25.0 WW . (a) What is the maximum potential difference that can be applied to the terminals aa and b?b? (b) For the voltage determined in part (a), what is the power delivered to each resistor? (c) What is the total power delivered to the combination of resistors?
• Why is the following situation impossible? A student is listening to the sounds from an air column that is 0.730 m long. He doesn’t know if the column is open at both ends or open at only one end. He hears resonance from the air column at frequencies 235 Hz and 587 Hz.
• A particle with charge of 12.0μCμC is placed at the center of a spherical shell of radius 22.0cm.22.0cm. What is the total electric flux through (a) the surface of the shell and (b) any hemispherical surface of the shell? (c) Do the results depend on the radius? Explain.
• Fifty turns of insulated wire 0.100 cm in diameter are tightly wound to form a flat spiral. The spiral fills a disk surrounding a circle of radius 5.00 cm and extending to a radius 10.00 cm at the outer edge. Assume the wire carries a current I at the center of its cross section. Approximate each turn of wire as a circle. Then a loop of current exists at radius 5.05 cm, another at 5.15 cm, and so on. Numerically calculate the magnetic field at the center of the coil.
• An 11.0 -W energy-efficient fluorescent lightbulb is designed to produce the same illumination as a conventional 40.0−W40.0−W incandescent lightbulb. Assuming a cost of $0.110/kWh$0.110/kWh for energy from the electric company, how much money does the user of the energy-efficient bulb save during 100 hh of use?
• A smooth cube of mass mm and edge length rr slides with speed vv on a horizontal surface with negligible friction. The cube then moves up a smooth incline that makes an angle θθ with the horizontal. A cylinder of mass mm and radius rolls without slipping with its center of mass moving with speed vv and encounters an incline of the same angle of inclination but with sufficient friction that the cylinder continues to roll without slipping. (a) Which object will go the greater distance up the incline? (b) Find the difference between the maximum distances the objects travel up the incline. (c) Explain what accounts for this difference in distances traveled.
• Why is the following situation impossible? Consider the lens-mirror combination shown in Figure P36.72. The lens has a focal length of $f_{\mathrm{L}}=0.200 \mathrm{m}$ , and the mirror has a focal length of $f_{\mathrm{M}}=0.500 \mathrm{m}$ An object is placed at $p=0.300 \mathrm{m}$ an object is placed at $p=0.300 \mathrm{m}$ from the lens. By moving a screen to various positions to the left of the lens, a student finds two different positions of the screen that produce a sharp image of the object. One of these positions corresponds to light leaving the object and traveling to the left through the lens. The other position corresponds to light traveling to the right from the object, reflecting from the mirror and then passing through the lens.
• Gas is confined in a tank at a pressure of 11.0 atmatm and a temperature of 25.0∘0∘C . If two-thirds of the gas is withdrawn and the temperature is raised to 75.0∘C,75.0∘C, what is the pressure of the gas remaining in the tank?
• A hydrogen atom is in its second excited state, corresponding to n=3.n=3. Find (a)(a) the radius of the electron’s Bohr orbit and (b)(b) the de Broglie wavelength of the electron in this orbit.
• Given a 2.50−μF2.50−μF capacitor, a 6.25−μF6.25−μF capacitor, and a 6.00−V6.00−V
battery, find the charge on each capacitor if you connect them (a) in series across the battery and (b)(b) in parallel across the battery.
• A sleeping area for a long space voyage consists of two cabins each connected by a cable to a central hub as shown in Figure P13.50. The cabins are set spinning around the hub axis, which is connected to the rest of the spacecraft to generate artificial gravity in the cabins. A space traveler lies in a bed parallel to the outer wall as shown in Figure P13.50. (a) With r=10.0m,r=10.0m, what would the angular speed of the 60.0 -kg traveler need to be if he is to experience half his normal Earth weight? (b) If the astronaut stands up perpendicular to the bed, without holding on to anything with his hands, will his head be moving at a faster, a slower, or the same tangential speed as his feet? Why? (c) Why is the action in part (b) dangerous?
• Use energy methods to calculate the distance of closest approach for a head-on collision between an alpha particle having an initial energy of 0.500 MeV and a gold nucleus (197Au)(197Au) at rest. Assume the gold nucleus remains at rest during the collision. (b) What minimum initial speed
must the alpha particle have to approach as close as 300 fmfm to the gold nucleus?
• A window washer pulls a rubber squeegee down a very tall vertical window. The squeegee has mass 160 g and is mounted on the end of a light rod. The coefficient of kinetic friction between the squeegee and the dry glass is 0.900. The window washer presses it against the window with a force having a horizontal component of 4.00 N. (a) If she pulls the squeegee down the window at constant velocity, what vertical force component must she exert? (b) The window washer increases the downward force component by 25.0%, while all other forces remain the same. Find the squeegee’s acceleration in this situation. (c) The squeegee is moved into a wet portion of the window, where its motion is resisted by a fluid drag force R proportional to its velocity according to R=−20.0v,R=−20.0v, where RR is in newtons and vv is in meters per second. Find the terminal velocity that the squeegee approaches, assuming the window washer exerts the same force described in part (b).
• A car travels due east with a speed of 50.0 km/hkm/h . Raindrops are falling at a constant speed vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of 60.0∘0∘ with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth.
• An electron has a kinetic energy of 3.00 eV. Find its wavelength. (b) What If? A photon has energy 3.00 eV. Find its wavelength.
• A spacecraft with a proper length of LpLp passes by an observer on the Earth. According to this observer, it takes a time interval ΔtΔt for the spacecraft to pass a fixed point. Determine the speed of the object as measured by the Earth-based observer.
• The elastic limit of a steel wire is 2.70×108Pa2.70×108Pa . What is the maximum speed at which transverse wave pulses can propagate along this wire without exceeding this stress? (The density of steel is 7.86×103kg/m3.)7.86×103kg/m3.)
• Consider a series RLCRLC circuit having the parameters R=200Ω,L=663mH,R=200Ω,L=663mH, and C=26.5μFC=26.5μF . The applied voltage has an amplitude of 50.0 VV and a frequency of 60.0 HzHz . Find (a) the current ImaxImax and its phase relative to the applied voltage Δv,(b)Δv,(b) the maximum voltage ΔVRΔVR across the resistor and its phase relative to the current, (c) the maximum voltage ΔVCΔVC across the capacitor and its phase relative to the current, and (d) the maximum voltage ΔVLΔVL across the inductor and its phase relative to the current.
• The compressibility κκ of a substance is defined as the fractional change in volume of that substance for a given change in pressure: κ=−1VdVdPκ=−1VdVdP
(a) Explain why the negative sign in this expression ensures κκ is always positive. (b) Show that if an ideal gas is compressed isothermally, its compressibility is given by κ1=κ1=
1/P/P . (c) What If? Show that if an ideal gas is compressed adiabatically, its compressibility is given by κ2=1/(γP)κ2=1/(γP) . Determine values for (d) κ1κ1 and (e) κ2κ2 for a monatomic ideal gas at a pressure of 2.00 atm.
• A 4.00 -g particle confined to a box of length LL has a speed of 1.00 mm/smm/s (a) What is the classical kinetic energy of the particle? (b) If the energy of the first excited state (n=2)(n=2) is equal to the kinetic energy found in part (a), what is the value of L?(c)L?(c) Is the result found in part (b)(b) realistic? Explain.
• The CO molecule makes a transition from the J=1J=1 to the J=2J=2 rotational state when it absorbs a photon of frequency 2.30×1011Hz2.30×1011Hz . (a) Find the moment of inertia of this molecule from these data. (b) Compare your answer with that obtained in Example 43.1 and comment on the significance of the two results.
• A skier leaves the ramp of a ski jump with a velocity of v=10.0m/sv=10.0m/s at θ=15.0∘θ=15.0∘ above the horizontal as shown in Figure P4.67.P4.67. The slope where she will land is inclined downward at ϕ=50.0∘,ϕ=50.0∘, and air resistance is negligible. Find (a) the distance from the end of the ramp to where the jumper lands and (b) her velocity components just before the landing. (c) Explain how you think the results might be affected if air resistance were included.
• Students allow a narrow beam of laser light to strike a water surface. They measure the angle of refraction for selected angles of incidence and record the data shown in the accompanying table. (a) Use the data to verify Snell’s law of refraction by plotting the sine of the angle of incidence versus the sine of the angle of refraction. (b) Explain what the shape of the graph demonstrates. (c) Use the resulting plot to deduce the index of refraction of water, explaining how you do so.
Angle of Incidence(degrees)10.020.030.040.050.060.070.080.0Angle of Refraction(degrees)7.515.122.328.735.240.345.347.7Angle of IncidenceAngle of Refraction(degrees)(degrees)10.07.520.015.130.022.340.028.750.035.260.040.370.045.380.047.7
• Figure P40.4 on page 1214 shows the spectrum of light emitted by a firefly. (a) Determine the temperature of a black body that would emit radiation peaked at the same wavelength. (b) Based on your result, explain whether firefly radiation is blackbody radiation.
• A 326-g object is attached to a spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 5.83 J, find (a) the maximum speed of the object, (b) the force constant of the spring, and (c) the amplitude of the motion.
• A 4.00-kg particle moves from the origin to position , having coordinates x=5.00m and y= 5.00 m (Fig. P7.43). One force on the particle is the gravitational force acting in the negative y direction. Using Equation 7.3, calculate the work dorce by the particle as it goes from O to along (a) the purple path, (b) the red path, and (c) the blue path. (d) Your results should all be identical. Why?
• For the system of four capacitors shown in Figure P26.19, find (a) the total energy stored in the system and (b) the energy stored by each capacitor. (c) Compare the sum of the answers in part (b) with your result to part (a) and explain your observation.
• A conductor consists of a circular loop of radius RR and two long, straight sections as shown in Figure P30.7.P30.7. The wire lies in the plane of the paper and carries a current II . (a) What is the direction of the magnetic field at the center rr . of the loop? (b) Find an expression for the magnitude of the magnetic field at the center of the loop.
• A particle of mass m= 1.18 kg is attached between two identical springs on a frictionless, horizontal tabletop. Both springs have spring constant k and are initially unstressed, and the particle is at x=0 . (a) The particle is pulled a distance x along a direction perpendicular to the initial configuration of the springs as shown in Figure P7.66 . Show that the force exerted by the springs on the particle is
→F=−2kx(1−L√x2+L2)ˆi
(b) Show that the potential energy of the system is
U(x)=kx2+2kL(L−√x2+L2)
(c) Make a plot of U(x) versus x and identify all equilibrium points. Assume L=1.20m and k=40.0N/m. (d) If the particle is pulled 0.500 m to the right and then released, what is its speed when it reaches x=0 ?
• An FM radio transmitter has a power output of 150 kW and operates at a frequency of 99.7 MHz. How many photons per second does the transmitter emit?
• Draw phasors to scale for the following voltages in SI units: (a) 25.0 sin ωtωt at ωt=90.0∘,ωt=90.0∘, (b) 30.0 sinωtsinωt at ωt=60.0∘ωt=60.0∘ , and (c)18.0(c)18.0 sin ωtωt at ωt=300∘.ωt=300∘.
• A dentist uses a spherical mirror to examine a tooth. The tooth is 1.00 $\mathrm{cm}$ in front of the mirror, and the image is formed 10.0 $\mathrm{cm}$ behind the mirror. Determine (a) the mirror’s radius of curvature and (b) the magnification of the image.
• One cubic meter of atomic hydrogen at 0∘C0∘C at atmospheric pressure contains approximately 2.70×10252.70×1025 atoms. The first excited state of the hydrogen atom has an energy of
2 eVeV above that of the lowest state, called the ground state. Use the Boltzmann factor to find the number of atoms in the first excited state (a) at 0∘C0∘C and at (b) (1.00×104)∘C(1.00×104)∘C .
• A block of mass M rests on a table. It is fastened to the lower end of a light, vertical spring. The upper end of the spring is fastened to a block of mass m. The upper block is pushed down by an additional force 3mg,3mg, so the spring compression is 4mg/k.4mg/k. In this configuration, the upper block is released from rest. The spring lifts the lower block off the table. In terms of m,m, what is the greatest possible
value for M?M?
• A nonconducting ring of radius 10.0 cm is uniformly charged with a total positive charge 10.0μCμC . The ring rotates at a constant angular speed 20.0 rad/srad/s about an axis through its center, perpendicular to the plane of the ring. What is the magnitude of the magnetic field on the axis of the ring 5.00 cmcm from its center?
• One particular plug-in power supply for a radio looks similar to the one shown in Figure 33.20 and is marked with the following information: Input 120 VAC8WVAC8W Output 9 VDCVDC 300 mAmA . Assume these values are accurate to two digits. (a) Find the energy efficiency of the device when the radio is operating. (b) At what rate is energy wasted in the device when the radio is operating? (c) Suppose the input power to the transformer is 8.00 WW when the radio is switched off and energy costs $0.110/kWh$0.110/kWh from the electric company. Find the cost of having six such transformers around the house, each plugged in for 31 days.
• Two solenoids A and B, spaced close to each other and sharing the same cylindrical axis, have 400 and 700 turns, respectively. A current of 3.50 A in solenoid A produces an average flux of 300μWb300μWb through each turn of A and a flux of 90.0μWb90.0μWb through each turn of B. (a) Calculate the mutual inductance of the two solenoids. (b) What is the inductance of A? (c) What emf is induced in B when the current in A changes at the rate of 0.500A/s0.500A/s ?
• Find the net electric flux through (a) the closed spherical surface in a uniform electric field shown in Figure P 24.18a and (b) the closed cylindrical surface shown in Figure P 24.18b. (c) What can you conclude about the charges, if any, inside the cylindrical surface?
• An inductor having inductance LL and a capacitor having capacitance CC are connected in series. The current in the circuit increases linearly in time as described by I=Kt,I=Kt, where KK is a constant. The capacitor is initially uncharged. Determine (a) the voltage across the inductor as a function of time, (b) the voltage across the capacitor as a function of time, and (c) the time when the energy stored in the capacitor first exceeds that in the inductor.
• Two objects attract each other with a gravitational force of magnitude 1.00×10−8N1.00×10−8N when separated by 20.0cm.20.0cm. If the total mass of the two objects is 5.00kg,5.00kg, what is the mass of each?
• Consider three capacitors C1,C2,C1,C2, and C3C3 and a battery. If only C1C1 is connected to the battery, the charge on C1C1 is 30.8μCμC . Now C1C1 is disconnected, discharged, and connected in series with C2.C2. When the series combination of C2C2 and C1C1 is connected across the battery, the charge on C1C1 is 23.1μCμC . The circuit is disconnected, and both capacitors are discharged. Next, C3,C1,C3,C1, and the battery are connected in series, resulting in a charge on C1C1 of 25.2μCμC . If, after being disconnected and discharged, C1,C2,C1,C2, and C3C3 , are connected in series with one another and with the battery, what is the charge on C1?C1?
• An early (incorrect) model of the hydrogen atom, suggested by J. J. Thomson, proposed that a positive cloud of charge ++ e was uniformly distributed throughout the volume of a sphere of radius R,R, with the electron (an equal-magnitude negatively charged particle −e)−e) at the center.(a) Using Gauss’s law, show that the electron would be in equilibrium at the center and, if displaced from the center a distance r<R,r<R, would experience a restoring force of the form F=−Kr,F=−Kr, where KK is a constant. (b) Show that K=kee2/R3.K=kee2/R3. (c) Find an expression for the frequency ff of simple harmonic oscillations that an electron of mass meme would undergo if displaced a small distance (<R)(<R) from the center and released. (d) Calculate a numerical value for RR that would result in a frequency of 2.47×1015Hz2.47×1015Hz , the frequency of the light radiated in the most intense line in the hydrogen spectrum.
• Coherent light rays of wavelength λλ strike a pair of slits separated by distance dd at an angle θ1θ1 with respect to the normal to the plane containing the slits as shown in Figure P37.17. The rays leaving the slits make an angle θ2θ2 with respect to the normal, and an interference maximum is formed by those rays on a screen that is a great distance from the slits. Show that the angle θ2θ2 is given by θ2=sin−1(sinθ1−mλd)wheremis an integer.θ2wheremis an integer.=sin−1(sinθ1−mλd)
• A cowboy stands on horizontal ground between two parallel, vertical cliffs. He is not midway between the cliffs. He fires a shot and hears its echoes. The second echo arrives 1.92 s after the first and 1.47 s before the third. Consider only the sound traveling parallel to the ground and reflecting from the cliffs. (a) What is the distance between the cliffs? (b) What If? If he can hear a fourth echo, how long after the third echo does it arrive?
• The positron is the antiparticle to the electron. It has the same mass and a positive electric charge of the same magnitude as that of the electron. Positronium is a hydrogen- like atom consisting of a positron and an electron revolving around each other. Using the Bohr model, find (a) the allowed distances between the two particles and (b) the allowed energies of the system.
• A car accelerates uniformly from rest and reaches a speed of 22.0 m/sm/s in 9.00 s. Assuming the diameter of a tire is 58.0cm,(a)58.0cm,(a) find the number of revolutions the tire makes during this motion, assuming that no slipping occurs. (b) What is the final angular speed of a tire in revolutions per second?
• Consider a one-dimensional chain of alternating singly-ionized positive and negative ions. Show that the potential energy associated with one of the ions and its interactions with the rest of this hypothetical crystal is
U(r)=−keαe2rU(r)=−keαe2r
where the Madelung constant is α=2ln2α=2ln⁡2 and rr is the distance between ions. Suggestion: Use the series expansion for ln(1+x).ln⁡(1+x).
• With reference to the dam studied in Example 14.4 and shown in Figure 14.5, (a) show that the total torque exerted by the water behind the dam about a horizontal axis through $O$ is $\frac{1}{6} \rho g w H^{3} .$ (b) Show that the effective line of action of the total force exerted by the water is at a distance $\frac{1}{3} H$ above $O$ .
• An idealized diesel engine operates in a cycle known as the air-standard diesel cycle shown in Figure P22.35. Fuel is sprayed into the cylinder at the point of maximum compression, B. Combustion occurs during the expansion B→C,B→C, which is modeled as an isobaric process. Show that the efficiency of an engine operating in this idealized diesel cycle is
ε=1−1γ(TD−TATC−TB)ε=1−1γ(TD−TATC−TB)
• Calculate the RR -value of a thermal window made of two single panes of glass each 0.125 in. thick and separated by a 0.250 -in. air space. (b) By what factor is the transfer of energy by heat through the window reduced by using the thermal window instead of the single-pane window? Include the contributions of inside and outside stagnant air layers.
• A light beam containing red and violet wavelengths is incident on a slab of quartz at an angle of incidence of 50.0∘.50.0∘. The index of refraction of quartz is 1.455 at 600 nmnm ( red light), and its index of refraction is 1.468 at 410 nmnm (violet light). Find the dispersion of the slab, which is defined as the difference in the angles of refraction for the two wavelengths.
• A 100-turn square coil of side 20.0 cm rotates about a vertical axis at 1.50×103 rev/min as indicated in Figure P31.36. The horizontal component of the Earth’s magnetic field at the coil’s location is equal to 2.00×10−5T . (a) Calculate the maximum emf induced in the coil by this field. (b) What is the orientation of the coil with respect to the magnetic field when the maximum emf occurs?
• A person walks 25.0° north of east for 3.10 km. How far would she have to walk due north and due east to arrive at the same location?
• A uniform cylindrical turntable of radius 1.90 m and mass 30.0 kg rotates counterclockwise in a horizontal plane with an initial angular speed of 4ππ rad/s. The fixed turn-table bearing is friction less. A lump of clay of mass 2.25 kg and negligible size is dropped onto the turntable from a small distance above it and immediately sticks to the turn-table at a point 1.80 m to the east of the axis. (a) Find the final angular speed of the clay and turntable. (b) Is the mechanical energy of the turntable–clay system constant in this process? Explain and use numerical results to verify your answer. (c) Is the momentum of the system constant in this process? Explain your answer.
• A nurse measures the temperature of a patient to be 41.5∘5∘C . (a) What is this temperature on the Fahrenheit scale? (b) Do you think the patient is seriously ill? Explain.
• A girl of mass mgmg is standing on a plank of mass mpmp Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity vgpvgp to the right relative to the plank. (The subscript gp denotes the girl relative to plank. (a) What is the velocity vpivpi of the plank relative to the surface of the ice? (b) What is the girl’s velocity vgivgi relative to the ice surface?
• Two identical conducting spheres each having a radius of 0.500 cm are connected by a light, 2.00-m-long conducting wire. A charge of 60.0μCμC is placed on one of the conductors. Assume the surface distribution of charge on each sphere is uniform. Determine the tension in the wire.
• A 20.0-kg floodlight in a park is supported at the end of a horizontal beam of negligible mass that is hinged to a pole as shown in Figure P12.18. A cable at an angle of θ=30.0∘θ=30.0∘ with the beam helps support the light. (a) Draw a force diagram for the beam. By computing torques about an axis at the hinge at the left-hand end of the beam, find (b) the tension in the cable, (c) the horizontal component of the force exerted by the pole on the beam, and (d) the vertical component of this force. Now solve the same problem from the force diagram from part (a) by computing torques around the junction between the cable and the beam at the right-hand end of the beam. Find (e) the vertical component of the force exerted by the pole on the beam, (f) the tension in the cable, and (g) the horizontal component of the force exerted by the pole on the beam. (h) Compare the solution to parts (b) through (d) with the solution to parts (e) through (g). Is either solution more accurate?
• A radio wave transmits 25.0 W/m2W/m2 of power per unit area. A flat surface of area AA is perpendicular to the direction of propagation of the wave. Assuming the surface is a perfect absorber, calculate the radiation pressure on it.
• Show that Equation 32.28 in the text is Kirchhoff’s loop rule as applied to the circuit in Figure P32.51P32.51 with the switch thrown to position bb .
• Why is the following situation impossible? An inventor comes to a patent office with the claim that her heat engine, which employs water as a working substance, has a thermodynamic efficiency of 0.110 . Although this efficiency is low compared with typical automobile engines, she explains that her engine operates between an energy reservoir at room temperature and a water-ice mixture at atmospheric pressure and therefore requires no fuel other than that to make the ice. The patent is approved, and working prototypes of the engine prove the inventor’s efficiency claim.
• A rectangular loop of area A=0.160m2 is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of the field is allowed to vary in time according to B=0.350e−t/2.00 , where B is in teslas and t is in seconds. The field has the constant value 0.350 T for t<0. What is the value for E at t=4.00s ?
• Why is the following situation impossible? An athlete tests her hand strength by having an assistant hang weights from her belt as she hangs onto a horizontal bar with her hands. When the weights hanging on her belt have increased to 80% of her body weight, her hands can no longer support her and she drops to the floor. Frustrated at not meeting her hand-strength goal, she decides to swing on a trapeze. The trapeze consists of a bar suspended by two parallel ropes, each of length ℓℓ allowing performers to swing in a vertical circular arc (Fig. P8. 68 ). The athlete holds the bar and steps off an elevated platform, starting from rest with the ropes at an angle θi=θi= 60.0∘0∘ with respect to the vertical. As she swings several times back and forth in a circular arc, she forgets her frustration related to the hand-strength test. Assume the size of the performer’s body is small compared to the length ℓℓ and air resistance is negligible.
• A charged cork ball of mass 1.00 g is suspended on a light string in the presence of a uniform electric field as shown in Figure P23.59.P23.59. When E→=(3.00i^+5.00j^)×105N/C,E→=(3.00i^+5.00j^)×105N/C, the ball is in equilibrium at θ=37.0∘.θ=37.0∘. Find (a)(a) the charge on the ball and (b)(b) the tension in the string.
• A small block of mass m=200gm=200g is released from rest at point @@ along the horizontal diameter on the inside of a frictionless, hemispherical bowl of radius R=30.0cmR=30.0cm (Fig. P8.41). Calculate (a) the gravitational potential energy of the block–Earth system when the block is at point relative to point , (b) the kinetic energy of the block at point , (c) its speed at point , and (d) its kinetic energy and the potential energy when the block is at point .
• When an uncharged conducting sphere of radius aa is placed at the origin of an xyzxyz coordinate system that lies in an initially uniform electric field E→=E0k,E→=E0k, the resulting electric potential is V(x,y,z)=V0V(x,y,z)=V0 for points inside the sphere and
V(x,y,z)=V0−E0z+E0a3z(x2+y2+z2)3/2V(x,y,z)=V0−E0z+E0a3z(x2+y2+z2)3/2
for points outside the sphere, where V0V0 is the (constant) electric potential on the conductor. Use this equation to determine the x,y,x,y, and zz components of the resulting electric field (a) inside the sphere and (b) outside the sphere.
• Identify the unknown nuclide or particle (X)(X)
(a) X→6528Ni+γ(b)21g214Po→X+α(c)X→5526Fe+e++νX→6528Ni+γ(b)21g214Po→X+α(c)X→5526Fe+e++ν
• The reaction π−+p→K0+Λ0π−+p→K0+Λ0 occurs with high probabil-
ity, whereas the reaction π−+p→K0+nπ−+p→K0+n never occurs.
Analyze these reactions at the quark level. Show that the first reaction conserves the total number of each type of quark and the second reaction does not.
• When a straight wire is warmed, its resistance is given by R=R0[1+α(T−T0)]R=R0[1+α(T−T0)] according to Equation 27.19,27.19, where αα is the temperature coefficient of resistivity. This expression needs to be modified if we include the change in dimensions of the wire due to thermal expansion. Find a more precise expression for the resistance, one that includes the effects of changes in the dimensions of the wire when it is warmed. Your final expression should be in terms of R0,T,T0,R0,T,T0, the temperature coefficient of resistivity α,α, and the coefficient of linear expansion α′.α′.
• The resistor RR in Figure P28.77P28.77 receives 20.0 WW of power. Determine the value of R.R.
• An AC source with ΔVrms=120V is connected between points a and d in Figure P33.24 . At what frequency will it deliver a power of 250 W ? Explain your answer.
• Show that for any object moving at less than one-tenth the speed of light, the relativistic kinetic energy agrees with the result of the classical equation K=12mu2K=12mu2 to within less than 1%% . Therefore, for most purposes, the classical equation is sufficient to describe these objects.
• One end of a light spring with force constant k=100N/mk=100N/m is attached to
a vertical wall. A light string is tied to the other end of the horizontal spring. As shown in Figure Pl5.61, thestring changes from horizontal to vertical as it passes over a pulley of mass MM in the shape of a solid disk of radius R=2.00cm.R=2.00cm. The pulley is free to turn on a fixed, smooth axle. The vertical section of the string supports an object of mass m=200gm=200g . The string does not slip at its contact with the pulley. The object is pulled downward a small distance and released. (a) What is the angular frequency ωω of oscillation of the object in terms of the mass M?M? (b) What is the highest possible value of the angular frequency of oscillation of the object? (C) What is the highest possible value of the angular frequency of oscillation of the object if the pulley radius
is doubled to R=4.00cmR=4.00cm ?
• What are the necessary conditions for equilibrium of the object shown in Figure P12.1? Calculate torques about an axis through point OO .
• The driver of a car slams on the brakes when he sees a tree blocking the road. The car slows uniformly with an acceleration of $-5.60 \mathrm{m} / \mathrm{s}^{2}$ for 4.20 $\mathrm{s}$ , making straight skid marks 62.4 $\mathrm{m}$ long, all the way to the tree. With what speed does the car then strike the tree?
• A 5.00−kg5.00−kg block is set into motion up an inclined plane with an initial speed of vi=8.00m/svi=8.00m/s (Fig. P8. 23 . The block comes to rest after traveling d=3.00md=3.00m along the plane, which is inclined at an angle of θ=30.0∘θ=30.0∘ to the horizontal. For this motion,
determine (a) the change in the block’s kinetic energy, (b) the change in the potential energy of the block-Earth system, and (c) the friction force exerted on the block (assumed to be constant). (d) What is
the coefficient of kinetic friction?
• A dance hall is built without pillars and with a horizontal ceiling 7.20 mm above the floor. A mirror is fastened flat against one section of the ceiling. Following an earthquake, the mirror is in place and unbroken. An engineer makes a quick check of whether the ceiling is sagging by directing a vertical beam of laser light up at the mirror and observing its reflection on the floor. (a) Show that if the mirror has rotated to make an angle ϕϕ with the horizontal, the normal to the mirror makes an angle ϕϕ with the vertical. (b) Show that the reflected laser light makes an angle 2ϕϕ with the vertical. (c) Assume the reflected laser light makes a spot floor 1.40 cmcm away from the point vertically below the laser. Find the angle ϕϕ .
• A flute is designed so that it produces a frequency of 261.6 Hz, middle C, when all the holes are covered and the temperature is 20.0°C. (a) Consider the flute as a pipe that is open at both ends. Find the length of the flute, assuming middle CC is the fundamental. (b) A second player, nearby in a colder room, also attempts to play middle CC on an identical flute. A beat frequency of 3.00 Hz is heard when both flutes are playing. What is the temperature of the second room?
• Two blocks of masses mm and 3mm are placed on a frictionless, horizontal surface. A light spring is attached to the more massive block, and the blocks are pushed together with the spring between them (Fig. P9.9). A cord initially holding the blocks together is burned; after that happens, the block of mass 3 mm moves to the right with a speed of 2.00 m/sm/s (a) What is the velocity of the block of mass mm ? (b) Find the system’s original elastic potential energy, taking m=0.350kgm=0.350kg . (c) Is the original energy in the explain your answer cord? (d) Explain your answer to part (c). (e) Is the momentum of the system conserved in the how that is possible considering
(f) there are large forces acting and (g) there is no motion beforehand and plenty of motion afterward?
• Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed bump (called the “Holly hump”) and had it installed. Suppose a 1 800-kg car passes over a hump in a roadway that follows the arc of a circle of radius 20.4 m as shown in Figure P6.40. (a) If the car travels at 30.0 km/h, what force does the road exert on the car as the car passes the highest point of the hump? (b) What If? What is the maximum speed the car can have without losing contact with the road as it passes this highest point?
• At what speed does a clock move if it is measured to run at a rate one-half the rate of a clock at rest with respect to an observer?
• A space vehicle is launched vertically upward from the Earth’s surface with an initial speed of vivi that is comparable to but less than the escape speed vescvesc . What maximum height does it attain? (b) A meteoroid falls toward the Earth. It is essentially at rest with respect to the Earth when it is at a height hh above the Earth’s surface. With what speed does the meteorite (a meteoroid that survives to impact the Earth’s surface) strike the Earth? (c) What If? Assume a baseball is tossed up with an initial speed that is very small compared to the escape speed. Show that the result from part (a) is consistent with Equation 4.12.
• Technetium-99 is used in certain medical diagnostic procedures. Assume 1.00×10−8g1.00×10−8g of 99Tc99Tc is injected into a 60.0-kg patient and half of the 0.140-MeV gamma rays are absorbed in the body. Determine the total radiation dose received by the patient.
• Assume a person bends forward to lift a load “with his back” as shown in Figure P12.48a. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, consider the model shown in Figure P12.48bP12.48b for a person bending for- ward to lift a 200−N200−N object. The spine and upper body are represented as a uniform horizontal rod of weight 350 NN , pivoted at the base of the spine. The erector spinalis muscle, attached at a point two-thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is θ=12.0∘.θ=12.0∘. Find (a) the tension TT in the back muscle and (b)(b) the compressional force in the spine. (c) Is this method a good way to lift a load? Explain your answer, using the results of parts (a) and (b). (d) Can you suggest a better method to lift a load?
• Two objects, A and B, are connected by hinges to a rigid rod that has a length $L .$ The objects
slide along perpendicular guide rails as shown in Figure $\mathrm{P} 2.63$ . Assume object A slides to the left with a constant speed $v .$ (a) Find the velocity $v_{\mathrm{B}}$ of object $\mathrm{B}$ as a function of the angle $\theta .$ (b) Describe $v_{\mathrm{B}}$ relative to $v .$ Is $v_{\mathrm{B}}$ always smaller than $v,$ larger than $v,$ or the same as $v,$ or does it have some other relationship?
• A pendulum with a cord of length r=1.00mr=1.00m swings in a vertical plane (Fig. P4.54). When the pendulum is in the two horizontal positions θ=θ= 90.0∘0∘ and θ=270∘θ=270∘ , its speed is 5.00 m/sm/s . Find the magnitude of (a) the radial acceleration and (b) the tangential acceleration for these positions. (c) Draw vector diagrams to determine the direction of the total acceleration for these two positions. (d) Calculate the magnitude and direction of the total acceleration at these two positions.
• A student, along with her backpack on the floor next to her, are in an elevator that is accelerating upward with acceleration aa . The student gives her backpack a quick kick at t=0t=0 , imparting to it speed vv and causing it to slide across the elevator floor. At time t,t, the backpack hits the opposite wall a distance LL away from the student. Find the coefficient of kinetic friction μkμk between the backpack and the elevator floor.
• The dimensions of a classroom are 4.20m×3.00m×4.20m×3.00m× 50 mm . (a) Find the number of molecules of air in the classroom at atmospheric pressure and 20.0∘C20.0∘C . (b) Find the mass of this air, assuming the air consists of diatomic molecules with molar mass 28.9g/mol.28.9g/mol. (c) Find the average kinetic energy of the molecules. (d) Find the rms molecular speed. (e) What If? Assume the molar specific heat of the air is independent of temperature. Find the change in internal energy of the air in the room as the temperature
is raised to 25.0∘C25.0∘C . (f) Explain how you could convince a fellow student that your answer to part (e) is correct, even though it sounds surprising.
• The highest note written for a singer in a published score was F-sharp above high C, 1.480 kHz, for Zerbinetta in the original version of Richard Strauss’s opera Ariadne auf Naxos. (a) Find the wavelength of this sound in air. (b) Suppose people in the fourth row of seats hear this note with level 81.0 dB. Find the displacement amplitude of the sound. (c) What If? In response to complaints, Strauss later transposed the note down to F above high C, 1.397 kHz. By what increment did the wavelength change?
• Why is the following situation impossible? In a large city with an air-pollution problem, a bus has no combustion engine. It runs over its citywide route on energy drawn from a large, rapidly rotating flywheel under the floor of the bus. The flywheel is spun up to its maximum rotation rate of 3 000 rev/min by an electric motor at the bus terminal. Every time the bus speeds up, the flywheel slows down slightly. The bus is equipped with regenerative braking so that the flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1 200 kg and radius 0.500 m. The bus body does work against air resistance and rolling resistance at the average rate of 25.0 hp as it travels its route with an average speed of 35.0 km/h.
• A seaplane of total mass mm lands on a lake with initial speed vivi i. The only horizontal force on it is a resistive force on its pontoons from the water. The resistive force is proportional to the velocity of the seaplane: R¯¯¯¯=−bv→R¯=−bv→ . Newton’s second law applied to the plane is −bvi^=m(dv/dt)i^−bvi^=m(dv/dt)i^ . From the fundamental theorem of calculus, this differential equation implies that the speed changes according to ∫vvidvv=−bm∫t0dt∫vivdvv=−bm∫0tdt
(a) Carry out the integration to determine the speed of the seaplane as a function of time. (b) Sketch a graph of the speed as a function of time. (c) Does the seaplane come to a complete stop after a finite interval of time? (d) Does the seaplane travel a finite distance in stopping?
• A car travels along a straight line at a constant speed of 60.0 $\mathrm{mi} / \mathrm{h}$ for a distance $d$ and then another distance $d$ in the same direction at another constant speed. The average velocity for the entire trip is 30.0 $\mathrm{mi} / \mathrm{h}$ . (a) What is the constant speed with which the car moved during the second distance $d ?$ (b) What If? Suppose the second distance $d$ were traveled in the opposite direction; you forgot something and had to return home at the same constant speed as found in part (a). What is the average velocity for this
trip? (c) What is the average speed for this new trip?
• In an experiment, a large number of electrons are fired at a sample of neutral hydrogen atoms and observations are made of how the incident particles scatter. The electron in the ground state of a hydrogen atom is found to be momentarily at a distance a0/2 from the nucleus in 1 000 of the observations. In this set of trials, how many times is the atomic electron observed at a distance 2a0 from the nucleus?
• A student wishes to measure the half-life of a radioactive substance using a small sample. Consecutive clicks of her Geiger counter are randomly spaced in time. The counter registers 372 counts during one 5.00 -min interval and 337 counts during the next 5.00 min. The average background rate is 15 counts per minute. Find the most probable value for the half-life. (b) Estimate the uncertainty in the half-life determination in part (a). Explain your reasoning.
• The effective spring constant associated with bonding in the N2N2 molecule is 2297N/m.2297N/m. The nitrogen atoms each have a mass of 2.32×10−26kg,2.32×10−26kg, and their nuclei are 0.120 nm apart. Assume the molecule is rigid. The first excited vibrational state of the molecule is above the vibrational ground state by an energy difference ΔE.ΔE. Calculate the JJ value of the rotational state that is above the rotational ground state by the same energy difference ΔEΔE .
• Analyze each of the following reactions in terms of con-
stituent quarks and show that each type of quark is con-
(a) π++p→K++Σ+π++p→K++Σ+ (b) K−+p→K++K0+K−+p→K++K0+
Ω−Ω− (c) Determine the quarks in the final particle for this
reaction: p+p→K0+p+π++?p+p→K0+p+π++? (d) In the reaction in
part (c), identify the mystery particle.
• The Balmer series for the hydrogen atom corresponds to electronic transitions that terminate in the state with quantum number n=2n=2 as shown in Figure P42.10P42.10 . Consider the photon of longest wavelength corresponding to a transition shown in the figure. Determine (a) its energy and (b) its wavelength. Consider the spectral line of shortest wavelength corresponding to a transition shown in the
Find (c)(c) its photon energy and (d)(d) its wavelength. (e) What is the shortest possible wavelength in the Balmer series?
• An electron has a kinetic energy of 12.0 eV. The electron is incident upon a rectangular barrier of height 20.0 eV and width 1.00nm. If the electron absorbed all the energy of a photon of green light (with wavelength 546 nm ) at the instant it reached the barrier, by what factor would the electron’s probability of tunneling through the barrier increase?
• In Section 17.2, we derived the speed of sound in a gas using the impulse–momentum theorem applied to the cylinder of gas in Figure 17.5. Let us find the speed of sound in a gas using a different approach based on the element of gas in Figure 17.3. Proceed as follows. (a) Draw a force diagram for this element showing the forces exerted on the left and right surfaces due to the pressure of the gas on either side of the element. (b) By applying Newton’s second law to the element, show that
−∂(ΔP)∂xAΔx=ρAΔx∂2s∂t2−∂(ΔP)∂xAΔx=ρAΔx∂2s∂t2
(c) By substituting ΔP=−(B∂s/∂x)(Eq.17.3)ΔP=−(B∂s/∂x)(Eq.17.3) , derive the following wave equation for sound:
Bρ∂2s∂x2=∂2s∂t2Bρ∂2s∂x2=∂2s∂t2
(d) To a mathematical physicist, this equation demonstrates the existence of sound waves and determines their speed. As a physics student, you must take another step or two. Substitute into the wave equation the trial solution s(x,t)=smaxcos(kx−ωt)s(x,t)=smaxcos(kx−ωt) . Show that this function satisfies the wave equation, provided ω/k=v=√B/ρ.ω/k=v=B/ρ−−−−√.
• A gamma ray (a high-energy photon) can produce an electron (e−)(e−) and a positron (e+)(e+) of equal mass when it enters the electric field of a heavy nucleus: γ→e++e−.γ→e++e−. What minimum gamma-ray energy is required to accomplish this task?
• Three charged particles are at the corners of an equilateral triangle as shown in Figure P23.13. (a) Calculate the electric field at the position of the 2.00−μC2.00−μC charge due to the 7.00−μC7.00−μC and −4.00−μC−4.00−μC charges. (b) Use your answer to part (a) to determine the force on the 2.00−μC2.00−μC charge.
• A cook puts 9.00 g of water in a 2.00−L2.00−L pressure cooker that is then warmed to 500∘C500∘C . What is the pressure inside the container?
• Two constant forces act on an object of mass m=5.00kg moving in the xy plane as shown in Figure P7.61. Force ¯F1 is 25.0 N at 35.0∘ , and force →F2 is 42.0 N at 150∘ . At time t=0, the object is at the origin and has velocity (4.00ˆi+2.50ˆj)m/s. (a) Express the two forces in unit-vector notation. Use unit-vector notation for your other answers. (b) Find the total force exerted on the object. (c) Find the object’s acceleration. Now, considering the instant t=3.00s, find (d) the object’s velocity, (e) its position, (f) its kinetic energy from 12mv2f, and (g) its kinetic energy from 12mv2i+∑→F⋅Δ→r. (h) What conclusion can you draw by comparing the answers to parts (f) and (g)?
• Why is the following situation impossible? A conducting rectangular loop of mass M=0.100kg , resistance R=1.00Ω and dimensions w=50.0cm by ℓ=90.0cm is held with its lower edge just above a region with a uniform magnetic field of magnitude B=1.00T as shown in Figure P31.52 The loop is released from rest. Just as the top edge of the loop reaches the region containing the field, the loop moves with a speed 4.00 m/s .
• Show that I=Iie−t/τI=Iie−t/τ is a solution of the differential equation
IR+LdIdt=0IR+LdIdt=0
where IiIi is the current at t=0t=0 and τ=L/Rτ=L/R
• A possible means for making an airplane invisible to radar is to coat the plane with an antireflective polymer. If radar waves have a wavelength of 3.00 cmcm and the index of refraction of the polymer is n=1.50,n=1.50, how thick would you make the coating?
• Write expressions for the electric and magnetic fields of a sinusoidal plane electromagnetic wave having an electric field amplitude of 300 V/m and a frequency of 3.00 GHz and traveling in the positive x direction.
• Assume if the shear stress in steel exceeds about 4.00×108N/m24.00×108N/m2 the steel ruptures. Determine the shearing force necessary to (a) shear a steel bolt 1.00 cmcm in diameter and (b) punch a 1.00 -cm-diameter hole in a steel plate 0.500 cmcm thick.
• One mediator of the weak interaction is the Z0Z0 boson, with mass 91GeV/c2.91GeV/c2. Use this information to find the order of magnitude of the range of the weak interaction.
• The lintel of prestressed reinforced concrete in Figure P12.40P12.40 is 1.50 mm long. The concrete encloses one steel reinforcing rod with cross-sectional area 1.50 cm2cm2 . The rod joins two strong end plates. The cross-sectional area of the concrete perpendicular to the rod is 50.0 cm2cm2 . Young’s modulus for the concrete is 30.0×109N/m230.0×109N/m2 . After the concrete cures and the original tension T1T1 in the rod is released, the concrete is to be under compressive stress 8.00×106N/m2.8.00×106N/m2. (a) By what distance will the rod compress the concrete when the original tension in the rod is released? (b) What is the new tension T2T2 in the rod? (c) The rod will then be how much longer than its unstressed length? (d) When the concrete was poured, the rod should have been stretched by what extension distance from its unstressed length? (e) Find the required original tension T1T1 in the rod.
• A pulse traveling along a string of linear mass density μμ is described by the wave function
y=[A0e−bx]sin(kx−ωt)y=[A0e−bx]sin⁡(kx−ωt)
where the factor in brackets is said to be the amplitude. (a) What is the power P(x)P(x) carried by this wave at a point (b) What is the power P(0)P(0) carried by this wave at the origin? (c) Compute the ratio P(x)/P(0)P(x)/P(0) .
• A ray of light travels from air into another medium, making an angle of θ1=θ1= 45.0∘0∘ with the normal as in Figure P35.11. Find the angle of refraction the angle of refraction is if the second medium is (a) fused quartz, (b) carbon disulfide, and (c) water.
• A free electron has a wave function
ψ(x)=Aei(5.00×1010x)ψ(x)=Aei(5.00×1010x)
where xx is in meters. Find its (a) de Broglie wavelength, (b) momentum, and (c) kinetic energy in electron volts.
• Figure P12.7P12.7 shows three uniform objects: a rod with m1=6.00kg,m1=6.00kg, a right triangle with m2=m2= 3.00 kgkg and a square with m3=5.00kgm3=5.00kg . Their coordinates in meters are given. Determine the center of gravity for the three-object system.
• An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 9.00m/s2.9.00m/s2. Determine the orbital period of the satellite.
• A proton is projected in the positive xx direction into a region of a uniform electric field E→=(−6.00×105)i^N/CE→=(−6.00×105)i^N/C at t=0.t=0. The proton travels 7.00 cmcm as it comes to rest. Determine (a) the acceleration of the proton, (b)(b) its initial speed, and (c)(c) the time interval over which the proton comes to rest.
• An 820−N820−N Marine in basic training climbs a 12.0−m12.0−m vertical rope at a constant speed in 8.00 ss . What is his power output?
• An uncharged, nonconducting, hollow sphere of radius 10.0 cm surrounds a 10.0−μC10.0−μC charge located at the origin of a Cartesian coordinate system. A drill with a radius of 1.00 mm is aligned along the zz axis, and a hole is drilled in the sphere. Calculate the electric flux through the hole.
• A spy satellite can consist of a large-diameter concave mirror forming an image on a digital-camera detector and sending the picture to a ground receiver by radio waves. In effect, it is an astronomical telescope in orbit, looking down instead of up. (a) Can a spy satellite read a license plate? (b) Can it read the date on a dime? Argue for your answers by making an order-of-magnitude calculation, specifying the data you estimate.
• As she picks up her riders, a bus driver traverses four successive displacements represented by the expression
(−6.30b)i^−(4.00bcos40∘)i^−(4.00bsin40∘)j^+(3.00bcos50∘)i^−(3.00bsin50∘)j^−(5.00b)j^(−6.30b)i^−(4.00bcos⁡40∘)i^−(4.00bsin⁡40∘)j^+(3.00bcos⁡50∘)i^−(3.00bsin⁡50∘)j^−(5.00b)j^
Here b represents one city block, a convenient unit of distance of uniform size; i^i^ is east; and j^j^ is north. The displacements at 40° and 50° represent travel on roadways in the city that are at these angles to the main east–west and north–south streets. (a) Draw a map of the successive displacements. (b) What total distance did she travel? (c) Compute the magnitude and direction of her total displacement. The logical structure of this problem and of several problems in later chapters was suggested by Alan Van Heuvelen and David Maloney, American Journal of Physics 67(3) 252–256, March 1999.
• The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.75 on page 432. The hydrofoil has a shape like that of an airplane wing. Its area projected onto a horizontal surface is A. When the boat is towed at sufficiently high speed, water of density $\rho$ moves in streamline flow so that its average speed at the top of the hydrofoil is $n$ times
larger than its speed $v_{b}$ below the hydrofoil. (a) Ignoring the buoyant force, show that the upward lift force exerted by the water on the hydrofoil has a magnitude
F≈13(n2−1)ρv2bAF≈13(n2−1)ρvb2A
(b) The boat has mass $M$ . Show that the liftoff speed is given by
v≈2Mg(n2−1)Aρ−−−−−−−−−−√v≈2Mg(n2−1)Aρ
• Two charged particles of equal magnitude are located along the yy axis equal distances above and below the xx axis as shown in Figure P25.24P25.24 . (a) Plot a graph of the electric potential at points along the xx axis over the interval −3a<x<3a−3a<x<3a . You should plot the potential in units of keQ/akeQ/a . (b) Let the charge of the particle located at y=−ay=−a be negative. Plot the potential along the yy axis over the interval −4a<y<4a−4a<y<4a
• When a 4.00 -kg object is hung vertically on a certain light spring that obeys Hooke’s law, the spring stretches 2.50cm. If the 4.00 -kg object is removed, (a) how far will the spring stretch if a 1.50−kg block is hung on it? (b) How much work must an external agent do to stretch the same spring 4.00 cm from its unstretched position?
• Two blocks connected by a rope of negligible mass are being dragged by a horizontal force (Fig. P5.47). Suppose F=68.0N,m1=12.0kgF=68.0N,m1=12.0kg m2=18.0kg,m2=18.0kg, and the coefficient of kinetic friction between each block and the surface is 0.100 . (a) Draw a free-body diagram for each block. Determine (b) the acceleration of the system and (c) the tension TT in the rope.
• A daring ranch hand sitting on a tree limb wishes to drop vertically onto a horse galloping under the tree. The constant speed of the horse is 10.0 $\mathrm{m} / \mathrm{s}$ , and the distance from the limb to the level of the saddle is 3.00 $\mathrm{m}$ . (a) What must be the horizontal distance between the saddle and limb when the ranch hand makes his move? (b) For what time interval is he in the air?
• Two waves on one string are described by the wave functions
y1=3.0cos(4.0x−1.6t)y2=4.0sin(5.0x−2.0t)y1=3.0cos(4.0x−1.6t)y2=4.0sin(5.0x−2.0t)
where xx and yy are in centimeters and tt is in seconds. Find the superposition of the waves y1+y2y1+y2 at the points (a)x=1.00,t=1.00;(b)x=1.00,t=0.500;(a)x=1.00,t=1.00;(b)x=1.00,t=0.500; and (c)x=0.500,t=0.(c)x=0.500,t=0. Note: Remember that the arguments of the trigonometric functions are in radians.
• For each of the following decays or reactions, name at least one conservation law that prevents it from occurring.
(a) π−+p→Σ++π0π−+p→Σ++π0
(b) μ−→π−+νeμ−→π−+νe
(c) p→π++π++π−p→π++π++π−
• At our distance from the Sun, the intensity of solar radiation is 1370 W/m2W/m2 . The temperature of the Earth is affected by the greenhouse effect of the atmosphere. This phenomenon describes the effect of absorption of infrared light emitted by the surface so as to make the surface
temperature of the Earth higher than if it were airless. For comparison, consider a spherical object of radius rr with no atmosphere at the same distance from the Sun as the Earth. Assume its emissivity is the same for all kinds of electromagnetic waves and its temperature is uniform over its surface. (a) Explain why the projected area over which it absorbs sunlight is πr2πr2 and the surface area over which it radiates is 4πr2.4πr2. (b) Compute its steady-state temperature. Is it chilly?
• Two rays traveling parallel to the principal axis strike a large plano-convex lens having a refractive index of 1.60 (Fig. P36.50). If the convex face is spherical, a ray near the edge does not pass through the focal point (spherical aberration occurs). Assume this face has a radius of curvature of $R=20.0 \mathrm{cm}$ and the two rays are at distances $h_{1}=$ 0.500 $\mathrm{cm}$ and $h_{2}=12.0 \mathrm{cm}$ from the principal axis. Find the difference $\Delta x$ in the positions where each crosses the principal axis.
• A 2,002,00 -L container has a center partition that divides it into two cqual parts as shown in Figure P22.41P22.41 . The left side contains 0.0440 molmol of H2H2 gas, and the right side contains 0.0440 molmol of O2O2 gas. Both gases are at room temperature and at atmospheric pressure. The partition is removed, and the gases are allowed to mix. What is the entropy increase of the system?
• A very long, thin rod carries electric charge with the linear density 35.0 nC/mnC/m . It lies along the xx axis and moves in the xx direction at a speed of 1.50×107m/s1.50×107m/s . (a) Find the electric field the rod creates at the point (x=0,y=(x=0,y= 20.0cm,z=0).20.0cm,z=0). (b) Find the magnetic field it creates at the same point. (c) Find the force exerted on an electron at this point, moving with a velocity of (2.40×108)ˆim/s(2.40×108)i^m/s
• A setup similar to the one shown in Figure P5.26 is often used in hospitals to support and apply a horizontal traction force to an injured leg. (a) Determine the force of tension in the rope supporting the leg. (b) What is the traction force exerted to the right on the leg?
• A heat pump has a coefficient of performance of 3.80 and operates with a power consumption of 7.03×103W7.03×103W . (a) How much energy does it deliver into a home during 8.00 hh of continuous operation? (b) How much energy does it extract from the outside air?
• Oxygen, modeled as an ideal gas, is in a container and has a temperature of 77.0∘0∘C . What is the rms-average magnitude of the momentum of the gas molecules in the container?
• A bimetallic strip of length LL is made of two ribbons of different metals bonded together. (a) First assume the strip is originally straight. As the strip is warmed, the metal with the greater average coefficient of expansion expands more than the other, forcing the strip into an arc with the outer radius having a greater circumference (Fig. Pl9.48). Derive an expression for the angle of bending θθ as a function of the initial length of the strips, their average coefficients of linear expansion, the change in temperature, and the separation of the centers of the strips (Δr=r2−r1)⋅(b) Show that the angle(Δr=r2−r1)⋅(b) Show that the angle of bending decreases to zero when ΔTΔT decreases to zero and also when the two average coefficients of expansion become equal. (c) What If? What happens if the strip is cooled?
• A nearsighted person cannot see objects clearly beyond 25.0 $\mathrm{cm}$ (her far point). If she has no astigmatism and contact lenses are prescribed for her, what (a) power and (b) type of lens are required to correct her vision?
• A uniform solid sphere of radius r is placed on the inside surface of a hemispherical bowl with radius R. The sphere is released from rest an angle θ to the vertical and rolls without slipping (Fig. P10.75). Determine the angular speed of the sphere when it reaches the bottom of the bowl.
• A large, hot pizza floats in outer space after being jettisoned as refuse from a spacecraft. What is the order of magnitude (a) of its rate of energy loss and (b) of its rate of temperature change? List the quantities you estimate and the value you estimate for each.
• A glass sphere (n 5 1.50) with a radius of 15.0 cm has a tiny air bubble 5.00 cm above its center. The sphere is viewed looking down along the extended radius containing the bubble. What is the apparent depth of the bubble below the surface of the sphere?
• A student measures the length of a brass rod with a steel tape at 20.0∘0∘C . The reading is 95.00cm.95.00cm. What will the tape indicate for the length of the rod when the rod and the tape are at (a) −15.0∘C−15.0∘C and (b) 55.0∘C?55.0∘C?
• Two insulating spheres have radii 0.300 cmcm and 0.500cm,0.500cm, masses 0.100 kgkg and 0.700 kgkg , and uniformly distributed charges −2.00μC−2.00μC and 3.00μCμC . They are released from rest when their centers are separated by 1.00 mm . (a) How fast will each be moving when they collide? (b) What If? If the spheres were conductors, would the speeds be greater or less than those calculated in part (a)? Explain.
• A point charge +2Q+2Q is at the origin and a point charge −Q−Q is located along the xx axis
at x=dx=d as in Figure P23.17P23.17 Find a symbolic expression for the net force on a third point charge +Q+Q located along the yy axis at y=dy=d.
• Strontium- 90 from the testing of nuclear bombs can still be found in the atmosphere. Each decay of 90 SrSr releases 1.10 MeVMeV of energy into the bones of a person who has had strontium replace his or her body’s calcium. Assume a 70.0 -kg person receives 1.00 ngng of 90 SrSr from contaminated milk. Take the half-life of 90 SrSr to be 29.1 yr. Calculate the
absorbed dose rate (in joules per kilogram) in one year.
• Two parallel rails with negligible resistance are 10.0 cm apart and are connected by a resistor of resistance R3= 5.00 \Omega. The circuit also contains two metal rods having resistances of R1=10.0Ω and R2=15.0Ω sliding along the rails (Fig. P31.31). The rods are pulled away from the resistor at constant speeds of v1=4.00m/s and v2=2.00m/s respectively. A uniform magnetic field of magnitude B= 0.0100 T is applied perpendicular to the plane of the rails. Determine the current in R3 .
• A spacecraft with a proper length of 300 m passes by an observer on the Earth. According to this observer, it takes 0.750μμ s for the spacecraft to pass a fixed point. Determine the speed of the spacecraft as measured by the Earth-based observer.
• This problem is about how strongly matter is coupled to radiation, the subject with which quantum mechanics began. For a simple model, consider a solid iron sphere 2.00 cm in radius. Assume its temperature is always uniform throughout its volume. (a) Find the mass of the sphere. (b) Assume the sphere is at 20.0°C and has emissivity 0.860. Find the power with which it radiates electromagnetic waves. (c) If it were alone in the Universe, at what rate would the sphere’s temperature be changing? (d) Assume Wien’s law describes the sphere. Find the wavelength lmax of electromagnetic radiation it emits most strongly. Although it emits a spectrum of waves having all different wavelengths, assume its power output is carried by photons of wavelength λmaxλmax Find (e)(e) the energy of one photon and (f) the number of photons it emits each second.
• The photon frequency that would be absorbed by the NO molecule in a transition from vibration state v=0v=0 to v=1v=1 with no change in rotation state, is 56.3 THz. The bond between the atoms has an effective spring constant of 1 530 N/m. (a) Use this information to calculate the reduced mass of the NO molecule. (b) Compute a value for μμ using Equation 43.4. (c) Compare your results to parts (a) and (b) and explain their difference, if any.
• Keilah, in reference frame SS , measures two events to be simultaneous. Event A occurs at the point (50.0m,0,0)(50.0m,0,0) at the instant 9:00:009:00:00 Universal time on January 15,201015,2010 . Event BB occurs at the point (150m,0,0)(150m,0,0) at the same moment. Torrey, moving past with a velocity of 0.800 cici , also observes the two events. In her reference frame SS , which event occurred first and what time interval elapsed between the events?
• The ship in Figure P 18.55 travels along a straight line parallel to the shore and a distance d=600md=600m from it. The ship’s radio receives signals of the same frequency from antennas AA and BB , separated by a distance L=800m.L=800m. The signals interfere constructively at point C,C, which is equidistant from AA and BB . The signal goes through the first minimum at point D,D, which is directly outward from the shore from point B.B. Determine the wavelength of the radio waves.
• A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s2m/s2 . (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer inside the train car. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth.
• A flea is at point on a horizontal turntable, 10.0 cm from the center. The turntable is rotating at 33.3 rev/min in the clockwise direction. The flea jumps straight up to a height of 5.00 cm. At takeoff, it gives itself no horizontal velocity relative to the turntable. The flea lands on the turntable at point . Choose the origin of coordinates to be at the center of the turntable and the positive x axis passing through at the moment of takeoff. Then the original position of the flea is 10.0ˆicmi^cm. (a) Find the position of point when the flea lands. (b) Find the position of point when the flea lands.
• The radio frequency at which a nucleus having a magnetic moment of magnitude μ displays resonance absorption between spin states is called the Larmor frequency and is given by
f=ΔEh=2μBh
Calculate the Larmor frequency for (a) free neutrons in a magnetic field of 1.00T, (b) free protons in a magnetic field of 1.00 T , and c ) free protons in the Earth’s magnetic field at a location where the magnitude of the field is 50.0μT.
• Q C A block of mass 0.500kg0.500kg is pushed against a horizontal spring of negligible mass until the spring is compressed a distance xx (Fig. P8.65P8.65 ). The force constant of the spring is 450N/m.450N/m. When it is released, the block travels along a frictionless, horizontal surface to point @@, the bottom of a vertical circular track of radius R=1.00m,R=1.00m, and continues to move up the track. The block’s speed at the bottom of the track is v@=12.0m/s,v@=12.0m/s, and the block experiences an average friction force of 7.00N7.00N while sliding up the track.
(a) What is xx ? (b) If the block were to reach the top of the track, what would be its speed at that point? (c) Does the block actually reach the top of the track, or does it fall off before reaching the top?
• As shown in Figure P35.65P35.65 , a light ray is incident normal to one face of a 30∘−60∘−90∘30∘−60∘−90∘ block of flint glass (a prism) that is immersed in water. (a) Determine the exit angle θ3θ3 of the ray. (b) A substance is dissolved in the water to increase the index of refraction n2.n2. At what value of n2n2 does total internal reflection cease at point P?P?
• As shown in Figure P8.18, a green bead of mass 25 g slides along a straight wire. The length of the wire from point to point is 0.600 m, and point is 0.200 m higher than point . A constant friction force
of magnitude 0.025 0 N acts on the bead. (a) If the bead is released from rest at point
, what is its speed at point ? (b) A red bead of mass 25 g slides along a curved wire, subject to a friction force with the same constant magnitude as that on the green bead. If the green and red beads are released simultaneously from rest at point , which bead reaches point with a higher speed? Explain.
• Owen and Dina are at rest in frame SS ‘, which is moving at 0.600 cc with respect to frame SS . They play a game of catch while Ed,Ed, at rest in frame SS , watches the action (Fig. P39.75).P39.75). Owen throws the ball to Dina at 0.800 cc (according to Owen), and their separation (measured in S′)S′) is equal to 1.80×1012m.1.80×1012m. (a) According to Dina, how fast is the ball moving? (b) According to Dina, what time interval is required for the ball to reach her? According to Ed, (c) how far apart are Owen and Dina, (d) how fast is the ball moving, and (e) what time interval is required for the ball to reach Dina?
• The deepest point in the ocean is in the Mariana Trench, about 11 kmkm deep, in the Pacific. The pressure at this depth is huge, about 1.13×108N/m21.13×108N/m2 . (a) Calculate the change in volume of 1.00 m3m3 of seawater carried from the surface to this deepest point. (b) The density of seawater at the surface is 1.03×103kg/m3.1.03×103kg/m3. Find its density at the bottom. (c) Explain whether or when it is a good approximation to think of water as incompressible.
• The following reactions or decays involve one or more neutrinos. In each case, supply the missing neutrino (νe,νμ,or(νe,νμ,or ντ)ντ) or antineutrino.
(a)π−→μ−+?(c)?+p→n+e+(e)?+n→p+μ−(b)K+→μ++?(d)?+n→P+e−(f)μ−→e−+?+?(a)π−→μ−+?(b)K+→μ++?(c)?+p→n+e+(d)?+n→P+e−(e)?+n→p+μ−(f)μ−→e−+?+?
• The index of refraction for violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular spread of visible light passing through a prism of apex angle 60.0∘0∘ if the angle of incidence is 50.0∘?50.0∘? See Figure P35.33.
• Why is the following situation impossible? In the Bohr model of the hydrogen atom, an electron moves in a circular orbit about a proton. The model states that the electron can exist only in certain allowed orbits around the proton: those whose radius rr satisfies r=n2(0.0529nm),r=n2(0.0529nm), where n=1,2n=1,2 3, \ldots. For one of the possible allowed states of the atom, the electric potential energy of the system is −13.6eV−13.6eV .
• If a K0SKS0 meson at rest decays in 0.900×10−10s,0.900×10−10s, how far
does a K05K50 meson travel if it is moving at 0.960 c?c?
• Evaluate Young’s modulus for the material whose stress–strain curve is shown in Figure 12.12.
• In a period of 1.00s,5.00×10231.00s,5.00×1023 nitrogen molecules strike a wall with an area of 8.00cm2.8.00cm2. Assume the molecules move with a speed of 300 m/sm/s and strike the wall head-on in elastic collisions. What is the pressure exerted on the wall? Note: The mass of one N2N2 molecule is 4.65×10−26kg4.65×10−26kg .
• A circular coil of five turns and a diameter of 30.0 cmcm is oriented in a vertical plane with its axis perpendicular to the horizontal component of the Earth’s magnetic field. A horizontal compass placed at the coil’s center is made to deflect 45.0∘0∘ from magnetic north by a current of 0.600 AA in the coil. (a) What is the horizontal component of the Earth’s magnetic field? (b) The current in the coil is switched off. A “dip needle” is a magnetic compass mounted so that it can rotate in a vertical north-south plane. At this location, a dip needle makes an angle of 13.0∘13.0∘ from the vertical. What is the total magnitude of the Earth’s magnetic field at this location?
• Vector →BB→ has x,y,x,y, and zz components of 4.00,6.00,4.00,6.00, and 3.00 units, respectively. Calculate (a) the magnitude of →BB→ and (b) the angle that →BB→ makes with each coordinate axis.
• Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest.
Pursued by ferocious wolves, you are in a sleigh with no horses, gliding without friction across an ice-covered lake. You take an action described by the equations (270kg)(7.50m/s)ˆi=(15.0kg)(−v1fˆi)+(255kg)(v2fˆi)v1f+v2f=8.00m/s(270kg)(7.50m/s)i^v1f+v2f=(15.0kg)(−v1fi^)+(255kg)(v2fi^)=8.00m/s
(a) Complete the statement of the problem, giving the data and identifying the unknowns. (b) Find the values of v1fv1f and v2fv2f . (c) Find the amount of energy that has been transformed from potential energy stored in your body to kinetic energy of the system.
• An object of mass m=m= 0.500 kgkg is suspended from the ceiling of an accelerating truck as shown in Figure P6.21P6.21 Taking a=3.00m/s2a=3.00m/s2 Taking a=3.00m/s2a=3.00m/s2 find (a)(a) the angle θθ that the string makes with the vertical and (b)(b) the tension TT in the string.
• A hawk is flying horizontally at 10.0 m/sm/s in a straight line, 200 mm above the ground. A mouse it has been carrying struggles free from its talons. The hawk continues on its path at the same speed for 2.00 s before attempting to retrieve its prey. To accomplish the retrieval, it dives in a straight line at constant speed and recaptures the mouse 3.00 mm above the ground. (a) Assuming no air resistance acts on the mouse, find the diving speed of the hawk. (b) What angle did the hawk make with the horizontal during its descent? (c) For what time interval did the mouse experience free fall?
• In 1887 in Bridgeport, Connecticut, C. J. Belknap built the water slide shown in Figure P8.73 (page 232).
A rider on a small sled, of total mass 80.0 kg, pushed off to start at the top of the slide (point ) with a speed of 2.50 m/s. The chute was 9.76 m high at the top and 54.3 m long. Along its length, 725 small wheels made friction negligible. Upon leaving the chute horizontally at its bottom end (point ), the rider skimmed across the water of Long Island Sound for as much as 50 m, “skipping along like a flat pebble,” before at last coming to rest and swimming ashore, pulling his sled after him. (a) Find the speed of the sled and rider at point . (b) Model the force of water friction as a constant retarding force acting on a particle. Find the magnitude of the friction force the water exerts on the sled. (c) Find the magnitude of the force the chute exerts on the sled at point . (d) At point , the chute is horizontal but curving in the vertical plane. Assume its radius of curvature is 20.0 m. Find the force the chute exerts on the sled at point.
• A technician wraps wire around a tube of length 36.0 cm having a diameter of 8.00 cm. When the windings are evenly spread over the full length of the tube, the result is a solenoid containing 580 turns of wire. (a) Find the inductance of this solenoid. (b) If the current in this solenoid increases at the rate of 4.00 A/s, find the self-induced emf in the solenoid.
• Three capacitors are connected to a battery as shown in Figure P26.20. Their capacitances are C1=3C,C2=C,C1=3C,C2=C, and C3=5C.C3=5C. (a) What is the equivalent capacitance of this set of capacitors? (b) State the ranking of the capacitors according to the charge they store from largest to smallest. ( c)c) Rank the capacitors according to the potential differences across them from largest to smallest. (d) What If? Assume C3C3 is increased. Explain what happens to the charge stored by each capacitor.
• The relationship between the heat capacity of a sample and the specific heat of the sample material is discussed in Section 20.2. Consider a sample containing 2.00 mol of an ideal diatomic gas. Assuming the molecules rotate but do not vibrate, find (a) the total heat capacity of the sample at constant volume and (b) the total heat capacity at constant pressure. (c) What If? Repeat parts (a) and (b), assuming the molecules both rotate and vibrate.
• Determine which of the following reactions can occur. For
those that cannot occur, determine the conservation law
(or laws) violated.
(a) p→π++π0p→π++π0 (b) p+p→p+p+π0p+p→p+p+π0
(c) p+p→p+π+p+p→p+π+ (d) π+→μ++νμπ+→μ++νμ
(e) n→p+e−+ν¯¯¯en→p+e−+ν¯e (f) π+→μ++nπ+→μ++n
• In the circuit diagrammed in Figure P32.16,P32.16, take E=E= 12.0 VV and R=24.0Ω.R=24.0Ω. Assume the switch is open for t<0t<0 and is closed at t=0.t=0. On a single set of axes, sketch graphs of the current in the circuit as a function of time for t≥t≥ 0,0, assuming (a) the inductance in the circuit is essentially zero, (b) the inductance has an intermediate value, and (c) the inductance has a very large value. Label the initial and final values of the current.
• The circuit in Figure P28.49P28.49 has been connected for several seconds. Find the current (a) in the 4.00−V4.00−V battery, (b) in the 3.00−Ω3.00−Ω resistor, (c)(c) in the 8.00−V8.00−V battery, and (d)(d) in the 3.00−V3.00−V battery. (e) Find the charge on the capacitor.
• A glider of length $\ell$ moves through a stationary photogate on an air track. A photogate (Fig. P2.31) is a device that measures the time interval $\Delta t_{d}$ during which the glider blocks a beam of infrared light passing across the photogate. The ratio $v_{d}=\ell / \Delta t_{d}$ is the average velocity of the glider over this part of its motion. Suppose the glider moves with constant acceleration. (a) Argue for or against the idea that $v_{d}$ is equal to the instantaneous velocity of the glider when it is halfway through the photogate in space. (b) Argue for or against the idea that $v_{d}$ is equal to the instantaneous velocity of the glider when it is halfway through the photogate in time.
• A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius 15.0 cm as shown in Figure P6.68. The circle is always in a vertical plane and rotates steadily about its vertical diameter with a period of 0.450 s. The position of the bead is described by the angle θθ that the radial line, from the center of the loop to the bead, makes with the vertical. (a) At what angle up from the bottom of the circle can the bead stay motionless relative to the turning circle? (b) What If? Repeat the problem, this time taking the period of the circle’s rotation as 0.850 s. (c) Describe how the solution to part (b) is different from the solution to part (a). (d) For any period or loop size, is there always an angle at which the bead can stand still relative to the loop? (e) Are there ever more than two angles? Arnold Arons suggested the idea for this problem.
• Find the radius of the 12 C nucleus. (b) Find the force of repulsion between a proton at the surface of a 16 C nucleus and the remaining five protons. (c) How much work (in MeV) has to
be done to overcome this electric repulsion in transporting the last proton from a large distance up to
the surface of the nucleus? (d) Repeat parts (a), and (c) for 23892U .
• An ideal gas initially at 300 KK undergoes an isobaric expansion at 2.50 kPakPa . If the volume increases from 1.00 m3m3 to 3.00 m3m3 and 12.5 kJkJ is transferred to the gas by heat, what are (a) the change in its internal energy and (b) its final temperature?
• A loaded ore car has a mass of 950 kgkg and rolls on rails with negligible friction. It starts from rest and is pulled up a mine shaft by a cable connected to a winch. The shaft is inclined at 30.0∘0∘ above the horizontal. The car accelerates uniformly to a speed of 2.20 m/sm/s in 12.0 ss and then continues at constant speed. (a) What power must the winch motor provide when the car is moving at constant speed?.
(b) What maximum power must the winch motor provide?
(c) What total energy has transferred out of the motor by work by the time the car moves off the end of the track, which is of length 1250 mm ?
• Three discrete spectral lines occur at angles of 10.1°, 13.7°, and 14.8° in the first-order spectrum of a grating spectrometer. (a) If the grating has 3 660 slits/cm, what are the wavelengths of the light? (b) At what angles are these lines found in the second-order spectrum?
• An inclined plane of angle θ has a spring of force constant k fastened securely at the bottom so that the spring is parallel to the surface. A block of mass m is placed on the plane at a distance d from the spring. From this position, the block is projected downward toward the spring with speed v as shown in Figure P7.63. By what distance is the spring compressed when the block momentarily comes to rest?
• Plot the wave function ψ1s(r)ψ1s(r) versus rr (see Eq. 42.22) and the radial probability density function P1s(r)P1s(r) versus rr (see Eq. 42.25 for hydrogen. Let rr range from 0 to 1.5a0,1.5a0, where a0a0 is the Bohr radius.
• At a certain distance from a charged particle, the magnitude of the electric field is 500 V/mV/m and the electric potential is −3.00kV−3.00kV . (a) What is the distance to the particle? (b) What is the magnitude of the charge?
• An athlete swings a ball, connected to the end of a chain, in a horizontal circle. The athlete is able to rotate the ball at the rate of 8.00 rev/srev/s when the length of the chain is 0.600m.0.600m. When he increases the length to 0.900m,0.900m, he is able to rotate the ball only 6.00 rev/s. (a) Which rate of rotation gives the greater speed for the ball? (b) What is the centripetal acceleration of the ball at 8.00 rev/srev/s ? (c) What is the centripetal acceleration at 6.00 rev/srev/s ?
• If An observer in a coasting spacecraft moves toward a mirror at speed vv relative to the reference frame labeled S in Figure P39.69P39.69 . The mirror is stationary with respect to S. A light pulse emitted by the spacecraft travels toward the mirror and is reflected back to the spacecraft. The spacecraft is a distance dd from the mirror (as measured by observers in S) at the moment the light pulse leaves
the spacecraft. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the spacecraft?
• The three charged particles in Figure P25.20 are at the vertices of an isosceles triangle (where d=2.00cm).d=2.00cm). Taking q=7.00μC,q=7.00μC, calculate the electric potential at point A,A, the midpoint of the base.
• For a particular transparent medium surrounded by air, find the polarizing angle θpθp in terms of the critical angle for total internal reflection θcθc .
• An optical fiber has an index of refraction nn and diameter d.d. It is surrounded by vacuum. Light is sent into the fiber along its axis as shown in Figure P35.43P35.43 on page 1036 . (a) Find the smallest outside radius RminRmin permitted for a bend in the fiber if no light is to escape. (b) What If? What result does part (a) predict as dd approaches zero? Is this behavior reasonable? Explain. (c) As nn increases? (d) As nn approaches 1 (e) Evaluate RminRmin assuming the fiber diameter is 100μmμm and its index of refraction is 1.40 .
• For hydrogen in the 1 state, what is the probability of finding the electron farther than 2.50a0a0 from the nucleus?
• A particle of mass mm and charge qq moves at high speed along the xx axis. It is initially near x=−∞,x=−∞, and it ends up near x=+∞.x=+∞. A second charge QQ is fixed at the point x=0x=0 , y=−d.y=−d. As the moving charge passes the stationary charge, its xx component of velocity does not change appreciably, but it acquires a small velocity in the yy direction. Determine the angle through which the moving charge is deflected from the direction of its initial velocity.
• A 2.00 -mol sample of oxygen gas is confined to a 5.00−L5.00−L vessel at a pressure of 8.00atm.8.00atm. Find the average translational kinetic energy of the oxygen molecules under these
• Expectant parents are thrilled to hear their unborn baby’s heartbeat, revealed by an ultrasonic detector that produces beeps of audible sound in synchronization with the fetal heartbeat. Suppose the fetus’s ventricular wall moves in simple harmonic motion with an amplitude of 1.80 mm and a frequency of 115 beats per minute. (a) Find the maximum linear speed of the heart wall. Suppose a source mounted on the detector in contact with the mother’s abdomen produces sound at 2 000 000.0 Hz, which travels through tissue at 1.50 km/s. (b) Find the maximum change in frequency between the sound that arrives at the wall of the baby’s heart and the sound emitted by the source. (c) Find the maximum change in frequency between the reflected sound received by the detector and that emitted by the source.
• State what the Fermi energy depends on according to the free-electron theory of metals and how the Fermi energy depends on that quantity. (b) Show that Equation 43.25 can be expressed as EF=(3.65×10−19)n2/3eEF=(3.65×10−19)ne2/3 where EFEF is in electron volts when nene is in electrons per cubic meter. (c) According to Table 43.2, by what factor does the free-electron concentration in copper exceed that in potassium? (d) Which of these metals has the larger Fermi energy? (e) By what factor is the Fermi energy larger? (f) Explain whether this behavior is predicted by Equation 43.25.
• A disk 8.00 cmcm in radius rotates at a constant rate of 1200 rev/minrev/min about its central axis. Determine (a) its angular speed in radians per second, (b) the tangential speed at a point 3.00 cmcm from its center, (c) the radial acceleration of a point on the rim, and (d) the total distance a point on the rim moves in 2.00 ss .
• A light spring has unstressed length 15.5 cm. It is described by Hooke’s law with spring constant 4.30N/m. One end of the horizontal spring is held on a fixed vertical axle, and the other end is attached to a puck of mass m that can move without friction over a horizontal surface. The puck is set into motion in a circle with a period of 1.30 s. (a) Find the extension of the spring x as it depends on m. Evaluate x for (b)m=0.0700kg, (c) m=0.140kg, (d) m= 0.180kg, and (e)m=0.190kg. (f) Describe the pattern of variation of x as it depends on m.
• A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length R.R.
That is, ω=√g/R.ω=g/R−−−−√.
• An object is released from rest at an altitude hh above the surface of the Earth. (a) Show that its speed at a distance rr from the Earth’s center, where RE≤r≤RE+h,RE≤r≤RE+h, is
v=2GME(1r−1RE+h)−−−−−−−−−−−−−−−−−−−√v=2GME(1r−1RE+h)
(b) Assume the release altitude is 500 km. Perform the integral
Δt=∫fidt=−∫fidrvΔt=∫ifdt=−∫ifdrv
to find the time of fall as the object moves from the release point to the Earth’s surface. The negative sign appears because the object is moving opposite to the radial direction, so its speed is v=−dr/dt.v=−dr/dt. Perform the integral numerically.
• A strong magnet is placed under a horizontal conducting ring of radius rr that carries current II as shown in Figure P 29.41. If the magnetic field →BB→ makes an angle θθ with the vertical at the ring’s location, what are (a) the magnitude and (b) the direction of the resultant magnetic force on the ring?
• Determine which decays can occur spontaneously.
(a) 4020Ca→e++4019K4020Ca→e++4019K
(b) 9844Ru→42He+9442Mo9844Ru→42He+9442Mo
(c) 14460Nd→42He+14058Ce14460Nd→42He+14058Ce
• Two flat, rectangular mirrors, both perpendicular to a horizontal sheet of paper, are set edge to edge with their reflecting surfaces perpendicular to each other. (a) A light ray in the plane of the paper strikes one of the mirrors at an arbitrary angle of incidence θ1.θ1. Prove that the final direction of the ray, after reflection from both mirrors, is opposite its initial direction. (b) What If? Now assume the paper is replaced with a third flat mirror, touching edges with the other two and perpendicular to both, creating a cornercube retroreflector (Fig. 35.8a).35.8a). A ray of light is incident from any direction within the octant of space bounded by the reflecting surfaces. Argue that the ray will reflect once from each mirror and that its final direction will be opposite its original direction. The Apollo 11 astronauts placed a panel of corner-cube retroreflectors on the Moon. Analysis of timing data taken with it reveals that the radius of the Moon’s orbit is increasing at the rate of 3.8 cm/yrcm/yr as it loses kinetic energy because of tidal friction.
• Two equal positively charged particles are at opposite corners of a trapezoid as shown in Figure P23.27. Find symbolic expressions for the total electric field at (a) the point PP and (b)(b) the point P′P′
• Figure P35.73 shows an overhead view of a room of square floor area and side L. At the center of the room is a mirror set in a vertical plane and rotating on a vertical shaft at angular speed ωω about an axis coming out of the page. A bright red laser beam enters from the center point on one wall of the room and strikes the mirror. As the mirror rotates, the reflected laser beam creates a red spot sweeping across the walls of the room. (a) When the spot of light on the wall is at distance xx from point OO , what is its speed? (b) What value of xx corresponds to the minimum value for the speed? (c) What is the minimum value for the speed? (d) What is the maximum speed of the spot on the wall? (e) In what time interval does the spot change from its minimum to its maximum speed?
• An LCLC circuit like that in Figure CQ32.8CQ32.8 consists of a 3.30−H3.30−H inductor and an 840−pF840−pF capacitor that initially carries a 105−μC105−μC charge. The switch is open for t<0t<0 and is then thrown closed at t=0.t=0. Compute the following quantities at t=2.00ms:(a)t=2.00ms:(a) the energy stored in the capacitor, (b)(b) the energy stored in the inductor, and (c)(c) the total energy in the circuit.
• The power of sunlight reaching each square meter of the Earth’s surface on a clear day in the tropics is close to 1 000 W. On a winter day in Manitoba, the power concentration of sunlight can be 100 W/m2. Many human activities are described by a power per unit area on the order of 102W/m2102W/m2 or less. (a) Consider, for example, a family of four paying $66$66 to the electric company every 30 days for 600 kWh of energy carried by electrical transmission to their house, which has floor dimensions of 13.0 m by 9.50 m. Compute the power per unit area used by the family. (b) Consider a car 2.10 m wide and 4.90 m long traveling at 55.0 mi/h using gasoline having “heat of combustion” 44.0 MJ/kg with fuel economy 25.0 mi/gal. One gallon of gasoline has a mass of 2.54 kg. Find the power per unit area used by the car. (c) Explain why direct use of solar energy is not practical for running a conventional automobile. (d) What are some uses of solar energy that are more practical?
• The intensity of light in a diffraction pattern of a single slit is described by the equation
I=Imaxsin2ϕϕ2I=Imaxsin2⁡ϕϕ2
where ϕ=(πasinθ)/λϕ=(πasin⁡θ)/λ . The central maximum is at ϕ=0ϕ=0 , and the side maxima are approximately at ϕ=(m+12)πϕ=(m+12)π for m=1,2,3,…m=1,2,3,… Determine more precisely (a) the location of the first side maximum, where m=1,m=1, and (b) the location of the second side maximum. Suggestion: Observe in Figure 38.6 a that the graph of intensity versus ϕϕ has a horizontal tangent at maxima and also at minima.
• A large man sits on a four-legged chair with his feet off the floor. The combined mass of the man and chair is 95.0 kg. If the chair legs are circular and have a radius of 0.500 cm at the bottom, what pressure does each leg exert on the floor?
• A car accelerates down a hill (Fig. P5.69), going from rest to 30.0 m/s in 6.00 s. A toy inside the car hangs by a string from the car’s ceiling. The ball in the figure represents the toy, of mass 0.100 kg. The acceleration is such that the string remains perpendicular to the ceiling. Determine (a) the angle θθ and (b) the tension in the string.
• Two speeding lead bullets, one of mass 12.0 g moving to the right at 300 m/s and one of mass 8.00 g moving to the left at 400 m/s, collide head-on, and all the material sticks together. Both bullets are originally at temperature 30.0°C. Assume the change in kinetic energy of the system appears entirely as increased internal energy. We would like to determine the temperature and phase of the bullets after the collision. (a) What two analysis models are appropriate for the system of two bullets for the time interval from before to after the collision? (b) From one of these models, what is the speed of the combined bullets after the collision? (c) How much of the initial kinetic energy has transformed to internal energy in the system after the collision? (d) Does all the lead melt due to the collision? (e) What is the temperature of the combined bullets after the collision? (f) What is the phase of the combined bullets after the collision?
• Singly ionized carbon is accelerated through a potential difference DV and passed into a mass spectrometer to determine the isotopes present (see Chapter 29). The magnitude of the magnetic field in the spectrometer is BB . The orbit radius for an isotope of mass m1m1 as it passes through the field is r1r1 . Find the radius of the orbit of an isotope of mass m2.m2.
• Young’s double-slit experiment underlies the instrument landing system used to guide aircraft to safe landings at some airports when the visibility is poor. Although real systems are more complicated than the example described here, they operate on the same principles. A pilot is trying to align her plane with a runway as suggested in Figure P37.20P37.20 .
• The potential difference across the filament of a lightbulb is maintained at a constant value while equilibrium temperature is being reached. The steady-state current in the bulb is only one-tenth of the current drawn by the bulb when it is first turned on. If the temperature coefficient of resistivity for the bulb at 20.0∘0∘C is 0.00450(∘C)−1(∘C)−1 and the resistance increases linearly with increasing temperature, what is the final operating temperature of the filament?
• A sample consisting of nn moles of an ideal gas undergoes a reversible isobaric expansion from volume ViVi to volume 3ViVi . Find the change in entropy of the gas by calculating ∫fidQ/T,∫ifdQ/T, where dQ=nCPdT.dQ=nCPdT.
• The first stage of a Saturn VV space vehicle consumed fuel and oxidizer at the rate of 1.50×104kg/s1.50×104kg/s with an exhaust speed of 2.60×103m/s2.60×103m/s . (a) Calculate the thrust produced by this engine. (b) Find the acceleration
the vehicle had just as it lifted off the launch pad on the Earth, taking the vehicle’s initial mass as 3.00×106kg3.00×106kg .
• A diffraction pattern is formed on a screen 120 cm away from a 0.400-mm-wide slit. Monochromatic 546.1-nm light is used. Calculate the fractional intensity I/ImaxI/Imax at a point on the screen 4.10 mm from the center of the principal maximum.
• Sketch (a) the wave function ψ(x) and (b) the probability density |ψ(x)|2 for the n=4 state of a quantum particle in a finite potential well. (See Active Fig. 41.7.)
• A thin wire ℓ=30.0cm long is held parallel to and d=80.0cm above a long, thin wire carrying I=200A and fixed in position (Fig. P31.68). The 30.0 -cm wire is released at the instant t=0 and falls, remaining parallel to the current-carrying wire as it falls. Assume the falling wire accelerates at 9.80 m/s2 . (a) Derive an equation for the emf induced in it as a function of time. (b) What is the minimum value of the emf? (c) What is the maximum value? (d) What is the induced emf 0.300 s after the wire is released?
• In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is 1.22 m. The mug slides off the counter and strikes the floor 1.40 m from the base of the counter. (a) With what velocity did the mug leave the counter? (b) What was the direction of the mug’s velocity just before it hit the floor?
• Suppose our Sun is about to explode. In an effort to bescape, we depart in a spacecraft at v 5 0.800c and head toward the star Tau Ceti, 12.0 ly away. When we reach the midpoint of our journey from the Earth, we see our Sun explode, and, unfortunately, at the same instant, we see Tau Ceti explode as well. (a) In the spacecraft’s frame of reference, should we conclude that the two explosions occurred simultaneously? If not, which occurred first? (b) What If? In a frame of reference in which the Sun and Tau Ceti are at rest, did they explode simultaneously? If not, which exploded first?
• Calculate the force required to pull a copper ball of radius 2.00 cm upward through a fluid at the constant speed 9.00 cm/s. Take the drag force to be proportional to the speed, with proportionality constant 0.950 kg/s. Ignore the buoyant force.
• Many cells are transparent and colorless. Structures of great interest in biology and medicine can be practically invisible to ordinary microscopy. To indicate the size and shape of cell structures, an interference microscope reveals a difference in index of refraction as a shift in interference fringes. The idea is exemplified in the following problem. An air wedge is formed between two glass plates in con-
tact along one edge and slightly separated at the opposite edge as in Figure P37.35P37.35 . When the plates are illuminated with monochromatic light from above, the reflected light has 85 dark fringes. Calculate the number of dark fringes that appear if water (n=1.33)(n=1.33) replaces the air between the plates.
• In a water pistol, a piston drives water through a large tube of area $A_{1}$ into a smaller tube of area $A_{2}$ as shown in Figure P14.70. The radius of the large tube is 1.00 cm and that of the small tube is 1.00 mm. The smaller tube is 3.00 cm above the larger tube. (a) If the pistol is fired horizontally at a height of 1.50 m, determine the time interval required for the water to travel from the nozzle to the ground. Neglect air resistance and assume atmospheric pressure is 1.00 atm. (b) If the desired range of the stream is $8.00 \mathrm{m},$ with what speed $v_{2}$ must the stream leave the nozzle? ( $\mathrm{c} )$ At what speed $v_{1}$ must the plunger be moved to achieve the desired range? (d) What is the pressure at the nozzle? (e) Find the pressure needed in the larger tube. (f) Calculate the force that must be exerted on the trigger to achieve the desired range. (The force that must be exerted is due to pressure over and above atmospheric pressure.)
• A prism that has an apex angle of 50.0∘0∘ is made of cubic zirconia. What is its minimum angle of deviation?
• A 2.00 -m-long cylindrical steel wire with a cross-sectional diameter of 4.00 mmmm is placed over a light, frictionless pulley. An object of mass m1=5.00kgm1=5.00kg is hung from one end of the wire and an object of mass m2=3.00kgm2=3.00kg from the other end as shown in Figure P12.35P12.35 . The objects are released and allowed to move freely. Compared with its length before the objects were attached, by how much has the wire stretched while the objects are in motion?
• A commercial capacitor is to be constructed as shown in Figure P26.46P26.46 . This particular capacitor is made from two strips of aluminum foil separated by a strip of paraffin- coated paper. Each strip of foil and paper is 7.00 cmcm wide. The foil is 0.00400 mmmm thick, and the paper is 0.0250 mmmm thick and has a dielectric constant of 3.70.3.70. What length should the strips have if a capacitance of 9.50×10−8F9.50×10−8F is desired before the capacitor is rolled up? (Adding a second strip of paper and rolling the capacitor would effectively double its capacitance by allowing charge storage on both sides of each strip of foil.
• How fast must a meterstick be moving if its length is measured to shrink to 0.500 m?
• A glider of length 12.4 cm moves on an air track with constant acceleration (Fig P2.31). A time interval of 0.628 s elapses between the moment when its front end passes a fixed point $\mathbb{Q}$ along the track and the moment when its back end passes this point. Next, a time interval of 1.39 $\mathrm{s}$ elapses between the moment when the back end of the glider passes the point $@$ and the moment when the front end of the glider passes a second point $(8 \text { ) farther down the }$ track. After that, an additional 0.431 s elapses until the back end of the glider passes point $(\text { a) Find the average }$ speed of the glider as it passes point $@$ . (b) Find the acceleration of the glider. (c) Explain how you can compute the acceleration without knowing the distance between points $@$ and $(\mathrm{B})$
• Use the component method to add the vectors →AA→ and →BB→ shown in Figure P 3.11. Both vectors have magnitudes of 3.00 m and vector →AA→ makes an angle of θ=30.0∘θ=30.0∘ with the xx axis. Express the resultant →A+→BA→+B→ in unit vector notation.
• An audio amplifier, represented by the AC source and resistor in Figure P33.7, delivers to the speaker alternating voltage at audio frequencies. If the source voltage has an amplitude of 15.0V,R=8.20Ω,15.0V,R=8.20Ω, and the speaker is equivalent to a resistance of 10.4Ω,10.4Ω, what is the time-averaged power transferred to it?
• An inductor that has an inductance of 15.0 HH and a resistance of 30.0ΩΩ is connected across a 100−V100−V battery. What is the rate of increase of the current (a) at t=0t=0 and (b)(b) at t=t= 1.50 ss ?
• In certain ranges of a piano keyboard, more than one string is tuned to the same note to provide extra loudness. For example, the note at 110 Hz has two strings at this frequency. If one string slips from its normal tension of 600 N to 540 N, what beat frequency is heard when the hammer strikes the two strings simultaneously?
• What is the rotational kinetic energy of the Earth about its spin axis? Model the Earth as a uniform sphere and use data from the end papers of this book. (b) The rotational kinetic energy of the Earth is decreasing steadily because of tidal friction. Assuming the rotational period decreases by 10.0μμ s each year, find the change in one day.
• Calculate the power delivered to each resistor in the circuit shown in Figure P28.17.
• John is pushing his daughter Rachel in a wheel-barrow when it is stopped by a brick 8.00 cmcm high (Fig. Pl2.21). The handles make an angle of θ=θ= 15.0∘0∘ with the ground. Due to the weight of Rachel and the wheelbarrow, a downward force of 400 NN is exerted at the center of the wheel, which has a radius of 20.0cm.20.0cm. (a) What force must John apply along the handles to just. start the wheel over the brick? (b) What is the force (magnitude and direction) that the brick exerts on the wheel just as the wheel begins to lift over the brick? In both parts, assume the brick remains fixed and does not slide along the ground. Also assume the force applied by John is directed exactly toward the center of the wheel.
• A package is dropped at time $t=0$ from a helicopter that is descending steadily at a speed $v_{i}$ . (a) What is the speed of the package in terms of $v_{i}, g,$ and $t$ ? (b) What vertical distance $d$ is it from the helicopter in terms of $g$ and $t$ ? (c) What are the answers to parts (a) and (b) if the helicopter is rising steadily at the same speed?
• Monochromatic ultraviolet light with intensity 550 W/m2W/m2 is incident normally on the surface of a metal that has a work function of 3.44 eV. Photoelectrons are emitted with a maximum speed of 420 km/skm/s . (a) Find the maximum possible rate of photoelectron emission from 1.00 cm2cm2 of the surface by imagining that every photon produces one photoelectron. (b) Find the electric current these electrons constitute. (c) How do you suppose the actual current compares with this maximum possible current?
• White light is spread out into its spectral components by a diffraction grating. If the grating has 2 000 grooves per centimeter, at what angle does red light of wavelength 640 nm appear in first order?
• A copper rod and a steel rod are different in length by 5.00 cmcm at 0∘0∘C. The rods are warmed and cooled together. (a) Is it possible that the length difference remains constant at all temperatures? Explain. (b) If so, describe the lengths at 0∘C0∘C as precisely as you can. Can you tell which rod is longer? Can you tell the lengths of the rods?
• Two pulses traveling on the same string are described by
y1=5(3x−4t)2+2y2=−5(3x+4t−6)2+2y1=5(3x−4t)2+2y2=−5(3x+4t−6)2+2
(a) In which direction does each pulse travel? (b) At what instant do the two cancel everywhere? (c) At what point do the two pulses always cancel?
• A 2.00 -kg block hangs from a rubber cord, being supported so that the cord is not stretched. The unstretched length of the cord is 0.500m,0.500m, and its mass is 5.00 g. The “spring constant” for the cord is 100 N/m. The block is released and stops momentarily at the lowest point. (a) Determine the tension in the cord when the block is at this lowest point. (b) What is the length of the cord in this “stretched” position? (c) If the block is held in this lowest position, find the speed of a transverse wave in the cord.
• Some physical systems possessing capacitance continuously distributed over space can be modeled as an infinite array of discrete circuit elements. Examples are a microwave wave guide and the axon of a nerve cell. To practice analysis of an infinite array, determine the equivalent capacitance C between terminals X and Y of the infinite set of capacitors represented in Figure P26.69P26.69 . Each capacitor has capacitance C0.C0. Suggestions: Imagine that the ladder is cut at the line ABAB and note that the equivalent capacitance of the infinite section to the right of ABAB is also C.C.
• Why is the following situation impossible? While at the bottom of a calm freshwater lake, a scuba diver sees the Sun at an apparent angle of 38.0∘0∘ above the horizontal.
• Heart–lung machines and artificial kidney machines employ electromagnetic blood pumps. The blood is confined to an electrically insulating tube, cylindrical in practice but represented here for simplicity as a rectangle of interior width ww and height h.h. Figure P 29.56 shows a rectangular section of blood within the tube. Two electrodes fit into the top and the bottom of the tube. The potential difference between them establishes an electric current through the blood, with current density JJ over the current through the blood, with current density JJ over the section of length LL shown in Figure P 29.56. A perpendicular magnetic field exists in the same region. (a) Explain why this arrangement produces on the liquid a force that is directed along the length of the pipe. (b) Show that the section of liquid in the magnetic field experiences a pressure increase JLBJLB . (c) After the blood leaves the pump, is it charged? (d) Is it carrying current? (e) Is it magnetized? (The same electromagnetic pump can be used for any fluid that conducts electricity, such as liquid sodium in a nuclear reactor.)
• A backyard swimming pool with a circular base of diameter 6.00 $\mathrm{m}$ is filled to depth 1.50 $\mathrm{m}$ (a) Find the absolute – pressure at the bottom of the pool. (b) Two persons with combined mass 150 $\mathrm{kg}$ enter the pool and float quietly there. No water overflows. Find the pressure increase at the bottom of the pool after they enter the pool and float.
• The voltage across an air-filled parallel-plate capacitor is measured to be 85.0 V. When a dielectric is inserted and completely fills the space between the plates as in Figure P26.42, the voltage drops to 25.0 V. (a) What is the dielectric constant of the inserted material? (b) Can you identify the dielectric? If so, what is it? (c) If the dielectric does not completely fill the space between the plates, what could you conclude about the voltage across the plates?
• Find the number of moles in one cubic meter of an ideal gas at 20.0∘0∘C and atmospheric pressure. (b) For air, Avogadro’s number of molecules has mass 28.9 gg . Calculate the mass of one cubic meter of air. (c) State how this result compares with the tabulated density of air at 20.0∘C20.0∘C .
• A particle starts from rest and accelerates as shown in Figure P2.16. Determine (a) the particle’s speed at $t=$ 10.0 $\mathrm{s}$ and at $t=20.0 \mathrm{s}$ , and (b) the distance traveled in the first 20.0 $\mathrm{s}$
• Figure P9.46aP9.46a shows an overhead view of the initial configuration of two pucks of mass mm on frictionless ice. The pucks are tied together with a string of length ℓℓ and negligible mass. At time t=0,t=0, a constant force of magnitude FF begins to pull to the right on the center point of the string. At time t,t, the moving pucks strike each other and stick together. At this time, the force has moved through a distance d,d, and the pucks have attained a speed v(Fig.P9.46b).v(Fig.P9.46b). (a) What is vv in terms of F,d,ℓ,F,d,ℓ, and m?m? (b) How much of the energy transferred into the system by work done by the force has been transformed to internal energy?
• As a certain sound wave travels through the air, it produces pressure variations (above and below atmospheric pressure) given by ΔP=1.27sin(πx−340πt)ΔP=1.27sin(πx−340πt) in SI units. Find (a) the amplitude of the pressure variations, (b) the frequency, (c) the wavelength in air, and (d) the speed of the sound wave.
• A solid, insulating sphere of radius aa has a uniform charge density throughout its volume and a total charge QQ . Concentric with this sphere is an uncharged, conducting, hollow sphere whose inner and outer radii are bb and cc as shown in Figure P 24.51. We wish to understand completely the charges and electric fields at all locations. (a) Find the charge contained within a sphere of radius r<ar<a (b) From this value, find the magnitude of the electric field for r<ar<a (c) What charge is contained within a sphere of radius rr when a<r<b?a<r<b? (d) From this value, find the magnitude of the electric field for rr when a<r<ba<r<b . (e) Now consider rr when b<r<b<r< c. What is the magnitude of the electric field for this range of values of r?(f)r?(f) From this value, what must be the charge on the inner surface of the hollow sphere? (g) From part (f), what must be the charge on the outer surface of the hollow sphere? (h) Consider the three spherical surfaces of radii a,b,a,b, and c.c. Which of these surfaces has the largest magnitude of surface charge density?
• A wire carrying a current II is bent into the shape of an equilateral triangle of side L.L. (a) Find the magnitude of the magnetic field at the center of the triangle. (b) At a point halfway between the center and any vertex, is the field stronger or weaker than at the center? Give a qualitative argument for your answer.
• The ρ−ρ− meson has a charge of −e,−e, a spin quantum number of 1,1, and a mass 1507 times that of the electron. The possible values for its spin magnetic quantum number are −1,0,−1,0, and 1.1. What If? Imagine that the electrons in atoms are replaced by ρ−ρ− mesons. List the possible sets of quantum numbers for ρ−ρ− mesons in the 3dd subshell.
• Why is the following situation impossible? A hypothetical metal has the following properties: its Fermi energy is 5.48 eV, its density is 4.90×103kg/m34.90×103kg/m3 , its molar mass is 100 g/molg/mol , and it has one free electron per atom.
• A small sphere of charge q1=0.800μCq1=0.800μC hangs from the end of a spring as in Figure P23.50aP23.50a . When another small sphere of charge q2=−0.600μCq2=−0.600μC is held beneath the first sphere asin Figure P23.50bP23.50b, the spring stretches by d=d= 3.50 cmcm from its original length and reaches a new equilibrium position with a separation between the charges of r=5.00cm.r=5.00cm. What is the force constant of the spring?
• Consider the two vectors →A=3ˆi−2ˆjA→=3i^−2j^ and →B=−ˆi−4ˆj.B→=−i^−4j^. Calculate (a) →A+→B,A→+B→, (b) →A−→B,A→−B→, (c) |→A+→B|,|A→+B→|, (d) |→A−→B|,|A→−B→|, and (e) the directions of →A+→BA→+B→ and →A−→BA→−B→ .
• In the What If? section of Example 37.2 , it was claimed that overlapping fringes in a two-slit interference pattern for two different wavelengths obey the following relation- ship even for large values of the angle θ:θ: m′m=λλ′m′m=λλ′ (a) Prove this assertion. (b) Using the data in Example 37.2,37.2, find the nonzero value of yy on the screen at which the fringes from the two wavelengths first coincide.
• A coil of Nichrome wire is 25.0 mm long. The wire has a diameter of 0.400 mmmm and is at 20.0∘0∘C . If it carries a current of 0.500 AA , what are (a) the magnitude of the electric field in the wire and (b) the power delivered to it? (c) What If? If the temperature is increased to 340∘C340∘C and the potential difference across the wire remains constant, what is the power delivered?
• Two resistors connected in series have an equivalent resistance of 690Ω.690Ω. When they are connected in parallel, their equivalent resistance is 150Ω.150Ω. Find the resistance of each resistor.
• The lens and mirror in Figure $\mathrm{P} 36.63$ are separated by $d=$ 1.00 $\mathrm{m}$ and have focal lengths of $+80.0 \mathrm{cm}$ and $-50.0 \mathrm{cm},$ respectively. An object is placed $p=1.00 \mathrm{m}$ to the left of the lens as shown. (a) Locate the final image, formed by light that has gone through the lens twice. (b) Determine the overall magnification of the image and $(\mathrm{c})$ state whether the image is upright or inverted.
• Lightning produces a maximum air temperature on the order of 104K104K , whereas a nuclear explosion produces a temperature on the order of 107K107K . ( a) Use Wien’s displacement law to find the order of magnitude of the wavelength of the thermally produced photons radiated with greatest
intensity by each of these sources. (b) Name the part of the electromagnetic spectrum where you would expect each to radiate most strongly.
• An airplane has a mass of $1.60 \times 10^{4} \mathrm{kg},$ and each wing has an area of $40.0 \mathrm{m}^{2} .$ During level flight, the pressure on the lower wing surface is $7.00 \times 10^{4} \mathrm{Pa}$ . (a) Suppose the lift on the airplane were due to a pressure difference alone. Determine the pressure on the upper wing surface. (b) More realistically, a significant part of the lift is due to deflection of air downward by the wing. Does the inclusion of this force mean that the pressure in part (a) is higher or lower? Explain.
• The energy gap for silicon at 300 K is 1.14 eV. (a) Find the lowest-frequency photon that can promote an electron from the valence band to the conduction band. (b) What is the wavelength of this photon?
• A sphere of mass MM is supported by a string that passes over a pulley at the end of a horizontal rod of length LL (Fig. P 18.25). The string makes an angle θθ with the rod. The fundamental frequency of standing waves in the portion of the string above the rod is ff . Find the mass of the portion of the string above the rod.
• Two small spheres hang in equilibrium at the bottom ends of threads, 40.0 cmcm long, that have their top ends tied to the same fixed point. One sphere has mass 2.40 gg and charge +300nC.+300nC. The other sphere has the same mass and charge +200+200 nC. Find the distance between the centers of the spheres.
• A glass optical fiber (n=1.50)(n=1.50) is submerged in water (n=(n= 1.33).1.33). What is the critical angle for light to stay inside the fiber?
• An object of mass mm hangs in equilibrium from a string with a total length LL and a linear mass density μμ . The string is wrapped around two light, friction less pulleys that are separated by a distance dd (Fig. P 18.71 a). (a) Determine the tension in the string. (b) At what frequency must the string between the pulleys vibrate to form the standing-wave pattern shown in Figure P 18.71 b?
• After a 0.300-kg rubber ball is dropped from a height of 1.75 m, it bounces off a concrete floor and
rebounds to a height of 1.50 m. (a) Determine the magnitude and direction of the impulse delivered to the ball by the floor. (b) Estimate the time the ball is in contact with the floor and use this estimate to calculate the average force the floor exerts on the ball.
• Interference effects are produced at point PP on a screen as a result of direct rays from a 500 -nm source and reflected rays from the mirror as shown in Figure P37.53. Assume the source is 100 mm to the left of the screen and 1.00 cmcm above the mirror. Find the distance yy to the first dark band above the mirror.
• Two concrete spans of a 250 -m-long bridge are placed end to end so that no room is allowed for expansion (Fig. Pl9.55a). If a temperature increase of 20.0∘0∘C occurs, what is
the height yy to which the spans rise when they buckle (Fig. Pl9. 55 b)b) ?
• A teapot with a surface area of 700 cm2cm2 is to be plated with silver. It is attached to the negative electrode of an electrolytic cell containing silver nitrate (Ag+NO−3).(Ag+NO−3). The cell is powered by a 12.0−V12.0−V battery and has a resistance of 1.80ΩΩ . If the density of silver is 10.5×103kg/m3,10.5×103kg/m3, over what time interval does a 0.133 -mm layer of silver build up on the teapot?
• The heating element of an electric coffee maker operates at 120 VV and carries a current of 2.00 AA . Assuming the water absorbs all the energy delivered to the resistor, calculate the time interval during which the temperature of 0.500 kgkg of water rises from room temperature (23.0∘C)(23.0∘C) to the boiling point.
• Suppose you fill two rubber balloons with air, suspend both of them from the same point, and let them hang down on strings of equal length. You then rub each with wool or on your hair so that the balloons hang apart with a noticeable separation between them. Make order-of-magnitude estimates of (a) the force on each, (b) the charge on each, (c) the field each creates at the center of the other, and (d) the total flux of electric field created by each balloon. In your solution, state the quantities you take as data and the values you measure or estimate for them.
• A triangular glass prism with apex angle ΦΦ has an index of refraction nn (Fig. P35.37). What is the smallest angle of incidence θ1θ1 for which a light ray can emerge from the other side?
• The oldest artificial satellite still in orbit is Vanguard I, launched March 3, 1958. Its mass is 1.60 kg. Neglecting atmospheric drag, the satellite would still be in its initial orbit, with a minimum distance from the center of the Earth of 7.02 Mm and a speed at this perigee point of 8.23 km/s. For this orbit, find (a) the total energy of the satellite–Earth system and (b) the magnitude of the angular momentum of the satellite. (c) At apogee, find the satellite’s speed and its distance from the center of the Earth. (d) Find the semimajor axis of its orbit. (e) Determine its period.
• The plane of a square loop of wire with edge length a=0.200m is oriented vertically and along an east-west axis. The Earth’s magnetic field at this point is of magnitude B=35.0μT and is directed northward at 35.0∘ below the horizontal. The total resistance of the loop and the wires connecting it to a sensitive ammeter is 0.500Ω . If theloop is suddenly collapsed by horizontal forces as shown in Figure P31.57 , what total charge enters one terminal of the ammeter?
• The intensity of solar radiation at the top of the Earth’s atmosphere is 1370 W/m2W/m2 . Assuming 60%% of the incoming solar energy reaches the Earth’s surface and you absorb 50%% of the incident energy, make an order-of-magnitude estimate of the amount of solar energy you absorb if you sunbathe for 60 minutes.
• If the magnitude of the drift velocity of free electrons in a copper wire is 7.84×10−4m/s7.84×10−4m/s , what is the electric field in the conductor?
• A toy rocket engine is securely fastened to a large puck that can glide with negligible friction over a horizontal surface, taken as the xyxy plane. The 4.00 -kg puck has a velocity of 3.00ˆim/si^m/s at one instant. Eight seconds later, its velocity is (8ˆi+10ˆj)m/s. Assuming the rocket engine exerts a con-
stant horizontal force, find (a) the components of the force and (b) its magnitude.
• A K0sKs0 particle at rest decays into a π+π+ and a π−.π−. The mass
of the K0SKS0 is 497.7MeV/c2,497.7MeV/c2, and the mass of each ππ meson is 139.6MeV/c2.139.6MeV/c2. What is the speed of each pion?
• A war-wolf or trebuchet is a device used during the Middle Ages to throw rocks at castles and now sometimes used to fling large vegetables and pianos as a sport. A simple trebuchet is shown in Figure P10.27. Model it as a stiff rod of negligible mass, 3.00 m long, joining particles of mass m1=0.120kgm1=0.120kg and m2=60.0kgm2=60.0kg at its ends. It can turn on a frictionless, horizontal axle perpendicular to the rod and 14.0 cmcm from the large-mass particle. The operator releases the trebuchet from rest in a horizontal orientation. (a) Find the maximum speed that the small-mass object attains. (b) While the small-mass object is gaining speed, does it move with constant acceleration? (C) Does it move with constant tangential acceleration? (d) Does the trebuchet move with constant angular acceleration? (e) Does it have constant momentum? (f) Does the trebuchet-Earth system have constant mechanical energy?
• Material with uniform resistivity pp is formed into a wedge as shown in Figure P27.72. Show that the resistance between face A and face B of this wedge is
R=ρLw(y2−y1)lny2y1R=ρLw(y2−y1)lny2y1
• A truck is moving with constant acceleration aa up a hill that makes an angle ϕϕ with the horizontal as in Figure P6.51. A small sphere of mass mm is suspended from the ceiling of the truck by a light cord. If the pendulum makes a constant angle θθ with the perpendicular to the ceiling, what is a?a?
• The total power per unit area radiated by a black body at a temperature TT is the area under the I(λ,T)I(λ,T) -versus-\lambda curve as shown in Active Figure 40.3.40.3. (a) Show that this power per unit area is
∫∞0I(λ,T)dλ=σT4∫0∞I(λ,T)dλ=σT4
where I(λ,T)I(λ,T) is given by Planck’s radiation law and σσ is a constant independent of TT . This result is known as Stefan’s law. (Section 20.7)) To carry out the integration, you
should make the change of variable x=hc/λkBTx=hc/λkBT and use
∫∞0x3dxex−1=π415∫0∞x3dxex−1=π415
(b) Show that the Stefan-Boltzmann constant σσ has the value
σ=2π5k4B15c2h3=5.67×10−8W/m2⋅K4σ=2π5kB415c2h3=5.67×10−8W/m2⋅K4
• The pupil of a cat’s eye narrows to a vertical slit of width 0.500 mm in daylight. Assume the average wavelength of the light is 500 nm. What is the angular resolution for horizontally separated mice?
• sound wave moves down a cylinder as in Active Figure 17.2. Show that the pressure variation of the wave is described by ΔP=±ρvω√s2max−s2,ΔP=±ρvωs2max−−−−√−s2, where s=s(x,t)s=s(x,t) is given by Equation 17.1.17.1.
• A curve in a road forms part of a horizontal circle. As a car goes around it at constant speed 14.0 m/s, the total horizontal force on the driver has magnitude 130 N. What is the total horizontal force on the driver if the speed on the same curve is 18.0 m/s instead?
• Two capacitors, C1=18.0μFC1=18.0μF and C2=36.0μF,C2=36.0μF, are connected in series, and a 12.0−V12.0−V battery is connected across the two capacitors. Find (a)(a) the equivalent capacitance and (b)(b) the energy stored in this equivalent capacitance. (c) Find the energy stored in each individual capacitor. (d) Show that the sum of these two energies is the same as the energy found in part (b). (e) Will this equality always be true, or does it depend on the number of capacitors and their capacitances? (f) If the same capacitors were connected in parallel, what potential difference would be required across them so that the combination stores the same energy as in part (a)? (g) Which capacitor stores more energy in this situation, C1C1 or C2?C2?
• A single conservative force acting on a particle within a system varies as →F=(−Ax+Bx2)ˆi, where A and B are constants, →F is in newtons, and x is in meters. (a) Calculate the potential energy function U(x) associated with this force for the system, taking U=0 at x=0. Find (b) the change in potential energy and (c) the change in kinetic energy of the system as the particle moves from x=2.00m to x=3.00m.
• For a hydrogen atom in its ground state, compute (a) the orbital speed of the electron, (b) the kinetic energy of the electron, and (c) the electric potential energy of the atom.
• A 40.0-cm-diameter circular loop is rotated in a uni- form electric field until the position of maximum electric flux is found. The flux in this position is measured to be 5.20×105N⋅m2/C.5.20×105N⋅m2/C. What is the magnitude of the electric field?
• A pendulum, comprising a light string of length L and a small sphere, swings in the vertical plane.
The string hits a peg located a distance d below the point of suspension (Fig. P8.67). (a) Show that if
the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg. (b) Show that if the pendulum is released from rest at the horizontal position (θ=90∘)θ=90∘)) and is to swing in a complete circle centered on the peg, the minimum value of d must be 3L/5L/5.
• An experimenter wishes to generate in air a sound wave that has a displacement amplitude of 5.50×10−6m.5.50×10−6m. The pressure amplitude is to be limited to 0.840 PaPa . What is the minimum wavelength the sound wave can have?
• Consider an infinite number of identical particles, each with charge q,q, placed along the xx axis at distances aa 2a,3a,4a,…2a,3a,4a,… from the origin. What is the electric field at the origin due to this distribution? Suggestion: Use
1+122+132+142+…=π261+122+132+142+…=π26
• A glider of mass mm is free to slide along a horizontal air track. It is pushed against a launcher at one
end of the track. Model the launcher as a light spring of force constant kk compressed by a distance xx . The glider is released from rest. (a) Show that the glider attains a speed of v=x(k/m)1/2v=x(k/m)1/2 . (b) Show that the magnitude of the impulse imparted to the glider is given by the expression I=x(km)1/2.I=x(km)1/2. (c) Is more work done on a cart with a large or a small mass?
• Two small beads having charges q1q1 and q2q2 of the same sign are fixed at the opposite ends of a horizontal insulating rod of length dd . The bead with charge q1q1 is at the origin. As shown in Figure P23.11P23.11 , a third small, charged bead is free to slide on the rod. (a) At what position xx is the third bead in equilibrium? (b) Can the equilibrium be stable?
• The wave function for an electron in the 2pp state of hydrogen is
ψ2p=1√3(2a0)3/2ra0e−r/2a0ψ2p=13–√(2a0)3/2ra0e−r/2a0
What is the most likely distance from the nucleus to find an electron in the 2pp state?
• How much work is done by the Moon’s gravitational field on a 1 000-kg meteor as it comes in from outer space and impacts on the Moon’s surface?
• From the scattering of sunlight, J. Thomson calculated the classical radius of the electron as having the value 2.82×10−15m2.82×10−15m . Sunlight with an intensity of 500 W/m2W/m2 falls on a disk with this radius. Assume light is a classical wave and the light striking the disk is completely absorbed. (a) Calculate the time interval required to accumulate 1.00 eV of energy. (b) Explain how your result for part (a) compares with the observation that photoelectrons are emitted promptly (within 10−9s).10−9s).
• A quantum simple harmonic oscillator consists of an electron bound by a restoring force proportional to its position relative to a certain equilibrium point. The proportionality constant is 8.99 N/m. What is the longest wavelength of light that can excite the oscillator?
• For the circuit shown in Figure P28.22P28.22 , calculate (a) the current in the 2.00−Ω2.00−Ω resistor and (b)(b) the potential difference between points aa and b.b.
• Two single-turn circular loops of wire have radii RR and r,r, with R>>rR>>r . The loops lie in the same plane and are concentric. (a) Show that the mutual inductance of the pair is approximately M=μ0πr2/2R.M=μ0πr2/2R. (b) Evaluate MM for r=r= 2.00 cmcm and R=20.0cm.R=20.0cm.
• On October 21, 2001, Ian Ashpole of the United Kingdom achieved a record altitude of 3.35 km (11 000 ft)
powered by 600 toy balloons filled with helium. Each filled balloon had a radius of about 0.50 m and an estimated mass of 0.30 kg. (a) Estimate the total buoyant force on the 600 balloons. (b) Estimate the net upward force on all 600 balloons. (c) Ashpole parachuted to the Earth after the balloons began to burst at the high altitude and the buoyant force decreased. Why did the balloons burst?
• A man claims that he can hold onto a 12.0 -kg child in a head-on collision as long as he has his seat belt on. Consider this man in a collision in which he is in one of two identical cars each traveling toward the other at 60.0 mi/hmi/h relative to the ground. The car in which he rides is brought
to rest in 0.10 s. (a) Find the magnitude of the average force needed to hold onto the child. (b) Based on your result to part (a), is the man’s claim valid? (c) What does the answer to this problem say about laws requiring the use of proper safety devices such as seat belts and special toddler seats?
• The electromagnetic power radiated by a nonrelativistic particle with charge qq moving with acceleration aa is
P=q2a26πϵ0c3P=q2a26πϵ0c3
where ϵ0ϵ0 is the permittivity of free space (also called the permittivity of vacuum) and cc is the speed of light in vacuum. (a) Show that the right side of this equation has units of watts. An electron is placed in a constant electric field of magnitude 100 N/CN/C . Determine (b) the acceleration of the magnitude 100 N/CN/C . Determine (b) the acceleration of the electron and (c)(c) the electromagnetic power radiated by this electron. (d) What If? If a proton is placed in a cyclotron with a radius of 0.500 mm and a magnetic field of magnitude 0.350T,0.350T, what electromagnetic power does this proton radi- ate just before leaving the cyclotron?
• A portion of Nichrome wire of radius 2.50 mmmm is to be used in winding a heating coil. If the coil must draw a current of 9.25 AA when a voltage of 120 VV is applied across its ends, find (a) the required resistance of the coil and (b) the length of wire you must use to wind the coil.
• Show that the energy–momentum relationship in Equation 39.27,E2=p2c2+(mc2)2,39.27,E2=p2c2+(mc2)2, follows from the expressions E=γmc2E=γmc2 and p=γmu.p=γmu.
• In Niels Bohr’s 1913 model of the hydrogen atom, the single electron is in a circular orbit of radius 5.29×10−11m5.29×10−11m and its speed is 2.19×106m/s2.19×106m/s . ( a) What is the magnitude of the magnetic moment due to the electron’s motion? (b) If the electron moves in a horizontal circle, counterclockwise as seen from above, what is the direction of this magnetic moment vector?
• A bead slides without friction around a loop-the-loop (Fig. P8.5). The bead is released from rest at a height
h=3.50Rh=3.50R (a) What is its speed at point @@ ? (b) How large is the normal force on the bead at point @@ if its mass is 5.00 g?
• A small particle of mass m is pulled to the top of a frictionless half-cylinder (of radius R) by a light cord that passes over the top of the cylinder as illustrated in Figure P7.29. (a) Assuming the particle moves at a constant speed, show that F=mgcosθ. Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times. (b) By directly integrating W=∫→F⋅d→r, find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder.
• An 8.40-kg object slides down a fixed, frictionless, inclined plane. Use a computer to determine and tabulate (a) the normal force exerted on the object and (b) its acceleration for a series of incline angles (measured from the horizontal) ranging from 0° to 90° in 5° increments. (c) Plot a graph of the normal force and the acceleration as functions of the incline angle. (d) In the limiting cases of 0° and 90°, are your results consistent with the known behavior?
• Find the maximum fractional energy loss for a 0.511 -MeV gamma ray that is Compton scattered from (a) a free electron and (b) a free proton.
• An inductor (L=400mH),(L=400mH), a capacitor (C=4.43μF)(C=4.43μF) and a resistor (R=500Ω)(R=500Ω) are connected in series. A 50.0−Hz50.0−Hz AC source produces a peak current of 250 mAmA in the circuit. (a) Calculate the required peak voltage ΔVmaxΔVmax (b) Determine the phase angle by which the current leads or lags the applied voltage.
• After determining that the Sun has existed for hundreds of millions of years, but before the discovery of
nuclear physics, scientists could not explain why the Sun has continued to burn for such a long time interval. For example, if it were a coal fire, it would have burned up in about 3000 yr. Assume the Sun, whose mass is equal to 1.99×1030kg , originally consisted entirely of hydrogen and its total power output is 3.85×1026W . (a) Assuming the energy-generating mechanism of the Sun is the fusion of hydrogen into helium via the net reaction
4(11H)+2(e−)→42He+2ν+γ
calculate the energy (in joules) given off by this reaction. (b) Take the mass of one hydrogen atom to be equal to 1.67×10−27kg . Determine how many hydrogen atoms constitute the Sun. (c) If the total power output remains constant, after what time interval will all the hydrogen be converted into helium, making the Sun die? (d) How does your answer to part (c) compare with current estimates of the expected life of the Sun, which are 4 billion to 7 billion years?
• Assume the Earth’s atmosphere has a uniform temperature of 20.0∘0∘C and uniform composition, with an effective molar mass of 28.9 g/molg/mol . (a) Show that the number density of molecules depends on height yy above sea level according to
nV(y)=n0e−m0gγ/kBTnV(y)=n0e−m0gγ/kBT Jetliners typically cruise at an altitude of 11.0km.11.0km. Find the ratio of the atmospheric density there to the density at sea level.
• In an assembly operation illustrated in Figure P 3.42, a robot moves an object first straight upward and then also to the east, around an arc forming one-quarter of a circle of radius 4.80 cm that lies in an east–west vertical plane. The robot then moves the object upward and to the north, through one-quarter of a circle of radius 3.70 cm that lies in a north–south vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical.
• A diode is a device that allows current to be carried in only one direction (the direction indicated by the arrow head in its circuit symbol). Find the average power delivered to the diode circuit of Figure P33.41P33.41 in terms of ΔVrmsΔVrms and R.R.
• The circuit shown in Figure P28.21 is connected for 2.00 min. (a) Determine the current in each branch of
the circuit. (b) Find the energy delivered by each battery. (c) Find the energy delivered to each resistor. (d) Identify the type of energy storage transformation that occurs in the operation of the circuit. (e) Find the total amount of energy transformed into internal energy in the resistors.
• Use Lenz’s law to answer the following questions concerning the direction of induced currents. Express your answers in terms of the letter labels a and b in each part of Figure P31.20. (a) What is the direction of the induced current in the resistor R in Figure P31.20 a when the bar magnet is moved to the left? (b) What is the direction of the current induced in the resistor R immediately after the switch S in Figure P31.20b is closed? (C) What is the direction of the induced current in the resistor R when the current I in Figure P31.20c decreases rapidly to zero?
• A uniform plank of length 2.00 mm and mass 30.0 kgkg is supported by three ropes as indicated by the blue vectors in Figure P12.25.P12.25. Find the tension in each rope when a 700−N700−N person is d=0.500md=0.500m from the left end.
• Water is forced out of a fire extinguisher by air pressure as shown in Figure $\mathrm{P} 14.59 .$ How much gauge air pressure in the tank is required for the water jet to have a speed of 30.0 $\mathrm{m} / \mathrm{s}$ when the water level is 0.500 $\mathrm{m}$ below the nozzle?
• A parallel-plate capacitor in air has a plate separation of ff 1.50 cmcm and a plate area of 25.0cm2.25.0cm2. The plates are charged to a potential difference of 250cm2.250cm2. The plates are charged the source. The capacitor is then immersed in distilled water. Assume the liquid is an insulator. Determine (a) the charge on the plates before and after immersion, (b) the capacitance and potential difference after immersion, and (c) the change in energy of the capacitor.
• For the purpose of measuring the electric resistance of shoes through the body of the wearer standing on a metal ground plate, the American National Standards Institute (ANSI) specifies the circuit shown in Figure P28.18P28.18 . The potential difference ΔVΔV across the 1.00−MΩ1.00−MΩ resistor is measured with an ideal voltmeter. (a) Show that the resistance of the footwear is Rshoes=50.0V−ΔVΔVRshoes=50.0V−ΔVΔV (b) In a medical test, a current through the human body should not exceed 150μAμA . Can the current delivered by the ANSI-specified circuit exceed 150μAμA ? To decide, consider a person standing barefoot on the ground plate.
• A hiker stands on an isolated mountain peak near sunset and observes a rainbow caused by water droplets in the air at a distance of 8.00 km along her line of sight to the most intense light from the rainbow. The valley is 2.00 km below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker?
• Two blocks of masses m1m1 and m2m2 are placed on a table in contact with each other as discussed in Example 5.7 and shown in Active Figure 5.12a5.12a. The coefficient of kinetic friction between the block of mass m1m1 and the table is μ1μ1, and that between the block of mass m2m2 and the table is μ2μ2. A horizontal force of magnitude FF is applied to the block of mass m1.m1. We wish to find P,P, the magnitude of the contact force between the blocks. (a) Draw diagrams showing the forces for each block. (b) What is the net force on the system of two blocks? (c) What is the net force acting on m1m1 ? (d) What is the net force acting on m2?m2? (e) Write Newton’s second law in the xx direction for each block. (f) Solve the two equations in two unknowns for the acceleration aa of the blocks in terms of the masses, the applied force FF, the coefficients of friction, and g.(g)g.(g) Find the magnitude PP of the contact force between the blocks in terms of the same quantities.
• A marble rolls back and forth across a shoebox at a constant speed of 0.8 m/s . Make an order-of-magnitude estimate of the probability of it escaping through the wall of the box by quantum tunneling. State the quantities you take as data and the values you measure or estimate for
• Refer to Section 39.4 . Prove that the Doppler shift in wavelength of electromagnetic waves is described by
λ′=λ1+v/c1−v/c−−−−−−−√λ′=λ1+v/c1−v/cwhere λ′λ′ is the wavelength measured by an observer moving at speed vv away from a source radiating waves of wave-length λ.λ.
• Note: In Problems 9 through 12, calculate numerical answers to three significant figures as usual.
Find the scalar product of the vectors in Figure P7.10.
• A 0.200-A current is charging a capacitor that has circular plates 10.0 cm in radius. If the plate separation is 4.00 mm, (a) what is the time rate of increase of electric field between the plates? (b) What is the magnetic field between the plates 5.00 cm from the center?
• Coherent microwaves of wavelength 5.00 cm enter a tall, narrow window in a building otherwise essentially opaque to the microwaves. If the window is 36.0 cm wide, what is the distance from the central maximum to the first-order minimum along a wall 6.50 m from the window?
• A car is parked on a steep incline, making an angle of 37.0∘0∘ below the horizontal and overlooking the ocean, when its brakes fail and it begins to roll. Starting from rest at t=0,t=0, the car rolls down the incline with a constant acceleration of 4.00 m/s2m/s2 , traveling 50.0 mm to the edge of a vertical cliff. The cliff is 30.0 mm above the ocean. Find (a) the speed of the car when it reaches the edge of the cliff, (b) the time interval elapsed when it arrives there, (c) the velocity of the car when it lands in the ocean, (d) the total time interval the car is in motion, and (e) the position of the car when it lands in the ocean, relative to the base of the cliff.
• A grating with 250 grooves/mm is used with an incandescent light source. Assume the visible spectrum to range in wavelength from 400 nm to 700 nm. In how many orders can one see (a) the entire visible spectrum and (b) the short-wavelength region of the visible spectrum?
• Three particles with equal positive charges qq are at the corners of an equilateral triangle of side aa as shown in Figure P25.26P25.26 . (a) At what point, if any, in the plane of the particles is the electric potential zero? (b) What is the electric potential at the position of one of the particles due to the other two particles in the triangle?
• The Bay of Fundy, Nova Scotia, has the highest tides in the world. Assume in mid ocean and at the mouth of the bay the Moon’s gravity gradient and the Earth’s rotation make the water surface oscillate with an amplitude of a few centimeters and a period of 12 h 24 min. At the head of the bay, the amplitude is several meters. Assume the bay has a length of 210 km and a uniform depth of 36.1 m. The speed of long-wavelength water waves is given by v=√gd,v=gd−−√, where dd is the water’s depth. Argue for or against the proposition that the tide is magnified by standing-wave resonance.
• An uncharged capacitor and a resistor are connected in series to a source of emf. If ε=9.00V,C=20.0μFε=9.00V,C=20.0μF , and R=100Ω,R=100Ω, find (a)(a) the time constant of the circuit, (b) the maximum charge on the capacitor, and (c)(c) the charge on the capacitor at a time equal to one time constant after the battery is connected.
• The energy flux carried by neutrinos from the Sun is estimated to be on the order of 0.400 W/m2W/m2 at the Earth’s surface. Estimate the fractional mass loss of the Sun over 109109 yr due to the emission of neutrinos. The mass of the Sun is 1.989×1030kg1.989×1030kg . The Earth- Sun distance is equal to 1.496×1011m.1.496×1011m.
• How many atoms of helium gas fill a spherical balloon of diameter 30.0 cmcm at 20.0∘0∘C and 1.00 atm2atm2 (b) What is the average kinetic energy of the helium atoms? (c) What is the rms speed of the helium atoms?
• The cosmic rays of highest energy are mostly protons, accelerated by unknown sources. Their spectrum shows a cutoff at an energy on the order of 10201020 eV. Above that energy, a proton interacts with a photon of cosmic microwave background radiation to produce mesons, for example, according to p+γ→p+π0p+γ→p+π0 . Demonstrate this fact by taking the following steps. (a) Find the minimum photon energy required to produce this reaction in the reference frame where the total momentum of the photon-proton system is zero. The reaction was observed experimentally in the 1950 s with photons of a few hundred MeV. (b) Use Wien’s displacement law to find the wavelength of a photon at the peak of the blackbody spectrum of the primordial microwave background radiation, with a temperature of 2.73 KK . (c) Find the energy of this photon. (d) Consider the reaction in part (a) in a moving reference frame so that the
photon is the same as that in part (c). Calculate the energy of the proton in this frame, which represents the Earth reference frame.
• A roller coaster at the Six Flags Great America amusement park in Gurnee, Illinois, incorporates some clever design technology and some basic physics. Each vertical loop, instead of being circular, is shaped like a teardrop (Fig. P6.19). The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure the cars remain on the track. The biggest loop is 40.0 m high. Suppose the speed at the top of the loop is 13.0 m/s and the corresponding centripetal acceleration of the riders is 2g. (a) What is the radius of the arc of the tear- drop at the top? (b) If the total mass of a car plus the riders is M, what force does the rail exert on the car at the top? (c) Suppose the roller coaster had a circular loop of radius 20.0 m. If the cars have the same speed, 13.0 m/s at the top, what is the centripetal acceleration of the riders at the top? (d) Comment on the normal force at the top in the situation described in part (c) and on the advantages of having teardrop-shaped loops.
• A tennis ball is a hollow sphere with a thin wall. It is set rolling without slipping at 4.03 m/s on a horizontal section of a track as shown in Figure P10.60. It rolls around the inside of a vertical circular loop of radius r=45.0cm.r=45.0cm. As the ball nears the bottom of the loop, the shape of the track deviates from a perfect circle so that the ball leaves the track at a point h=20.0cmh=20.0cm below the horizontal section. (a) Find the ball’s speed at the top of the loop. (b) Demonstrate that the ball will not fall from the track at the top of the loop. (c) Find the ball’s speed as it leaves the track at the bottom. What If? (d) Suppose that static friction between ball and track were negligible so that the ball slid instead of rolling. Would its speed then be higher, lower, or the same at the top of the loop? (e) Explain your answer to part (d).
• Two circular loops are parallel, coaxial, and almost in contact, with their centers 1.00 mm apart (Fig. P30.60). Each loop is 10.0 cmcm in radius. The top loop carries a clockwise current of I=140AI=140A . The bottom loop carries a counterclockwise current of I=140AI=140A . (a) Calculate the magnetic force exerted by the bottom loop on the top loop. (b) Suppose a student thinks the first step in solving part (a) is to use Equation 30.7 to find the magnetic field created by one of the loops. How would you argue for or against this idea? (c) The upper loop has a mass of 0.021 0 kg. Calculate its acceleration, assuming the only forces acting on it are the force in part (a) and the gravitational force.
• A pulsed ruby laser emits light at 694.3 nmnm . For a 14.0 -ps pulse containing 3.00 JJ of energy, find (a) the physical length of the pulse as it travels through space and (b) the
number of photons in it. (c) The beam has a circular cross section of diameter 0.600cm.0.600cm. Find the number of photons per cubic millimeter.
• Two small silver spheres, each with a mass of 10.0 gg , are separated by 1.00 mm . Calculate the fraction of the electrons in one sphere that must be transferred to the other to produce an attractive force of 1.00×104N1.00×104N (about 1 ton) between the spheres. The number of electrons per atom of silver is 47.47.
• Write the expression for yy as a function of xx and tt in SI units for a sinusoidal wave traveling along a rope in the negative xx direction with the following characteristics; A=A= 8.00cm,λ=80.0cm,f=3.00Hz8.00cm,λ=80.0cm,f=3.00Hz , and y(0,t)=0y(0,t)=0 at t=t= 0.0. (b) What If? Write the expression for yy as a function of xx and tt for the wave in part (a) assuming y(x,0)=0y(x,0)=0 at the point x=10.0cm.x=10.0cm.
• Assume a hydrogen atom is a sphere with diameter 0.100 nm and a hydrogen molecule consists of two such spheres in contact. (a) What fraction of the space in a tank of hydrogen gas at 0°C and 1.00 atm is occupied by the hydrogen molecules themselves? (b) What fraction of the space within one hydrogen atom is occupied by its nucleus, of radius 1.20 fm?
• A 200-kg object and a 500-kg object are separated by 4.00 m. (a) Find the net gravitational force exerted by these objects on a 50.0-kg object placed midway between them. (b) At what position (other than an infinitely remote one) can the 50.0-kg object be placed so as to experience a net force of zero from the other two objects?
• A 10.0 -kg block of metal measuring 12.0 $\mathrm{cm}$ by 10.0 $\mathrm{cm}$ by 10.0 $\mathrm{cm}$ is suspended from a scale and immersed in water as shown in Figure $\mathrm{P} 14.24 \mathrm{b}$ . The 12.0 -cm dimension is vertical, and the top of the block is 5.00 $\mathrm{cm}$ below the surface of the water. (a) What are the magnitudes of the forces acting on the top and on the bottom of the block due to the surrounding water? (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block.
• At high noon, the Sun delivers 1000 WW to each square meter of a blacktop road. If the hot asphalt transfers energy only by radiation, what is its steady-state temperature?
• A uniform circular disk of mass m=24.0gm=24.0g and radius r=r= 40.0 cmcm hangs vertically from a fixed, frictionless, horizontal hinge at a point on its circumference as shown in Figure
35aP34.35a . A beam of electromagnetic radiation with intensity 10.0 MW/m2MW/m2 is incident on the disk in a direction perpendicular to its surface. The disk is perfectly absorbing, and the resulting radiation pressure makes the disk rotate. Assuming the radiation is always perpendicular to the surface of the disk, find the angle θθ through which the disk rotates from the vertical as it reaches its new equilibrium position shown in Figure 34.35 bb.
• An electron is in the nn th Bohr orbit of the hydrogen atom. (a) Show that the period of the electron is T=n3t0T=n3t0 and determine the numerical value of t0.t0. (b) On average, an electron remains in the n=2n=2 orbit for approximately 10μμ s before it jumps down to the n=1n=1 (ground-state) orbit. How many revolutions does the electron make in the excited state? (c) Define the period of one revolution as an electron year, analogous to an Earth year being the period of the Earth’s motion around the Sun. Explain whether we should think of the electron in the n=2n=2 orbit as “living for a long time.”
• A parcel of air moving in a straight tube with a constant acceleration of $-4.00 \mathrm{m} / \mathrm{s}^{2}$ has a velocity of 13.0 $\mathrm{m} / \mathrm{s}$ at $10 : 05 : 00$ a.m. (a) What is its velocity at $10 : 05 : 01$ a.m. (b) At $10 : 05 : 04$ a.m.? (c) At $10 : 04 : 59$ a.m. (d) Describe the shape of a graph of velocity versus time for this parcel of air. (e) Argue for or against the following statement: “Knowing the single value of an object’s constant acceleration is like knowing a whole list of values for its velocity.”
• Consider a cube of gold 1.00 mm on an edge. Calculate the approximate number of conduction electrons in this cube whose energies lie in the range 4.000 to 4.025 eV.
• Two identical blocks resting on a frictionless, horizontal surface are connected by a light spring having a spring constant k=100N/mk=100N/m and an unstretched length Li=0.400mLi=0.400m as shown in Figure P23.67aP23.67a . A charge QQ is slowly placed on each block, causing the spring to stretch to an equilibrium length L=0.500mL=0.500m as shown in Figure P23.67bP23.67b . Determine the value of Q,Q, modeling the blocks as charged particles.
• A piston in a gasoline engine is in simple harmonic motion. The engine is running at the rate of 3600 rev/min. Taking the extremes of its position relative to its center point as ±5.00cm,±5.00cm, find the magnitudes of the (a) maximum velocity and (b) maximum acceleration of the piston.
• A wad of sticky clay of mass mm is hurled horizontally at a wooden block of mass MM initially at rest on a horizontal surface. The clay sticks to the block. After impact, the block slides a distance dd before coming to rest. If the coefficient of friction between the block and the surface is μμ what was the speed of the clay immediately before impact?
• A particle is subject to a force Fx that varies with position as shown in Figure P7.15 . Find the work done by the force on the particle as it moves (a) from x=0 to x=5.00m, (b) from x=5.00m to x=10.0m, and (c) from x=10.0m to x=15.0m. (d) What is the total work done by the force over the distance x=0 to x=15.0m ?
• A sound wave from a police siren has an intensity of 100.0 W/m2W/m2 at a certain point; a second sound wave from a nearby ambulance has an intensity level that is 10 dbdb greater than the police siren’s sound wave at the same point. What is the sound level of the sound wave due to the ambulance?
• Before 1960, people believed that the maximum attainable coefficient of static friction for an automobile tire on a roadway was μs=1.μs=1. Around 1962 , three companies independently developed racing tires with coefficients of 1.6.1.6. This problem shows that tires have improved further since then. The shortest time interval in which a piston-engine car initially at rest has covered a distance of one-quarter mile is about 4.43 s. (a) Assume the car’s rear wheels lift the front wheels off the pavement as shown in Figure P5.42. What minimum value of μμ , is necessary to achieve the record time? (b) Suppose the driver were able to increase his or her engine power, keeping other things equal. How would this change affect the elapsed time?
• Figure $\mathrm{P} 36.65$ shows a piece of glass with index of refraction $n=1.50$ surrounded by air. The ends are hemi- spheres with radii $R_{1}=2.00 \mathrm{cm}$ and $R_{2}=4.00 \mathrm{cm},$ and the centers of the hemispherical ends are separated by a distance of $d=8.00 \mathrm{cm} .$ A point object is in air, a distance $p=1.00 \mathrm{cm}$ from the left end of the glass. (a) Locate the image of the object due to refraction at the two spherical surfaces. (b) Is the final image real or virtual?
• Strong magnetic fields are used in such medical procedures as magnetic resonance imaging, or MRI. A technician wearing a brass bracelet enclosing area 0.00500 m2 places her hand in a solenoid whose magnetic field is 5.00 T directed perpendicular to the plane of the bracelet. The electrical resistance around the bracelet’s circumference is 0.02000Ω . An unexpected power failure causes the field to drop to 1.50 T in a time interval of 20.0 ms . Find (a) the current induced in the bracelet and (b) the power delivered to the bracelet. Note: As this problem implies, you should not wear any metal objects when working in regions of strong magnetic fields.
• For the system of four capacitors shown in Figure P26.19P26.19 , find (a) the equivalent capacitance of the system, (b) the charge on each capacitor, and (c) the potential difference across each
• Two astronauts (Fig. P 11.55), each having a mass MM, are connected by a rope of length dd having negligible mass. They are isolated in space, orbiting their center of mass at speeds vv. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum of the two-astronaut system and (b) the rotational energy of the system. By pulling on the rope, one of the astronauts shortens the distance between them to d/2.d/2. (c) What is the new angular momentum of the system? (d) What are the astronauts’ new speeds? (e) What is the new rotational energy of the system? (f) How much chemical potential energy in the body of the astronaut was converted to mechanical energy in the system when he shortened the rope?
• Two infinite, nonconducting sheets of charge are parallel to each other as shown in Figure P 24.54. The sheet on the left has a uniform surface charge density σ,σ, and the one on the right has a uniform charge density −σ−σ . Calculate the electric field at points (a) to the left of, (b)(b) in between, and (c)(c) to the right of the two sheets. (d) What If? Find the electric fields in all three regions if both sheets have positive uniform surface charge densities of value σσ .
• Damping is negligible for a 0.150 -kg object hanging from a light, 6.30−N/m6.30−N/m spring. A sinusoidal force with an amplitude of 1.70 NN drives the system. At what frequency
will the force make the object vibrate with an amplitude of 0.440 mm ?
• Show that the variation of atmospheric pressure with altitude is given by $P=P_{0} e^{-\alpha y},$ where $\alpha=\rho_{0} g / \rho P_{0}, P_{0}$ is atmospheric pressure at some reference level $y=0,$ and $\rho_{0}$ is the atmospheric density at this level. Assume the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform over the infinitesimal change) can be expressed from Equation 14.4 as $d P=-\rho g d y .$ Also assume the density of air is proportional to the pressure, which, as we will see in Chapter $20,$ is equivalent to assuming the temperature of the air is the same at all altitudes.
• From Gauss’s law, the electric field set up by a uniform line of charge is
E→=(λ2πϵ0r)r^E→=(λ2πϵ0r)r^
where r^r^ is a unit vector pointing radially away from the line and λλ is the linear charge density along the line. Derive an expression for the potential difference between r=r1r=r1 and r=r2r=r2
• An LCLC circuit like the one in Figure CQ32.8CQ32.8 contains an 82.0−mH82.0−mH inductor and a 17.0−μF17.0−μF capacitor that initially carries a 180−μC180−μC charge. The switch i open for t<0t<0 and is then thrown closed at t=0.t=0. (a) Find the frequency (in hertz) of the resulting oscillations. At t=1.00mst=1.00ms , find (b) the charge on the capacitor and (c)(c) the current in the circuit.
• A uniformly charged insulating rod of length 14.0 cm is bent into the shape of a semicircle as shown in Figure P23.35. The rod has a total charge of −7.50μC−7.50μC . Find (a)(a) the magnitude and (b)(b) the direction of the electric field at OO , the center of the semicircle.
• A bat, moving at 5.00 m/s, is chasing a flying insect. If the bat emits a 40.0-kHz chirp and receives back an echo at 40.4 kHz, (a) what is the speed of the insect? (b) Will the bat be able to catch the insect? Explain.
• At what frequency does the inductive reactance of a 57.0−μH57.0−μH inductor equal the capacitive reactance of a 57.0−μF57.0−μF capacitor?
• A pirate has buried his treasure on an island with five trees located at the points (30.0 m, 20.0 m), (60.0 m, 80.0 m), (10.0 m, 10.0 m), (40.0 m, 30.0 m), and (70.0 m, 60.0 m), all measured relative to some origin, as shown in Figure P 3.64. His ship’s log instructs you to start at tree AA and move toward tree B,B, but to cover only one-half the distance between AA and BB . Then move toward tree C,C, covering one-third the distance between your current location and C.C. Next move toward tree D,D, covering one-fourth the distance between where you are and D.D. Finally move toward tree E,E, covering one-fifth the distance between you and E,E, stop, and dig. (a) Assume you have correctly determined the order in which the pirate labeled the trees as A,B,C,D,A,B,C,D, and EE as shown in the figure. What are the coordinates of the point where his treasure is buried? (b) What If? What if you do not really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order of the trees, for instance, to BB (30 m, 20 m), AA (60 m, 80 m), EE (10 m, 10 m), CC (40 m, 30 m), and DD (70 m, 60 m)? State reasoning to show that the answer does not depend on the order in which the trees
are labeled.
• Massive stars ending their lives in supernova explosions produce the nuclei of all the atoms in the bottom half of the periodic table by fusion of smaller nuclei. This problem roughly models that process. A particle of mass mm moving along the xx axis with a velocity component
+u+u collides head-on and sticks to a particle of mass m/3m/3 moving along the xx axis with the velocity component −u−u . (a) What is the mass MM of the resulting particle? (b) Evaluate the expression from part (a) in the limit u→0u→0 . (c) Explain whether the result agrees with what you should expect from nonrelativistic physics.
• Regarding the Earth and a cloud layer 800 mm above the Earth as the “plates” of a capacitor, calculate the capacitance of the Earth-cloud layer system. Assume the cloud layer has an area of 1.00 km2km2 and the air between the cloud and the ground is pure and dry. Assume charge builds up on the cloud and on the ground until a uniform electric field of 3.00×106N/C3.00×106N/C throughout the space between them makes the air break down and conduct electricity as a
lightning bolt. (b) What is the maximum charge the cloud can hold?
• How much work is required to assemble eight identical charged particles, each of magnitude q,q, at the corners of a cube of side ss ?
• In the potassium iodide (KI) molecule, assume the K and I atoms bond ionically by the transfer of one electron from K to I. (a) The ionization energy of K is 4.34 eV, and the electron affinity of I is 3.06 eV. What energy is needed to transfer an electron from KK to I, to form K+K+ and I−I− ions from neutral atoms? This quantity is sometimes called the activation energy EaEa (b) A model potential energy function for the KI molecule is the Lennard-Jones potential:
U(r)=4ϵ[(σr)12−(σr)6]+EaU(r)=4ϵ[(σr)12−(σr)6]+Ea
where rr is the inter nuclear separation distance and ϵϵ and σσ are adjustable parameters. The EaEa term is added to ensure the correct asymptotic behavior at large rr . At the equilibrium separation distance, r=r0=0.305nm,U(r)r=r0=0.305nm,U(r) is a minimum, and dU/dr=0.dU/dr=0. In addition, U(r0)U(r0) is the negative of the dissociation energy: U(r0)=−3.37U(r0)=−3.37 eV. Find σσ and ϵϵ (c) Calculate the force needed to break up a KI molecule. (d) Calculate the force constant for small oscillations about r=r0.r=r0. Suggestion: Set r=r0+s,r=r0+s, where s/r0<<1s/r0<<1 , and expand U(r)U(r) in powers of s/r0s/r0 up to second-order terms.
• Find the direction of the magnetic field acting on a positively charged particle moving in the various situations.
• Assume dark matter exists throughout space with a uniform density of 6.00×10−28kg/m3.6.00×10−28kg/m3. (a) Find the amount of such dark matter inside a sphere centered on the Sun, having the Earth’s orbit as its equator. (b) Explain whether the gravitational field of this dark matter would
have a measurable effect on the Earth’s revolution.
• What A cylinder contains a mixture of helium and argon gas in equilibrium at 150∘C150∘C (a) What is the average kinetic energy for each type of gas molecule? (b) What is the rms speed of each type of molecule?
• A volumetric flask made of Pyrex is calibrated at 20.0∘20.0∘C. It is filled to the 100−mL100−mL mark with 35.0∘C35.0∘C acetone. After the flask is filled, the acetone cools and the flask warms so that the combination of acetone and flask reaches a uniform temperature of 32.0∘C.32.0∘C. The combination is then cooled back to 20.0∘C20.0∘C . (a) What is the volume of the acetone when it cools to 20.0∘C20.0∘C ? (b) At the temperature of 32.0∘C,32.0∘C, does the level of acetone lie above or below the 100 -mL mark on the flask? Explain.
• A long string carries a wave; a 6.00 -m segment of the string contains four complete wavelengths and has a mass of 180 g. The string vibrates sinusoidally with a frequency of 50.0 HzHz and a peak-to-valley displacement of 15.0cm.15.0cm. (The “peak-to-valley” distance is the vertical distance from the farthest positive position to the farthest negative position. (a) Write the function that describes this wave traveling in the positive xx direction. (b) Determine the power being supplied to the string.
• Consider the mass spectrometer shown schematically in Active Figure 29.14. The magnitude of the electric field between the plates of the velocity selector is 2.50×103V/m2.50×103V/m , and the magnetic field in both the velocity selector and the deflection chamber has a magnitude of 0.0350 T. Calculate the radius of the path for a singly charged ion having a mass m=2.18×10−26kgm=2.18×10−26kg .
• A ruby laser delivers a 10.0-ns pulse of 1.00-MW average power. If the photons have a wavelength of 694.3 nm, how many are contained in the pulse?
• A wooden block of mass MM resting on a friction less, horizontal surface is attached to a rigid rod of length ℓℓ and of negligible mass (Fig. P 11.37). The rod is pivoted at the other end. A bullet of mass mm traveling parallel to the horizontal surface and perpendicular to the rod with speed vv hits the block and becomes embedded in it. (a) What is the angular momentum of the bullet–block system about a vertical axis through the pivot? (b) What fraction of the original kinetic energy of the bullet is converted into internal energy in the system during the collision?
• A cylindrical habitat in space 6.00 km in diameter and 30.0 km long has been proposed (by G. K. O’Neill, 1974). Such a habitat would have cities, land, and lakes on the inside surface and air and clouds in the center. They would all be held in place by rotation of the cylinder about its long axis. How fast would the cylinder have to rotate to imitate the Earth’s gravitational field at the walls of the cylinder?
• Why is the following situation impossible? A gamma-ray photon with energy 1.05 MeV strikes a stationary electron, causing the following reaction to occur:
γ−+e−→e−+e−+e+γ−+e−→e−+e−+e+
Assume all three final particles move with the same speed in the same direction after the reaction.
• Find the normalization constant A for a wave function made up of the two lowest states of a quantum particle in a box extending from x=0 to x=L :
ψ(x)=A[sin(πxL)+4sin(2πxL)]
(b) A particle is described in the space −a≤x≤a by the wave function
ψ(x)=Acos(πx2a)+Bsin(πxa)
Determine the relationship between the values of A and B required for normalization.
• An object of mass m1=m1= 4.00 kgkg is tied to an object of mass m2=3.00kgm2=3.00kg with String II of length ℓ=ℓ= 0.500m.0.500m. The combination is swung in a vertical cirring, String 2,2, of length ℓ=0.500m.ℓ=0.500m. During the motion, the two strings are collinear at all times as shown in Figure P6.44. At the top of its motion, m2m2 is traveling at v=4.00m/sv=4.00m/s . (a) What is the tension in String 1 at this instant? (b) What is the tension in String 2 at this instant? (c) Which string will break first if the combination is rotated faster and faster?
• A flowerpot is knocked off a balcony from a height dd above the sidewalk as shown in Figure P17.11. It falls toward an unsuspecting man of height hh who is standing below. Assume the man requires a time interval of ΔtΔt to respond to the warning. How close to the sidewalk can the flowerpot fall before it is too late for a warning shouted from the balcony to reach the man in time? Use the symbol vv for the speed of sound.
• An elevator moves downward in a tall building at a constant speed of 5.00 m/s. Exactly 5.00 s after the top of the elevator car passes a bolt loosely attached to the wall of the elevator shaft, the bolt falls from rest. (a) At what time does the bolt hit the top of the still-descending elevator? (b) In what way is this problem similar to Example 2.8? (c) Estimate the highest floor from which the bolt can fall if the elevator reaches the ground floor before the bolt hits the top of the elevator.
• What must be the contact area between a suction cup (completely evacuated) and a ceiling if the cup is to support the weight of an 80.0-kg student?
• To understand why plasma containment is necessary, consider the rate at which an unconfined plasma would be lost. (a) Estimate the rms speed of deuterons in a plasma at a temperature of 4.00×108K.4.00×108K. (b) What If? Estimate the order of magnitude of the time interval during which such a plasma would remain in a 10.0 -cm cube if no steps were taken to contain it.
• A 60.0 -kg person bends his knees and then jumps straight up. After his feet leave the floor, his motion is unaffected by air resistance and his center of mass rises by a maximum of 15.0 cm. Model the floor as completely solid and motionless. (a) Does the floor impart impulse to the person? (b) Does the floor do work on the person? (c) With what momentum does the person leave the floor? (d) Does it make sense to say that this momentum came from the floor? Explain. (e) With what kinetic energy does the person leave the floor? (f) Does it make sense to say that this energy came from the floor? Explain.
• A large hall in a museum has a niche in one wall. On the floor plan, the niche appears as a semicircular indentation of radius 2.50 $\mathrm{m}$ . A tourist stands on the centerline of the niche, 2.00 $\mathrm{m}$ out from its deepest point, and whispers “Hello.” Where is the sound concentrated after reflection from the niche?
• Devise a table similar to that shown in Figure 42.18 for atoms containing 11 through 19 electrons. Use Hund’s rule and educated guesswork.
• A disk with moment of inertia I1I1 rotates about a friction less, vertical axle with angular speed ωi.ωi. A second disk, this one having moment of inertia I2I2 and initially not rotating, drops onto the first disk (Fig. P 11.30). Because of friction between the surfaces, the two eventually reach the same angular speed ωfωf (a) Calculate ωfωf . (b) Calculate the ratio of the final to the initial rotational energy.
• The wave function for a wave on a taut string is
y(x,t)=0.350sin(10πt−3πx+π4)y(x,t)=0.350sin⁡(10πt−3πx+π4)
where xx and yy are in meters and tt is in seconds. If the linear mass density of the string is 75.0g/m,75.0g/m, (a) what is the average rate at which energy is transmitted along the string? (b) What is the energy contained in each cycle of the wave?
• Two strings are vibrating at the same frequency of 150 Hz. After the tension in one of the strings is decreased, an observer hears four beats each second when the strings vibrate together. Find the new frequency in the adjusted string.
• A police car traveling at 95.0 km/hkm/h is traveling west, chasing a motorist traveling at 80.0 km/hkm/h . ( a) What is the velocity of the motorist relative to the police car? (b) What is the velocity of the police car relative to the motorist? (c) If they are originally 250 mm apart, in what time interval will the police car overtake the motorist?
• A stepladder of negligible weight is constructed as shown in Figure P12.56P12.56 , with AC=AC= BC=ℓ.BC=ℓ. A painter of mass mm stands on the ladder a distance dd from the bottom. Assuming the floor is frictionless, find (a) the tension in the horizontal bar DEDE connecting the two halves of the ladder, (b) the normal forces at AA and BB , and (c)(c) the components of the reaction force at the single hinge CC that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also treat each half of the ladder separately.
• A puck of mass m1m1 is tied to a string and allowed to revolve in a circle of radius RR on a frictionless, horizontal table. The other end of table. The other end of the string passes through a small hole in the cen-object of mass m2m2 is tied to it (Fig. P 6.54).6.54). The suspended object remains in equilibrium while the puck on the tabletop revolves. Find symbolic expressions for (a) the tension in the string, (b) the radial force acting on the puck, and (c) the speed of the puck. (d) Qualitatively describe what will happen in the motion of the puck if the value of m2m2 is increased by placing a small additional load on the puck. (e) Qualitatively describe what will happen in the motion of the puck if the value of m2m2 instead decreased by removing a part of the hanging load.
• The magnification of the image formed by a refracting surface is given by
M=−n1qn2pM=−n1qn2p
where $n_{1}, n_{2}, p,$ and $q$ are defined as they are for Figure 36.17 and Equation 36.8 . A paperweight is made of a solid glass hemisphere with index of refraction $1.50 .$ The radius of the circular cross section is $4.00 \mathrm{cm} .$ The hemisphere is placed on its flat surface, with the center directly over a 2.50 -mm-long line drawn on a sheet of paper. What is the length of this line as seen by someone looking vertically down on the hemisphere?
• Starting with Equation 43.17, show that the ionic cohesive energy of an ionically bonded solid is given by Equation 43.18.
• The 14 CC isotope undergoes beta decay according to the process given by Equation 44.21.44.21. Find the QQ value for this process.
• The rest energy of an electron is 0.511 MeV. The rest energy of a proton is 938 MeV. Assume both particles have kinetic energies of 2.00 MeV. Find the speed of (a) the electron and (b) the proton. (c) By what factor does the speed of the electron exceed that of the proton? (d) Repeat the calculations in parts (a) through (c) assuming both particles have kinetic energies of 2000 MeV.
• If An ideal gas with specific heat ratio γγ confined to a cylinder is put through a closed cycle. Initially, the gas is at Pi,Vi,Pi,Vi, and TiTi First, its pressure is tripled under constant volume. It then expands adiabatically to its original pres- sure and finally is compressed isobarically to its original volume. (a) Draw a PVdiagram of this cycle. (b) Determine the volume at the end of the adiabatic expansion. Find
(c) the temperature of the gas at the start of the adiabatic expansion and (d) the temperature at the end of the cycle. (e) What was the net work done on the gas for this cycle?
• Suppose a flutist plays a 523-Hz C note with first harmonic displacement amplitude A1=100nm.A1=100nm. From Figure 18.19 bb read, by proportion, the displacement amplitudes of harmonics 2 through 7 . Take these as the values A2A2 through A7A7 in the Fourier analysis of the sound and assume B1=B1= B2=⋯=B7=0.B2=⋯=B7=0. Construct a graph of the waveform of the sound. Your waveform will not look exactly like the flute waveform in Figure 18.18 b because you simplify by ignoring cosine terms; nevertheless, it produces the same sensation to human hearing.
• Why is the following situation impossible? A piece of transparent material having an index of refraction n=1.50n=1.50 is cut into the shape of a wedge as shown in Figure P37.60.P37.60. Both the top and bottom surfaces of the wedge are in contact with air. Monochromatic light of wavelength λ=632.8nmλ=632.8nm is normally incident from above, and the wedge is viewed from above. Let h=1.00mmh=1.00mm represent the height of the wedge and ℓ=0.500mℓ=0.500m its length. A thin-film interference pattern appears in the wedge due to reflection from the top and bottom surfaces. You have been given the task of counting the number of bright fringes that appear in the entire length ℓℓ of the wedge. You find this task tedious, and your concentration is broken by a noisy distraction after accurately counting 5000 bright fringes.
• A solid insulating sphere of radius RR has a nonuniform charge density that varies with rr according to the expression ρ=Ar2,ρ=Ar2, where AA is a constant and r<Rr<R is measured from the center of the sphere. (a) Show that the magnitude of the electric field outside (r>R)(r>R) the sphere is E⃗=AR5/5ϵ0r2.E→=AR5/5ϵ0r2. (b) Show that the magnitude of the electric field inside (r<R)(r<R) the sphere is E=Ar3/5ϵ0.E=Ar3/5ϵ0. Note: The volume element dVdV for a spherical shell of radius rr and thickness drdr is equal to 4πr2dr.4πr2dr.
• The 3pp level of sodium has an energy of −3.0eV,−3.0eV, and the 3dd level has an energy of −1.5−1.5 eV. (a) Determine ZZ eff or each of these states. (b) Explain the difference.
• In a location where the speed of sound is 343 m/s, a 2 000-Hz sound wave impinges on two slits 30.0 cm apart. (a) At what angle is the first maximum of sound intensity located? (b) What If? If the sound wave is replaced by 3.00-cm microwaves, what slit separation gives the same angle for the first maximum of microwave intensity?(c) What If? If the slit separation is 1.00 mm, what frequency of light gives the same angle to the first maximum of light intensity?
• A string with a mass m=8.00gm=8.00g and a length L=5.00mL=5.00m has one end attached to a wall; the other end is draped over a small, fixed pulley a distance d=4.00md=4.00m from the wall and attached to a hanging object with a mass M=4.00kgM=4.00kg as in Figure P18.21.P18.21. If the horizontal part of the string is plucked, what is the fundamental frequency of its vibration?
• Assuming T=300K,T=300K, (a) for what value of the bias voltage ΔVΔV in Equation 43.27 does I=9.00I0?I=9.00I0? (b) What If? What if I=−0.900I0?I=−0.900I0?
• An inductor in the form of a solenoid contains 420 turns and is 16.0 cm in length. A uniform rate of decrease of current through the inductor of 0.421 A/sA/s induces an emf of 175μVμV . What is the radius of the solenoid?
• A certain orthodontist uses a wire brace to align a patient’s crooked tooth as in Figure P5.4. The tension in the wire is adjusted to have a magnitude of 18.0 N . Find the magnitude of the net force exerted by the wire on the crooked tooth.
• Figure P35.58 shows a top view of a square enclosure. The inner surfaces are plane mirrors. A ray of light enters a small hole in the center of one mirror. (a) At what angle θθ must the ray enter if it exits through the hole after being reflected once by each of the other three mirrors? (b) What If? Are there other values of θθ for which the ray can exit after multiple reflections? If so, sketch one of the ray’s paths.
• Current Versus Potential Difference ΔVcdΔVcd Measured in a Bulk Ceramic Sample of YBa2Cu3O7−δYBa2Cu3O7−δ at Room Temperature
• The dielectric material between the plates of a parallelplate capacitor always has some nonzero conductivity σσ . Let AA represent the area of each plate and dd the distance between them. Let κκ represent the dielectric constant of the material. (a) Show that the resistance RR and the capacitance CC of the capacitor are related by
RC=κϵ0σRC=κϵ0σ
(b) Find the resistance between the plates of a 14.0−nF14.0−nF capacitor with a fused quartz dielectric.
• A ball dropped from a height of 4.00 m makes an elastic collision with the ground. Assuming no mechanical energy is lost due to air resistance, (a) show that the ensuing motion is periodic and (b) determine the period of the motion. (c) Is the motion simple harmonic? Explain.
• What is the maximum current in a 2.20−μF2.20−μF capacitor when it is connected across (a) a North American electrical outlet having ΔVrms=120VΔVrms=120V and f=60.0Hzf=60.0Hz and (b)(b) a European electrical outlet having ΔVrms=240VΔVrms=240V and f=50.0Hz2f=50.0Hz2
• An object of mass m=1.00kgm=1.00kg is observed to have an acceleration →aa→ with a magnitude of 10.0 m/s2m/s2 in a direction 60.0∘0∘ east of north. Figure P5.27P5.27 shows
a view of the object from above. The force →F2F→2 acting on the object
has a magnitude of 5.00 NN and is directed north. Determine the magnitude and direction of the one other horizontal force →F1F→1 acting on the object.
• A packed bundle of 100 long, straight, insulated wires forms a cylinder of radius R=0.500cm.R=0.500cm. If each wire carries 2.00A,2.00A, what are (a)(a) the magnitude and (b)(b) the direction of the magnetic force per unit length acting on a wire located 0.200 cm from the center of the bundle? (c) What If? Would a wire on the outer edge of the bundle experience a force greater or smaller than the value calculated in parts (a) and (b)? Give a qualitative argument for your answer.
• When gamma rays are incident on matter, the intensity of the gamma rays passing through the material varies with depth xx as I(x)=I0e−μx,I(x)=I0e−μx, where I0I0 is the intensity of the radiation at the surface of the material (at x=0)x=0) and μμ is the linear absorption coefficient. For low-energy gamma rays in steel, take the absorption coefficient to be 0.720 mm−1mm−1 . (a) Determine the “half-thickness” for steel, that is, the thickness of steel that would absorb half the incident gamma rays. (b) In a steel mill, the thickness of sheet steel passing into a roller is measured by monitoring the intensity of gamma radiation reaching a detector below the rapidly moving metal from a small source immediately above the metal. If the thickness of the sheet changes from 0.800 mmmm to 0.700mm,0.700mm, by what percentage does the gamma-ray intensity change?
• Consider the two nuclear reactions
A+B→C+EC+D→F+G
(a) Show that the net disintegration energy for these two reactions (Qnet=Q1+QI) is identical to the disintegration energy for the net reaction
A+B+D→E+F+G
(b) One chain of reactions in the proton-proton cycle in the Sun’s core is
11H+11H→21H+01e+ν01e+0−1e→2γ
11H+21H→32He+γ
11H+32He→42He+01e+ν
01e+0−1e→2γ
Based on part (a), what is Qnet for this sequence?
• Lasers have been used to suspend spherical glass beads in the Earth’s gravitational field. ( a) A black bead has a radius of 0.500 mmmm and a density of 0.200g/cm3.0.200g/cm3. Determine the e the radiation intensity needed to support the bead. (b) What is the minimum power required for this laser?
• A very slow neutron (with speed approximately equal to zero) can initiate the reaction
10n+105B→73Li+42He10n+105B→73Li+42He
The alpha particle moves away with speed 9.25×106m/s9.25×106m/s . Calculate the kinetic energy of the lithium nucleus. Use nonrelativistic equations.
• Two identical beads each have a mass mm and charge q.q. When placed in a hemispherical bowl of radius RR with frictionless, nonconducting walls, the beads move, and at equilibrium, they are a distance dd apart (Fig. P23.72). (a) Determine the charge q on each bead. (b) Determine the charge required for dd to become equal to 2RR .
• In the AC circuit shown in Figure P33.5, R=70.0ΩR=70.0Ω and the output voltage of the AC source is ΔVmaxΔVmax sin ωt.ωt. (a) If ΔVR=0.250ΔVmaxΔVR=0.250ΔVmax for the first time at t=0.0100s,t=0.0100s, what is the angular frequency of the source? (b) What is the next value of tt for which ΔVR=0.250ΔVmax?ΔVR=0.250ΔVmax?
• In the circuit shown in Figure P33.72, assume all parameters except C are given. Find (a) the current in the circuit as a function of time and (b) the power delivered to the circuit. (c) Find the current as a function of time after only switch 1 is opened. (d) After switch 1 is and voltage are in phase. Find the capacitance C . Find (e) the impedance of the circuit when both switches are open, (f) the maximum energy stored in the capacitor during oscillations, and (g) the maximum energy stored in the inductor during oscillations. (h) Now the frequency of the voltage source is doubled. Find the phase difference between the current and the voltage. (i) Find the frequency that makes the inductive reactance one-half the capacitive reactance.
• Suppose seawater exerts an average frictional drag force of 1.00×105N1.00×105N on a nuclear-powered ship. The fuel consists of enriched uranium containing 3.40%% of the fissionable isotope $\underset{92}{235} \mathrm{U},and the ship’s reactor has an efficiency of 20.0%. Assuming 200 MeV is released per fission event, how far can the ship travel per kilogram of fuel? • A scanning tunneling microscope (STM) can precisely determine the depths of surface features because the current through its tip is very sensitive to differences in the width of the gap between the tip and the sample surface. Assume the electron wave function falls off exponentially in this direction with a decay length of 0.100 nm , that is, with C=10.0nm−1. Determine the ratio of the current when the STM tip is 0.500 nm above a surface feature to the current when the tip is 0.515 nm above the surface. • Around the core of a nuclear reactor shielded by a large pool of water, Cerenkov radiation appears as a blue glow. (See Fig. Pl7.38 on page 507.)507.) Cerenkov radiation than the speed of light in that medium. It is the electromagnetic equivalent of a bow wave or a sonic boom. An electron is traveling through water at a speed 10.0%% faster than the speed of light in water. Determine the electron’s (a) total energy, (b) kinetic energy, and (c) momentum. (d) Find the angle between the shock wave and the electron’s direction of motion. • Why is the following situation impossible? A thin brass ring has an inner diameter 10.00 cmcm at 20.0∘20.0∘C. A solid aluminum cylinder has diameter 10.02 cmcm at 20.0∘C20.0∘C . Assume the average coefficients of linear expansion of the two metals are constant. Both metals are cooled together to a temperature at which the ring can be slipped over the end of the cylinder. • A battery has an emf of 15.0 V. The terminal voltage of the battery is 11.6 V when it is delivering 20.0 W of power to an external load resistor R. (a) What is the value of R ? (b) What is the internal resistance of the battery? • A ray of light is incident on a flat surface of a block of crown glass that is surrounded by water. The angle of refraction is 19.6∘.19.6∘. Find the angle of reflection. • A ball of mass m=0.275kgm=0.275kg swings in a vertical circular path on a string L=0.850mL=0.850m long as in Figure P6.45P6.45 (a) What are the forces acting on the ball at any point on the path? (b) Draw force diagrams for the ball when it is at the bottom of the circle and when it is at the top. (c) If its speed is 5.20 m/sm/s at the top of the circle, what is the tension in the string there? (d) If the string breaks when its tension exceeds 22.5N,22.5N, what is the maximum speed the ball can have at the bottom before that happens? • A thin rod of length ℓℓ and uniform charge per unit length λλ lies along the xx axis as shown in Figure P23.37P23.37 . (a) Show that the electric field at P,P, a distance dd from the rod along its perpendicular bisector, has no xx component and is given by E=2keλsinθ0/d.E=2keλsin⁡θ0/d. (b) What If? Using your result to part (a), show that the field of a rod of infinite length is E=E= 2keλ/dkeλ/d . • A guitar’s steel string vibrates (see Fig. 31.5). The component of magnetic field perpendicular to the area of a pickup coil nearby is given by B=50.0+3.20sin1046πt where B is in milliteslas and t is in seconds. The circular pickup coil has 30 turns and radius 2.70mm. Find the emf induced in the coil as a function of time. • A parallel-plate capacitor of plate separation d is charged to a potential difference ΔV0.ΔV0. A dielectric slab of thickness dd and dielectric constant κκ is introduced between the plates while the battery remains connected to the plates. (a) Show that the ratio of energy stored after the dielectric is introduced to the energy stored in the empty capacitor is U/U0=κU/U0=κ (b) Give a physical explanation for this increase in stored energy. (c) What happens to the charge on the capacitor? Note: This situation is not the same as in Example 26.5,26.5, in which the battery was removed from the circuit before the dielectric was introduced. • A pendulum with a length of 1.00 mm is released from an initial angle of 15.0∘.15.0∘. After 1000 ss , its amplitude has been reduced by friction to 5.50∘.5.50∘. What is the value of b/2mb/2m ? • A spherical aluminum ball of mass 1.26$\mathrm{kg}$contains an empty spherical cavity that is concentric with the ball. The ball barely floats in water. Calculate (a) the outer radius of the ball and (b) the radius of the cavity. • In Figure P 38.71, suppose the transmission axes of the left and right polarizing disks are perpendicular to each other. Also, let the center disk be rotated on the common axis with an angular speed ωω . Show that if unpolarized light is incident on the left disk with an intensity Imax,Imax, the intensity of the beam emerging from the right disk is I=116Imax(1−cos4ωt)I=116Imax(1−cos⁡4ωt) This result means that the intensity of the emerging beam is modulated at a rate four times the rate of rotation of the center disk. Suggestion: Use the trigonometric identities cos2θ=12(1+cos2θ)cos2⁡θ=12(1+cos⁡2θ) and sin2θ=12(1−cos2θ)sin2⁡θ=12(1−cos⁡2θ). • A block having mass mm and charge +Q+Q is connected to an insulating spring having a force constant k.k. The block lies on a frictionless, insulating, horizontal track, and the system is immersed in a uniform electric field of magnitude EE directed as shown in Figure P25.8P25.8 (page 734 . The block is released from rest when the spring is unstretched (at x=0).x=0). We wish to show that the ensuing motion of the block is simple harmonic. (a) Consider the system of the block, the spring, and the electric field. Is this system isolated or nonisolated? (b) What kinds of potential energy exist within this system? (c) Call the initial configuration of the system that existing just as the block is released from rest. The final configuration is when the block momentarily comes to rest again. What is the value of xx when the block comes to rest momentarily? (d) At some value of xx we will call x=x0x=x0 , the block has zero net force on it. What analysis model describes the particle in this situation? (e) What is the value of x0?x0? (f) Define a new coordinate system x′x′ such that x′=x−x0x′=x−x0 . Show that x′x′ satisfies a differential equation for simple harmonic motion. (g) Find the period of the simple harmonic motion. (h) How does the period depend on the electric field magnitude? • Two parallel wires separated by 4.00 cm repel each other with a force per unit length of 2.00×10−4N/m2.00×10−4N/m . The current in one wire is 5.00 AA . ( a) Find the current in the other wire. (b) Are the currents in the same direction or in opposite directions? (c) What would happen if the direction of one current were reversed and doubled? • An accelerating voltage of 2.50×103V2.50×103V is applied to an electron gun, producing a beam of electrons originally traveling horizontally north in vacuum toward the center of a viewing screen 35.0 cm away. What are (a) the magnitude and (b) the direction of the deflection on the screen caused by the Earth’s gravitational field? What are (c) the magnitude and (d) the direction of the deflection on the screen caused by the vertical component of the Earth’s magnetic field, taken as 20.0μTμT down? (e) Does an electron in this vertical magnetic field move as a projectile, with constant vector acceleration perpendicular to a constant northward component of velocity? (f) Is it a good approximation to assume it has this projectile motion? Explain. • A wire carrying a current I is bent into the shape of an exponential spiral, r=eθ,r=eθ, from θ=0θ=0 to θ=2πθ=2π as sug- gested in Figure P30.73P30.73 . To complete a loop, the ends of the spiral are connected by a straight wire along the xx axis. (a) The angle ββ between a radial line and its tangent line at any point on a curve r=f(θ)r=f(θ) is related to the function by tanβ=rdr/dθtan⁡β=rdr/dθ Use this fact to show that β=π/4.β=π/4. (b) Find the magnetic field at the origin. • A 200 -g block is pressed against a spring of force constant 1.40 kN/mkN/m until the block compresses the spring 10.0cm.10.0cm. The spring rests at the bottom of a ramp inclined at 60.0∘0∘ to the horizontal. Using energy considerations, determine how far up the incline the block moves from its initial position before it stops (a) if the ramp exerts no friction force on the block and (b) if the coefficient of kinetic friction is 0.400 . • The alpha-emitter plutonium- 238 (23894Pu atomic mass 238.049560u, half-life 87.7 yr) was used in a nuclear energy source on the Apollo Lunar Surface Experiments Package (Fig. P45.49). The energy source, called the Radioisotope Thermoelectric Generator, is the small gray object to the left of the gold-shrouded Central Station in the photograph. Assume the source contains 3.80 kg of ^{238\mathrm{Pu} and the efficiency for conversion of radioactive decay energy to energy transferred by electrical transmission is 3.20% . Determine the initial power output of the source. • The work done by an engine equals one-fourth the energy it absorbs from a reservoir. (a) What is its thermal efficiency? (b) What fraction of the energy absorbed is expelled to the cold reservoir? • Convert the following temperatures to their values on the Fahrenheit and Kelvin scales: (a) the sublimation point of dry ice, −78.5∘C;−78.5∘C; (b) human body temperature, 37.0∘37.0∘C. • In a Young’s double-slit experiment, two parallel slits with a slit separation of 0.100 mm are illuminated by light of wavelength 589 nm, and the interference pattern is observed on a screen located 4.00 m from the slits. (a) What is the difference in path lengths from each of the slits to the location of the center of a third-order bright fringe on the screen? (b) What is the difference in path lengths from the two slits to the location of the center of the third dark fringe away from the center of the pattern? • According to its design specification, the timer circuit delaying the closing of an elevator door is to have a capacitance of 32.0μFμF between two points AA and B.B. When one circuit is being constructed, the inexpensive but durable capacitor installed between these two points is found to have capacitance 34.8μFμF . To meet the specification, one additional capacitor can be placed between the two points. (a) Should it be in series or in parallel with the 34.8−μF34.8−μF . capacitor? (b) What should be its capacitance? (c) What If? The next circuit comes down the assembly line with capacitance 29.8μFμF between AA and BB . To meet the specification, what additional capacitor should be installed in series or in parallel in that circuit? • The position of a particle is given by the expression x=x= 4.00cos(3.00πt+π),4.00cos(3.00πt+π), where xx is in meters and tt is in seconds. Determine (a) the frequency and (b) period of the motion, (c) the amplitude of the motion, (d) the phase constant, and (e) the position of the particle at t=0.250st=0.250s . • A solid cube of wood of side 2aa and mass MM is resting on a horizontal surface. The cube is constrained to rotate about a fixed axis AB (Fig. P 11.62). A bullet of mass mm and speed vv is shot at the face opposite ABCDABCD at a height of 4a/3.4a/3. The bullet becomes embedded in the cube. Find the minimum value of vv required to tip the cube so that it falls on face ABCD.ABCD. Assume m<<M.m<<M. • Model the tungsten filament of a lightbulb as a black body at temperature 2 900 K. (a) Determine the wavelength of light it emits most strongly. (b) Explain why the answer to part (a) suggests that more energy from the lightbulb goes into infrared radiation than into visible light. • Assume the intensity of solar radiation incident on the upper atmosphere of the Earth is 1370 W/m2W/m2 and use data from Table 13.2 as necessary. Determine (a) the intensity of solar radiation incident on Mars, (b) the total power incident on Mars, and (c) the radiation force that acts on that planet if it absorbs nearly all the light. (d) State how this force compares with the gravitational attraction exerted by the Sun on Mars. (e) Compare the ratio of the gravitational force to the light-pressure force exerted on the Earth and the ratio of these forces exerted on Mars, found in part (d). • Spiderman, whose mass is 80.0kg,80.0kg, is dangling on the free end of a 12.0 -m-long rope, the other end of which is fixed to a tree limb above. By repeatedly bending at the waist, he is able to get the rope in motion, eventually getting it to swing enough that he can reach a ledge when the rope makes a 60.0∘ angle with the vertical. How much work was done by the gravitational force on Spiderman in this maneuver? • Astronomers detect a distant meteoroid moving along a straight line that, if extended, would pass at a distance 3RERE from the center of the Earth, where RERE is the Earth’s radius. What minimum speed must the meteoroid have if it is not to collide with the Earth? • Calculate the inductance of an LCLC circuit that oscillates at 120 HzHz when the capacitance is 8.00μFμF . • A motor in normal operation carries a direct current of 0.850 A when connected to a 120−V power supply. The resistance of the motor windings is 11.8Ω . While in normal operation, (a) what is the back emf generated by the motor? (b) At what rate is internal energy produced in the windings? (c) What If? Suppose a malfunction stops the motor shaft from rotating. At what rate will internal energy be produced in the windings in this case? (Most motors have a thermal switch that will turn off the motor to prevent overheating when this stalling occurs.) • An ideal gas is taken through a Carnot cycle. The isothermal expansion occurs at 250∘C250∘C , and the isothermal compression takes place at 50.0∘0∘C . The gas takes in 1.20×103J1.20×103J , of energy from the hot reservoir during the isothermal expansion. Find (a) the energy expelled to the cold reservoir in each cycle and (b) the net work done by the gas in each cycle. • A 2.00-m-long wire having a mass of 0.100 kg is fixed at both ends. The tension in the wire is maintained at 20.0 N. (a) What are the frequencies of the first three allowed modes of vibration? (b) If a node is observed at a point 0.400 m from one end, in what mode and with what frequency is it vibrating? • A steam engine is operated in a cold climate where the exhaust temperature is 0∘C0∘C (a) Calculate the theoretical maximum efficiency of the engine using an intake steam temperature of 100∘C100∘C . (b) If, instead, superheated steam at 200∘C200∘C is used, find the maximum possible efficiency. • As an astronaut, you observe a small planet to be spherical. After landing on the planet, you set off, walking always straight ahead, and find yourself returning to your spacecraft from the opposite side after completing a lap of 25.0 km. You hold a hammer and a falcon feather at a height of 1.40 m, release them, and observe that they fall together to the surface in 29.2 s. Determine the mass of the planet. • Two objects are connected by a light string passing over a light, frictionless pulley as shown in Figure P8.7. The object of mass m1m1 is released from rest at height hh above the table. Using the isolated system model, (a) determine the speed of m2m2 just as m1m1 hits the table and (b) find the maxi- mum height above the table to which m2m2 rises. • High-power lasers in factories are used to cut through cloth and metal (Fig. P34.25). One such laser has a beam diameter of 1.00 mmmm and generates an electric field having an amplitude of 0.700 MV/m at the target. Find (a) the amplitude of the magnetic field produced, (b) the intensity of the laser, and (c) the power delivered by the laser. • 00μCμC placed at the center of curvature PP • A radioactive nucleus has half-life T1/2.T1/2. A sample containing these nuclei has initial activity R0R0 at t=0.t=0. Calculate the number of nuclei that decay during the interval between the later times t1t1 and t2t2 . • A slab of insulating material has a nonuniform positive charge density ρ=Cx2,ρ=Cx2, where xx is measured from the center of the slab as shown in Figure P 24.59 and CC is a constant. The slab is infinite in the yy and zz directions. Derive expressions for the electric field in (a) the exterior regions (|x|>d/2)(|x|>d/2) and (b)(b) the interior region of the slab (−d/2<x<(−d/2<x< d/2).d/2). • In 1962 , measurements of the magnetic field of a large tornado were made at the Geophysical Observatory in Tulsa, Oklahoma. If the magnitude of the tornado’s field was B=1.50×10−8B=1.50×10−8 T pointing north when the tornado was 9.00 kmkm east of the observatory, what current was carried up or down the funnel of the tornado? Model the vortex as a long, straight wire carrying a current. • Seawater contains 3.00 mg of uranium per cubic meter. (a) Given that the average ocean depth is about 4.00 km and water covers two-thirds of the Earth’s surface, estimate the amount of uranium dissolved in the ocean. (b) About 0.700% of naturally occurring uranium is the fissionable isotope 285 UU Estimate how long the uranium in the oceans could supply the world’s energy needs at the current usage of 1.50×1013J/s1.50×1013J/s . (c) Where does the dissolved uranium come from? (d) Is it a renewable energy source? • A rod of mass mm and radius RR rests on two parallel rails (Fig. P 29.37) that are a distance dd apart and have a length L.L. The rod carries a current II in the direction shown and rolls along the rails without slipping. A uniform magnetic field BB is directed perpendicular to the rod and the rails. If it starts from rest, what is the speed of the rod as it leaves the rails? • A square plate of copper with 50.0-cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicularly to the plate. Find (a) the charge density of each face of the plate and (b) the total charge on each face. • A block of mass 3.00 kgkg is pushed up against a wall by a force →PP→ that makes an angle of θ=50.0∘θ=50.0∘ with the horizontal as shown in Figure P5.48. The coefficient of static friction between the block and the wall is 0.250 . (a) Determine the possible values for the magnitude of →PP→ that allow the block to remain stationary. (b) Describe what happens if |P||P| has a larger value and what happens if it is smaller. (c) Repeat parts (a)(a) and (b),(b), assuming the force makes an angle of θ=13.0∘θ=13.0∘ with the horizontal. • A sinusoidal wave in a rope is described by the wave function y=0.150sin(0.800x−50.0t)y=0.150sin⁡(0.800x−50.0t) where xx and yy are in meters and tt is in seconds. The mass per length of the string is 12.0g/m.12.0g/m. (a) Find the maximum transverse acceleration of an element of this string. (b) Determine the maximum transverse force on a 1.00 cmcm segment of the string. (c) State how the force found in part (b) compares with the tension in the string. • A freshly prepared sample of a certain radioactive isotope has an activity of 10.0 mCimCi . After 4.00 hh , its activity is 8.00 mCimCi . Find (a)(a) the decay constant and (b)(b) the half-life. (c) How many atoms of the isotope were contained in the freshly prepared sample? (d) What is the sample’s activity 30.0 hh after it is prepared? • A vertical spring stretches 3.9 cm when a 10-g object is hung from it. The object is replaced with a block of mass 25 g that oscillates up and down in simple harmonic motion. Calculate the period of motion. • The quantity of charge qq (in coulombs) that has passed through a surface of area 2.00 cm2cm2 varies with time according to the equation q=4t3+5t+6,q=4t3+5t+6, where tt is in seconds. (a) What is the instantaneous current through the surface at t=1.00s2t=1.00s2 (b) What is the value of the current density? • An HCl molecule is excited to its second rotational energy level, corresponding to J=2.J=2. If the distance between its nuclei is 0.1275 nm, what is the angular speed of the molecule about its center of mass? • To find the total angular displacement during the playing time of the compact disc in part (B) of Example 10.2, the disc was modeled as a rigid object under constant angular acceleration. In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h.h. Show that the radius rr of a given portion of the track is given by r=ri+hθ2πr=ri+hθ2π where riri is the radius of the innermost portion of the track and θθ is the angle through which the disc turns to arrive at the location of the track of radius rr . (b) Show that the rate of change of the angle θθ is given by dθdt=vri+(hθ/2π)dθdt=vri+(hθ/2π) where vv is the constant speed with which the disc surface passes the laser. (c) From the result in part (b),(b), use integration to find an expression for the angle θθ as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time. • Ultrasound is used in medicine both for diagnostic imaging (Fig. P17.9) and for therapy. For diagnosis, short pulses of ultrasound are passed through the patient’s body. An echo reflected from a structure of interest is recorded, and the distance to the structure can be determined from the time delay for the echo’s return. To reveal detail, the wavelength of the reflected ultrasound must be small compared to the size of the object reflecting the wave. The speed of ultrasound in human tissue is about 1 500 m/s (nearly the same as the speed of sound in water). (a) What is the wavelength of ultrasound with a frequency of 2.40 MHz? (b) In the whole set of imaging techniques, frequencies in the range 1.00 MHz to 20.0 MHz are used. What is the range of wavelengths corresponding to this range of frequencies? • The intensity on the screen at a certain point in a double-slit interference pattern is 64.0%% of the maximum value. (a) What minimum phase difference (in radians) between sources produces this result? (b) Express this phase difference as a path difference for 486.1 -nm light. • Four capacitors are connected as shown in Figure P26.23. (a) Find the equivalent capacitance between points aa and b.b. (b) Calculate the charge on each capacitor, taking ΔVab=15.0V.ΔVab=15.0V. . • A 1.00 -L insulated bottle is full of tea at 90.0∘0∘C . You pour out one cup of tea and immediately screw the stopper back on the bottle. Make an order-of-magnitude estimate of the change in temperature of the tea remaining in the bottle that results from the admission of air at room temperature. State the quantities you take as data and the values you measure or estimate for them. • A wire having a linear mass density of 1.00 g/cm is placed on a horizontal surface that has a coefficient of kinetic friction of 0.200. The wire carries a current of 1.50 A toward the east and slides horizontally to the north at constant velocity. What are (a) the magnitude and (b) the direction of the smallest magnetic field that enables the wire to move in this fashion? • The sound intensity at a distance of 16 mm from a noisy generator is measured to be 0.25W/m2.0.25W/m2. What is the sound intensity at a distance of 28 mm from the generator? • In a game of American football, a quarterback takes the ball from the line of scrimmage, runs backward a distance of 10.0 yards, and then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a forward pass downfield 50.0 yards perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement? • A 0.100-A current is charging a capacitor that has square plates 5.00 cm on each side. The plate separation is 4.00 mm. Find (a) the time rate of change of electric flux between the plates and (b) the displacement current between the plates. • Model air as a diatomic ideal gas with M=28.9g/molM=28.9g/mol . A cylinder with a piston contains 1.20 kgkg of air at 25.0∘0∘C and 2.00×105Pa2.00×105Pa . Energy is transferred by heat into the system as it is permitted to expand, with the pressure rising to 4.00×1054.00×105 Pa. Throughout the expansion, the relationship between pressure and volume is given by P=CV1/2P=CV1/2 where CC is a constant. Find (a) the initial volume, (b) the final volume, (c) the final temperature, (d) the work done on the air, and (e) the energy transferred by heat. • Transcranial magnetic stimulation (TMS) is a noninvasive technique used to stimulate regions of the human brain. In TMS, a small coil is placed on the scalp and a brief burst of current in the coil produces a rapidly changing magnetic field inside the brain. The induced emf can stimulate neuronal activity. (a) One such device generates an upward magnetic field within the brain that rises from zero to 1.50 T in 120 ms. Determine the induced emf around a horizontal circle of tissue of radius 1.60 mm. (b) What If? The field next changes to 0.500 T downward in 80.0 ms. How does the emf induced in this process compare with that in part (a)? • A wire having a uniform linear charge density λλ is bent into the shape shown in Figure P25.44P25.44 . Find the electric potential at point O.O. • We have seen that a long solenoid produces a uniform magnetic field directed along the axis of a cylindrical region. To produce a uniform magnetic field directed parallel to a diameter of a cylindrical region, however, one can use the saddle coils illustrated in Figure P30.70. The loops are wrapped over a long, somewhat flattened tube. Figure P30.70a shows one wrapping of wire around the tube. This wrapping is continued in this manner until the visible side has many long sections of wire carrying current to the left in Figure P30.70a and the back side has many lengths carrying current to the right. The end view of the tube in Figure P30.70b shows these wires and the currents they carry. By wrapping the wires carefully, the distribution of wires can take the shape suggested in the end view such that the overall current distribution is approximately the superposition of two overlapping, circular cylinders of radius R (shown by the dashed lines) with uniformly distributed current, one toward you and one away from you. The current density JJ is the same for each cylinder. The center of one cylinder is described by a position vector d→d→ relative to the center of the other cylinder. Prove that the magnetic field inside the hollow tube is μ0Jd/2μ0Jd/2 downward. Suggestion: The use of vector methods simplifies the calculation. • Four objects are situated along the yy axis as follows: a 2.00−kg2.00−kg object is at +3.00m,+3.00m, a 3.00−kg3.00−kg object is at +2.50m,+2.50m, a 2.50 -kg object is at the origin, and a 4.00−kg4.00−kg object is at −0.500m.−0.500m. Where is the center of mass of these objects? • Two lightbulbs have cylindrical filaments much greater in length than in diameter. The evacuated bulbs are identical except that one operates at a filament temperature of 2 100°C and the other operates at 2 000°C. (a) Find the ratio of the power emitted by the hotter lightbulb to that emitted by the cooler lightbulb. (b) With the bulbs operating at the same respective temperatures, the cooler lightbulb is to be altered by making its filament thicker so that it emits the same power as the hotter one. By what factor should the radius of this filament be increased? • When an aluminum bar is connected between a hot reservoir at 725 KK and a cold reservoir at 310K,2.50kJ310K,2.50kJ of energy is transferred by heat from the hot reservoir to the cold reservoir. In this irreversible process, calculate the change in entropy of (a) the hot reservoir, (b) the cold reservoir, and (c) the Universe, neglecting any change in entropy of the aluminum rod. • If A moving beltway at an airport has a speed v1v1 and a length L.L. A woman stands on the beltway as it moves from one end to the other, while a man in a hurry to reach his flight walks on the beltway with a speed of v2v2 relative to the moving beltway. (a) What time interval is required for the woman to travel the distance L? (b) What time interval is required for the man to travel this distance? (c) A second beltway is located next to the first one. It is identical to the first one but moves in the opposite direction at speed v1⋅v1⋅ Just as the man steps onto the beginning of the beltway and begins to walk at speed v2v2 relative to his beltway, a child steps on the other end of the adjacent beltway. The child stands at rest relative to this second beltway. How long after stepping on the beltway does the man pass the child? • The effective spring constant describing the potential energy of the HI molecule is 320 N/m and that for the HF molecule is 970 N/m. Calculate the minimum amplitude of vibration for (a) the HI molecule and (b) the HF molecule. • A wheel 2.00 m in diameter lies in a vertical plane and rotates about its central axis with a constant angular acceleration of 4.00 rad/s2rad/s2 . The wheel starts at rest at t=0,t=0, and the radius vector of a certain point PP on the rim makes an angle of 57.3∘3∘ with the horizontal at this time. At t=2.00st=2.00s , find (a) the angular speed of the wheel and, for point P,P, (b) the tangential speed, (c) the total acceleration, and (d) the angular position. • For a certain transverse wave, the distance between two successive crests is 1.20m,1.20m, and eight crests pass a given point along the direction of travel every 12.0 ss . Calculate the wave speed. • Figure P40.61P40.61 shows the stopping potential versus the incident photon frequency for the photoelectric effect for sodium. Use the graph to find (a) the work function of sodium, (b) the ratio h/e,h/e, and (c)(c) the cutoff wavelength. The data are taken from R. A. Millikan, Physical Review 7:362(1916).7:362(1916). • A force acting on a particle moving in the xy plane is given by →F=(2yˆi+x2ˆj), where →F is in newtons and x and y are in meters. The particle moves from the origin to a final position having coordinates x=5.00m and y=5.00m as shown in Figure P7.43 . Calculate the work done by →F on the particle as it moves along (a) the purple path, (b) the red path, and (c) the blue path. (d) Is →F conservative or nonconservative? (c) Explain your answer to part (d). • An alien spaceship traveling at 0.600cc toward the Earth launches a landing craft. The landing craft travels in the same direction with a speed of 0.800 c relative to the mother ship. As measured on the Earth, the spaceship is 0.200 ly from the Earth when the landing craft is launched. (a) What speed do the Earth-based observers measure for the approaching landing craft? (b) What is the distance to the Earth at the moment of the landing craft’s launch as measured by the aliens? (c) What travel time is required by for the landing craft to reach the Earth as measured by the aliens on the mother ship? (d) If the landing craft has a mass of 4.00×105kg4.00×105kg , what is its kinetic energy as measured in the Earth reference frame? • Consider the deuterium–tritium fusion reaction with the tritium nucleus at rest: 21H+31H→42He+10n21H+31H→42He+10n (a) Suppose the reactant nuclei will spontaneously fuse if their surfaces touch. From Equation 44.1, determine the required distance of closest approach between their centers. (b) What is the electric potential energy (in electron volts) at this distance? (c) Suppose the deuteron is fired straight at an originally stationary tritium nucleus with just enough energy to reach the required distance of closest approach. What is the common speed of the deuterium and tritium nuclei, in terms of the initial deuteron speed vi,vi, as they touch? (d) Use energy methods to find the minimum initial deuteron energy required to achieve fusion. (e) Why does the fusion reaction actually occur at much lower deuteron energies than the energy calculated in part (d)? • An electron moves in a circular path perpendicular to a constant magnetic field of magnitude 1.00 mT. The angular momentum of the electron about the center of the circle is 4.00 ×10−25kg⋅m2/s×10−25kg⋅m2/s . Determine (a) the radius of the circular path and (b) the speed of the electron. • Find the number of electrons and the number of each species of quarks in 1 LL of water. (b) Make an order-of- magnitude estimate of the number of each kind of fun- damental matter particle in your body. State your assump- tions and the quantities you take as data. • A sample of a diatomic ideal gas has pressure P and volume V. When the gas is warmed, its pressure triples and its volume doubles. This warming process includes two steps, the first at constant pressure and the second at constant volume. Determine the amount of energy transferred to the gas by heat. • In 1816, Robert Stirling, a Scottish clergyman, patented the Stirling engine, which has found a wide variety of applications ever since, including the solar power application discussed on the cover of this textbook. Fuel is burned externally to warm one of the engine’s two cylinders. A fixed quantity of inert gas moves cyclically between the cylinders, expanding in the hot one and contracting in the cold one. Figure P22.57 represents a model for its thermodynamic cycle. Consider n moles of an ideal monatomic gas being taken once through the cycle, consisting of two isothermal processes at temperatures 3TiTi and TiTi and two constant- volume processes. Let us find the efficiency of this engine. (a) Find the energy transferred by heat into the gas during the isovolumetric process AB. (b) Find the energy transferred by heat into the gas during the isothermal process BC. (c) Find the energy transferred by heat into the gas during the isovolumetric process CD. (d) Find the energy transferred by heat into the gas during the isothermal process DA. (e) Identify which of the results from parts (a) through (d) are positive and evaluate the energy input to the engine by heat. (f) From the first law of thermodynamics, find the work done by the engine. (g) From the results of parts (e) and (f), evaluate the efficiency of the engine. A Stirling engine is easier to manufacture than an internal combustion engine or a turbine. It can run on burning garbage. It can run on the energy transferred by sunlight and produce no material exhaust. Stirling engines are not currently used in automobiles due to long startup times and poor acceleration response. • Why is the following situation impossible? Two narrow slits are separated by 8.00 mm in a piece of metal. A beam of microwaves strikes the metal perpendicularly, passes through the two slits, and then proceeds toward a wall some distance away. You know that the wavelength of the radiation is 1.00 cm 65%, but you wish to measure it more precisely. Moving a microwave detector along the wall to study the interference pattern, you measure the position of the m 5 1 bright fringe, which leads to a successful measurement of the wavelength of the radiation. • The nucleus of an atom is on the order of 10−14m10−14m in diameter. For an electron to be confined to a nucleus, its de Broglie wavelength would have to be on this order of magnitude or smaller. (a) What would be the kinetic energy of an electron confined to this region? (b) Make an order of magnitude estimate of the electric potential energy of a system of an electron inside an atomic nucleus. (c) Would you expect to find an electron in a nucleus? Explain. • Water in an electric teakettle is boiling. The power absorbed by the water is 1.00 kW. Assuming the pressure of vapor in the kettle equals atmospheric pressure, determine the speed of effusion of vapor from the kettle’s spout if the spout has a cross-sectional area of 2.00 cm2cm2. Model the steam as an ideal gas. • A certain molecule has ff degrees of freedom. Show that an ideal gas consisting of such molecules has the following properties: (a) its total internal energy is fnRT/2fnRT/2 , (b) its molar specific heat at constant volume is fR/2,fR/2, (c) its molar specific heat at constant pressure is (f+2)R/2,(f+2)R/2, and (d) its specific heat ratio is γ=CP/CV=(f+2)/fγ=CP/CV=(f+2)/f • A concrete slab is 12.0 cmcm thick and has an area of 5.00 m2m2 . Electric heating coils are installed under the slab to melt the ice on the surface in the winter months. What minimum power must be supplied to the coils to maintain a temperature difference of 20.0°C between the bottom of the slab and its surface? Assume all the energy transferred is through the slab. • After a 0.800 -nm x-ray photon scatters from a free electron, the electron recoils at 1.40×106m/s1.40×106m/s . (a) What is the Compton shift in the photon’s wavelength? (b) Through what angle is the photon scattered? • Express in unit-vector notation the following vectors, each of which has magnitude 17.0 cm. (a) Vector →EE→ is directed 27.0∘0∘ counter clockwise from the positive xx axis. (b) Vector →FF→ is directed 27.0∘27.0∘ counterclockwise from the positive yy axis. (c) Vector →GG→ is directed 27.0∘27.0∘ clockwise from the negative yy axis. • 00 -nF capacitor with an initial charge of 5.10μCμC is dis- charged through a 1.30−kΩ1.30−kΩ resistor. (a) Calculate the current in the resistor 9.00μμ s after the resistor is connected across the terminals of the capacitor. (b) What charge remains on the capacitor after 8.00μsμs ? (c) What is the maximum current in the resistor? • The displacement vectors →AA→ and →BB→ shown in Figure P 3.11 both have magnitudes of 3.00m.3.00m. The direction of vector →AA→ is θ=30.0∘.θ=30.0∘. Find graphically (a) →A+→B,A→+B→, (b) →A−→B,A→−B→, (c) →B−→A,B→−A→, and (d) →A−2→BA→−2B→ (Report all angles counter clock-wise from the positive xx axis.) • Problems 69 through 72 apply ideas from this and earlier chapters to some properties of superconductors, which were introduced in Section 27.5. A novel method of storing energy has been proposed. A huge underground superconducting coil, 1.00 kmkm in diameter, would be fabricated. It would carry a maximum current of 50.0 kAkA through each winding of a 150−150− turn Nb3SnNb3Sn solenoid. (a) If the inductance of this huge coil were 50.0H,50.0H, what would be the total energy stored? (b) What would be the compressive force per unit length acting between two adjacent windings 0.250 mm apart? • Three equal positive charges qq are at the corners of an equilateral triangle of side aa as shown in Figure P23.42P23.42 . Assume the three charges together create an electric field. ( a) Sketch the field lines in the plane of the charges. (b) Find the where the electric field is zero. What are (c) the magnitude and (d) the direction of the electric field at PP due to the two charges at the base? • In 1990, Walter Arfeuille of Belgium lifted a 281.5 -kg object through a distance of 17.1 cm using only his teeth. (a) How much work was done on the object by Arfeuille in this lift, assuming the object was lifted at constant speed? (b) What total force was exerted on Arfeuille’s teeth during the lift? • Why is the following situation impossible? Starting from rest, a charging rhinoceros moves 50.0$\mathrm{m}$in a straight line in 10.0 s. Her acceleration is constant during the entire motion, and her final speed is 8.00$\mathrm{m} / \mathrm{s}$. • A positively charged disk has a uniform charge per unit area σσ as described in Example 23.8 . Sketch the electric field lines in a plane perpendicular to the plane of the disk passing through its center. • The tank in Figure$\mathrm{P} 14.13$is filled with water of depth$d=2.00 \mathrm{m} .$At the bottom of one sidewall is a rectangular hatch of height$h=1.00 \mathrm{m}$and width$w=2.00 \mathrm{m}$that is hinged at the top of the hatch. (a) Determine the magnitude of the force the water exerts on the hatch. (b) Find the magnitude of the torque exerted by the water about the hinges. • A firefighter, a distance d from a burning building, directs a stream of water from a fire hose at angle θi above the horizontal as shown in Figure P4.15. If the initial speed of the stream is vi at what height h does the water strike the building? • Let the polar coordinates of the point (x,y)(x,y) be (r,θ)(r,θ) Determine the polar coordinates for the points (a) (−x,y),(−x,y), (b) (−2x,−2y),(−2x,−2y), and (c)(3x,−3y)(c)(3x,−3y) . • An object of mass m1=5.00kg placed on a frictionless, horizontal table is connected to a string that passes over a pulley and then is fastened to a hanging object of mass m2=9.00kg as shown in Figure P. 28 . (a) Draw free-body diagrams of both objects. Find (b) the magnitude of the acceleration of the objects and (c) the tension in the string. • A transverse wave on a string is described by the wave function y=0.120sin(π8x+4πt)y=0.120sin⁡(π8x+4πt) where xx and yy are in meters and tt is in seconds. Determine (a) the transverse speed and (b) the transverse acceleration at t=0.200t=0.200 s for an element of the string located at x=x= 1.60m.1.60m. What are (c)(c) the wavelength, (d)(d) the period, and (c) the speed of propagation of this wave? • A taut string has a length of 2.60 m and is fixed at both ends. (a) Find the wavelength of the fundamental mode of vibration of the string. (b) Can you find the frequency of this mode? Explain why or why not. • Two small beads having positive charges q1=3qq1=3q and q2=qq2=q are fixed at the opposite ends of a horizontal insulating rod of length d=1.50m.d=1.50m. The bead with charge q1q1 is at the origin. As shown in Figure P23.11,P23.11, a third small, charged bead is free to slide on the rod. (a) At what position xx is the third bead in equilibrium? (b) Can the equilibrium be stable? • When a nucleus decays, it can leave the daughter nucleus in an excited state. The 9343 Te nucleus (molar mass 92.9102 g/mol) in the ground state decays by electron capture and e+ emission to energy levels of the daughter (molar mass 92.9068 g/mol in the ground state) at 2.44MeV,2.03MeV,1.48MeV, and 1.35 MeV . (a) Identify the daughter nuclide. (b) To which of the listed levels of the daughter are electron capture and e+ decay of 9343Tc allowed? • Figure P39.28 shows a jet of material (at the upper right) being ejected by galaxy M87M87 (at the lower left). Such jets are believed to be evidence of supermassive black holes at the center of a galaxy. Suppose two jets of material from the center of a galaxy are ejected in opposite directions. Both jets move at 0.750 crelative to the galaxy center. Determine the speed of one jet relative to the other. • Why is the following situation impossible? An RLC circuit is used in a radio to tune into a North American AM commercial radio station. The values of the circuit components are R=15.0Ω,L=2.80μH,R=15.0Ω,L=2.80μH, and C=0.910pF.C=0.910pF. • A black aluminum glider floats on a film of air above a level aluminum air track. Aluminum feels essentially no force in a magnetic field, and air resistance is negligible. A strong magnet is attached to the top of the glider, forming a total mass of 240 g. A piece of scrap iron attached to one end stop on the track attracts the magnet with a force of 0.823 N when the iron and the magnet are separated by 2.50 cm. (a) Find the acceleration of the glider at this instant. (b) The scrap iron is now attached to another green glider, forming total mass 120 g. Find the acceleration of each glider when the gliders are simultaneously released at 2.50-cm separation. • A quantum particle of mass m moves in a potential well of length 2L. Its potential energy is infinite for x<−L and for x>+L. In the region −L<x<L, its potential energy is given by U(x)=−ℏ2x2mL2(L2−x2) In addition, the particle is in a stationary state that is described by the wave function ψ(x)=A(1−x2/L2) for −L<x<+L and by ψ(x)=0 elsewhere. (a) Determine the energy of the particle in terms of ℏ,m, and L. (b) Determine the normalization constant A. (c) Determine the probability that the particle is located between x=−L/3 and x=+L/3. • A wooden block of volume$5.24 \times 10^{-4} \mathrm{m}^{3}$floats in water, and a small steel object of mass$m$is placed on top of the block. When$m=0.310 \mathrm{kg}$, the system is in equilibrium and the top of the wooden block is at the level of the water. (a) What is the density of the wood? (b) What happens to the block when the steel object is replaced by an object whose mass is less than 0.310$\mathrm{kg}$? (c) What happens to the block when the steel object is replaced by an object whose mass is greater than 0.310$\mathrm{kg}$? • Consider a freely moving quantum particle with mass mm and speed u.u. Its energy is E=K=12mu2.E=K=12mu2. (a) Determine the phase speed of the quantum wave representing the particle and (b)(b) show that it is different from the speed at which the particle transports mass and energy. • A uniform beam of mass mm is inclined at an angle θθ to the horizontal. Its upper end (point P)P) produces a 90∘90∘ bend in a very rough rope tied to a wall, and its lower end rests on a rough floor (Fig. Pl2.51). Let μsμs represent the coefficient of static friction between beam and floor. Assume μsμs is less than the cotangent of θ.θ. (a) Find an expression for the maximum mass MM that can be suspended from the top before the beam slips. Determine (b) the magnitude of the reaction force at the floor and (c) the magnitude of the force exerted by the beam on the rope at PP in terms of mm , M,M, and μsμs . • A one-dimensional harmonic oscillator wave function is ψ=Axe−bx2 (a) Show that ψ satisfies Equation 41.24. (b) Find b and the total energy E.(c) Is this wave function for the ground state or for the first excited state? • Eight charged particles, each of magnitude q,q, are located on the corners of a cube of edge ss as shown in Figure P23.77P23.77 . ( a) Determine the x,y,x,y, and zz components of the total force exerted by the other charges on the charge located at point A.A. What are (b) the magnitude and (c) the direction of this total force? • The compression ratio of an Otto cycle as shown in Active Figure 22.12 is VA/VB=8.00.VA/VB=8.00. At the beginning AA of the compression process, 500 cm3cm3 of gas is at 100 kPakPa and 20.0∘0∘C . At the beginning of the adiabatic expansion, the temperature is TC=750∘CTC=750∘C . Model the working fluid as an ideal gas with γ=1.40.γ=1.40. (a) Fill in this table to follow the states of the gas: TABLE CANT COPY (b) Fill in this table to follow the processes: TABLE CANT COPY (c) Identify the energy input |Qk|,|Qk|, (d) the energy exhaust |Q|,|Q|, and (e) the net output work WcngWcng (f) Calculate the thermal efficiency. (g) Find the number of crankshaft revolutions per minute required for a one-cylinder engine to have an output power of 1.00kW=1.34hp.1.00kW=1.34hp. Note: The thermodynamic cycle involves four piston strokes. • A thin rod of superconducting material 2.50 cm long is placed into a 0.540-T magnetic field with its cylindrical axis along the magnetic field lines. (a) Sketch the directions of the applied field and the induced surface current. (b) Find the magnitude of the surface current on the curved surface of the rod. • A large, flat, horizontal sheet of charge has a charge per unit area of 9.00μC/m2.9.00μC/m2. Find the electric field just above the middle of the sheet. • A transverse sinusoidal wave on a string has a period T=T= 25.0 msms and travels in the negative xx direction with a speed of 30.0 m/sm/s . At t=0,t=0, an element of the string at x=0x=0 has a transverse position of 2.00 cmcm and is traveling downward with a speed of 2.00 m/sm/s . (a) What is the amplitude of the wave? (b) What is the initial phase angle? (c) What is the maximum transverse speed of an element of the string? (d) Write the wave function for the wave. • A convex spherical mirror has a focal length of magnitude$8.00 \mathrm{cm} .$( a) What is the location of an object for which the magnitude of the image distance is one-third the magnitude of the object distance? (b) Find the magnification of the image and (c) state whether it is upright or inverted. • Why is the following situation impossible? At the Summer Olympics, an athlete runs at a constant speed down a straight track while a spectator near the edge of the track blows a note on a horn with a fixed frequency. When the athlete passes the horn, she hears the frequency of the horn fall by the musical interval called a minor third. That is, the frequency she hears drops to five-sixths its original value. • The peak of the graph of nuclear binding energy per nucleon occurs near 56Fe,whichiswhyironisprominentinthespectrumoftheSunandstars.Showthat56Fe,whichiswhyironisprominentinthespectrumoftheSunandstars.Showthat_{}^{56} \mathrm{Fe}has a higher binding energy per nucleon than its neighbors 55Mnand55Mnand_{}^{59} \mathrm{Co}. • In Active Figure 2.11b, the area under the velocity-versus-time graph and between the vertical axis and time$t$(vertical dashed line) represents the displacement. As shown, this area consists of a rectangle and a triangle. (a) Compute their areas. (b) Explain how the sum of the two areas compares with the expression on the right hand side of Equation 2.16 . • A by-product of some fission reactors is the isotope 239 Pu , an alpha emitter having a half-life of 24120 yr: 239Pu→23592U+α Consider a sample of 1.00 kg of pure 299 Pu at t=0. Calculate (a) the number of 239Pu nuclei present at t=0 and (b) the initial activity in the sample. (c) What If? For what time interval does the sample have to be stored if a “safe” activity level is 0.100 Bq ? • In Active Figures 18.20a and 18.20b, notice that the amplitude of the component wave for frequency ff is large, that for 3ff is smaller, and that for 5 f smaller still. How do we know exactly how much amplitude to assign to each frequency component to build a square wave? This problem helps us find the answer to that question. Let the square wave in Active Figure 18.20c have an amplitude AA and let t=0t=0 be at the extreme left of the figure. So, one period TT of the square wave is described by y(t)={A−A0<t<T2T2<t<Ty(t)={A0<t<T2−AT2<t<T Express Equation 18.13 with angular frequencies: y(t)=∑n(Ansinnωt+Bncosnωt)y(t)=∑n(Ansin⁡nωt+Bncos⁡nωt) Now proceed as follows. (a) Multiply both sides of Equation 18.13 by sin mωtmωt and integrate both sides over one period TT . Show that the left-hand side of the resulting equation is equal to 0 if mm is even and is equal to 4A/mωA/mω if mm is odd. (b) Using trigonometric identities, show that all terms on the right-hand side involving BnBn are equal to zero. (c) Using trigonometric identities, show that all terms on the right-hand side involving AnAn are equal to zero except for the one case of m=n.m=n. (d) Show that the entire right-hand side of the equation reduces to 12AmT.12AmT. (e) Show that the Fourier series expansion for a square wave is y(t)=∑n4Anπsinnωty(t)=∑n4Anπsin⁡nωt • A sample of an ideal gas goes through the process shown in Figure P20.30.P20.30. From AA to B,B, the process is adiabatic; from BB to C,C, it is isobaric with 100 kJkJ of energy entering the system by heat; from CC to D,D, the process is isothermal; and from DD to A,A, it is isobaric with 150 kJkJ of energy leaving the system by heat. Determine the difference in internal energy Eint,B−Eint,,AEint,B−Eint,,A • In a women’s 100 -m race, accelerating uniformly, Laura takes 2.00 s and Healan 3.00 s to attain their maximum speeds, which they each maintain for the rest of the race. They cross the finish line simultaneously, both setting a world record of 10.4 s. (a) What is the acceleration of each sprinter? (b) What are their respective maximum speeds? (c) Which sprinter is ahead at the 6.00 -s mark, and by how much? (d) What is the maximum distance by which Healan is behind Laura, and at what time does that occur? • A space station is constructed in the shape of a hollow ring of mass 5.00×104kg5.00×104kg . Members of the crew walk on a deck formed by the inner surface of the outer cylindrical wall of the ring, with radius r=100m.r=100m. At rest when constructed, the ring is set rotating about its axis so that the people inside experience an effective free-fall acceleration equal to g.g. (See Fig. P 11.29.) The rotation is achieved by firing two small rockets attached tangentially to opposite points on the rim of the ring. (a) What angular momentum does the space station acquire? (b) For what time interval must the rockets be fired if each exerts a thrust of 125 N? • The potential in a region between x=0x=0 and x=6.00mx=6.00m is V=a+bx,V=a+bx, where a=10.0Va=10.0V and b=−7.00V/m.b=−7.00V/m. Determine (a) the potential at x=0,3.00m,x=0,3.00m, and 6.00 mm and (b) the magnitude and direction of the electric field at x=x= 0,3.00m,0,3.00m, and 6.00m.6.00m. • Assume a transparent rod of diameter d=2.00μmd=2.00μm has an index of refraction of 1.36 Determine the maximum angle θθ for which the light rays incident on the end of the rod in Figure P35.39 are subject to total internal reflection along the walls of the rod. Your answer defines the size of the cone of acceptance for the rod. • A 1.00-m-diameter circular mirror focuses the Sun’s rays onto a circular absorbing plate 2.00 cm in radius, which holds a can containing 1.00 LL of water at 20.0∘0∘C . (a) If the solar intensity is 1.00kW/m2,1.00kW/m2, what is the intensity on the absorbing plate? At the plate, what are the maximum magnitudes of the fields (b) →EE→ and (c)→B(c)B→ ? (d) If 40.0 %% of the energy is absorbed, what time interval is required to bring the water to its boiling point? • Tension is maintained in a string as in Figure P16.29.P16.29. The observed wave speed is v=24.0m/sv=24.0m/s when the suspended mass is m=m= 3.00 kgkg . (a) What is the mass per unit length of the string? (b) What is the wave speed when the suspended mass is m=2.00kgm=2.00kg ? • The truck in Figure P39.1 is moving at a speed of 10.0 m/s relative to the ground. The person on the truck throws a baseball in the backward direction at a speed of 20.0 m/s relative to the truck. What is the velocity of the baseball as measured by the observer on the ground? • A 0.900−V0.900−V potential difference is maintained across a 1.50 -m length of tungsten wire that has a cross-sectional area of 0.600mm2.0.600mm2. What is the current in the wire? • An RLCRLC circuit is used in a radio to tune into an FMFM station broadcasting at f=99.7MHzf=99.7MHz . The resistance in the circuit is R=12.0Ω,R=12.0Ω, and the inductance is L=1.40μHL=1.40μH . What capacitance should be used? • The Apollo 11 astronauts set up a panel of efficient corner-cube retroreflectors on the Moon’s surface (Fig. 35.8a). The speed of light can be found by measuring the time interval required for a laser beam to travel from the Earth, reflect from the panel, and return to the Earth. Assume this interval is measured to be 2.51 s at a station where the Moon is at the zenith and take the center-to-center distance from the Earth to the Moon to be equal to 3.84×108m.3.84×108m. (a) What is the measured speed of light? (b) Explain whether it is necessary to consider the sizes of the Earth and the Moon in your calculation. • A nuclear power plant operates by using the energy released in nuclear fission to convert 20∘C water into 400∘C steam. How much water could theoretically be converted to steam by the complete fissioning of 1.00 g of 35U at 200 MeV/ fission? • A minivan travels straight north in the right lane of a divided highway at 28.0 m/s. A camper passes the minivan and then changes from the left lane into the right lane. As it does so, the camper’s path on the road is a straight displacement at 8.50° east of north. To avoid cutting off the minivan, the north–south distance between the camper’s back bumper and the minivan’s front bumper should not decrease. (a) Can the camper be driven to satisfy this requirement? (b) Explain your answer. • Figure P4.32P4.32 represents the total acceleration of a particle moving clockwise in a circle of radius 2.50 mm at a certain instant of time. For that instant, find (a) the radial acceleration of the particle, (b) the speed of the particle, and (c) its tangential acceleration. • The system shown in Figure P8.11 consists of a light, inextensible cord, light, frictionless pulleys, and blocks of equal mass. Notice that block B is attached to one of the pulleys. The system is initially held at rest so that the blocks are at the same height above the ground. The blocks are then released. Find the speed of block A at the moment the vertical separation of the blocks is h.h. • Strontium- 90 is a particularly dangerous fission product of 235U235U because it is radioactive and it substitutes for calcium in bones. What other direct fission products would accompany it in the neutron-induced fission of 255U ? Note: This reaction may release two, three, or four free neutrons. • In Chapter$9,$we will define the center of mass of an object and prove that its motion is described by the particle under constant acceleration model when constant forces act on the object. A gymnast jumps straight up, with her center of mass moving at 2.80$\mathrm{m} / \mathrm{s}$as she leaves the ground. How high above this point is her center of mass (a) 0.100$\mathrm{s}$, (b)$0.200 \mathrm{s},$(c) 0.300$\mathrm{s}$, and (d) 0.500$\mathrm{s}$thereafter? • From the Maxwell-Boltzmann speed distribution, show that the most probable speed of a gas molecule is given by Equation 21.27 . Note: The most probable speed corresponds to the point at which the slope of the speed distribution curve dNv/dvdNv/dv is zero. • An electron that has an energy of approximately 6 eV moves between infinitely high walls 1.00 nm apart. Find (a) the quantum number n for the energy state the electron occupies and (b) the precise energy of the electron. • A 1.00 -kg glider attached to a spring with a force constant of 25.0 N/mN/m oscillates on a frictionless, horizontal air track. At t=0,t=0, the glider is released from rest at x=−3.00cmx=−3.00cm (that is, the spring is compressed by 3.00 cmcm ). Find (a) the period of the glider’s motion, (b) the maximum values of its speed and acceleration, and (c)(c) the position, velocity, and acceleration as functions of time. • A bag of cement weighing 325 NN hangs in equilibrium from three wires as suggested in Figure P5.24P5.24 . Two of the wires make angles θ1=θ1= 0∘60.0∘ and θ2=40.0∘θ2=40.0∘ with the horizontal. Assuming the system is in equilibrium, find the tensions T1T1 , T2,T2, and T3T3 in the wires. • A uniformly charged disk of radius 35.0 cmcm carries charge with a density of 7.90×10−3C/m2.7.90×10−3C/m2. Calculate the electric field on the axis of the disk at (a) 5.00cm,(b)10.0cm,5.00cm,(b)10.0cm, (c) 50.0cm,50.0cm, and (d)200cm(d)200cm from the center of the disk. • The inner conductor of a coaxial cable has a radius of 0.800mm,0.800mm, and the outer conductor’s inside radius is 3.00 mmmm . The space between the conductors is filled with polyethylene, which has a dielectric constant of 2.30 and a dielectric strength of 18.0×106V/m.18.0×106V/m. What is the maxi- mum potential difference this cable can withstand? • An electric scooter has a battery capable of supplying 120 WhWh of energy. If friction forces and other losses account for 60.0%% of the energy usage, what altitude change can a rider achieve when driving in hilly terrain if the rider and scooter have a combined weight of 890 N?N? • Suppose you build a two-engine device with the exhaust energy output from one heat engine supplying the input energy for a second heat engine. We say that the two engines are running in series. Let e1e1 and e2e2 represent the efficiencies of the two engines. (a) The overall efficiency of the two-engine device is defined as the total work output divided by the energy put into the first engine by heat. Show that the overall efficiency ee is given by e=e1+e2−e1e2e=e1+e2−e1e2 What If? For parts (b) through (e) that follow, assume the two engines are Carnot engines. Engine 1 operates between temperatures ThTh and TiTi. The gas in engine 2 varies in temperature between TiTi and Tc.Tc. In terms of the temperatures, (b) what is the efficiency of the combination engine? (c) Does an improvement in net efficiency result from the use of two engines instead of one? (d) What value of the intermediate temperature TiTi results in equal work being done by each of the two engines in series? (e) What value of TiTi result in each of the two engines in series having the same efficiency? • An infinitely long line charge having a uniform charge per unit length λλ lies a distance dd from point OO as shown in Figure P 24.15 Determine the total electric flux through the surface of a sphere of radius RR centered at OO resulting from this line charge. Consider both cases, where (a) R<dR<d and (b) R>dR>d . • Write an expression that describes the pressure variation as a function of position and time for a sinusoidal sound wave in air. Assume the speed of sound is 343m/s,λ=343m/s,λ= 0.100m,0.100m, and ΔPmax=0.200PaΔPmax=0.200Pa. • Two hypothetical planets of masses m1m1 and m2m2 and radii r1r1 and r2,r2, respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course. (a) When their center-to-center separation is dd , find expressions for the speed of each planet and for their relative speed. (b) Find the kinetic energy of each planet just before they collide, taking m1=2.00×1024kg,m2=m1=2.00×1024kg,m2= 8.00 ×1024kg,r1=3.00×106m,×1024kg,r1=3.00×106m, and r2=5.00×106m.r2=5.00×106m. Note: Both the energy and momentum of the isolated two-planet system are constant. • The human brain and spinal cord are immersed in the cerebrospinal fluid. The fluid is normally continuous between the cranial and spinal cavities and exerts a pressure of 100 to 200$\mathrm{mm}$of$\mathrm{H}_{2} \mathrm{O}$above the prevailing atmospheric pressure. In medical work, pressures are often measured in units of millimeters of$\mathrm{H}_{2} \mathrm{O}$because body fluids, including the cerebrospinal fluid, typically have the same density as water. The pressure of the cerebrospinal fluid can be measured by means of a spinal tap as illustrated in Figure P14.18. A hollow tube is inserted into the spinal column, and the height to which the fluid rises is observed. If the fluid rises to a height of$160 \mathrm{mm},$we write its gauge pressure as 160$\mathrm{mm} \mathrm{H}_{2} \mathrm{O}$. ( a ) Express this pressure in pascals, in atmospheres, and in millimeters of mercury. (b) Some conditions that block or inhibit the flow of cerebrospinal fluid can be investigated by means of Queckenstedt’s test. In this procedure, the veins in the patient’s neck are compressed to make the blood pressure rise in the brain, which in turn should be transmitted to the cerebrospinal fluid. Explain how the level of fluid in the spinal tap can be used as a diagnostic tool for the condition of the patient’s spine. • Given the vectors →A=2.00ˆi+6.00ˆjA→=2.00i^+6.00j^ and →B=3.00ˆi−2.00ˆj,B→=3.00i^−2.00j^, (a) draw the vector sum →C=→A+→BC→=A→+B→ and the vector difference →D=→A−→B⋅D→=A→−B→⋅ (b) Calculate →CC→ and →D,D→, in terms of unit vectors. (c) Calculate →CC→ and →DD→ in terms of polar coordinates, with angles measured with respect to the positive xx axis. • A 40.0-mA current is carried by a uniformly wound air-core solenoid with 450 turns, a 15.0-mm diameter, and 12.0-cm length. Compute (a) the magnetic field inside A 40.0-mA current is carried by a uniformly wound air-core solenoid with 450 turns, a 15.0-mm diameter, and 12.0-cm length. Compute (a) the magnetic field inside • Show that the difference between decibel levels β1β1 and β2β2 of a sound is related to the ratio of the distances r1r1 and r2r2 from the sound source by β2−β1=20log(r1r2)β2−β1=20log(r1r2) • A liquid has a density ρ.ρ. (a) Show that the fractional change in density for a change in temperature ΔTΔT is Δρ/ρ=Δρ/ρ= −βΔT−βΔT . (b) What does the negative sign signify? (c) Fresh water has a maximum density of 1.0000 g/cm3g/cm3 at 4.0∘4.0∘C. At 10.0∘C,10.0∘C, its density is 0.9997g/cm3.0.9997g/cm3. What is ββ for water over this temperature interval? (d) At 0∘C0∘C , the density of water is 0.9999g/cm3.0.9999g/cm3. What is the value for ββ over the temperature range 0∘C0∘C to 4.00∘C4.00∘C ? • Consider two conducting spheres with radii R1R1 and R2R2 separated by a distance much greater than either radius. A total charge QQ is shared between the spheres. We wish to show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero. The total charge QQ is equal to q1+q2,q1+q2, where q1q1 represents the charge on the first sphere and q2q2 the charge on the second. Because the spheres are very far apart, you can assume the charge of each is uniformly distributed over its surface. (a) Show that the energy associated with a single conducting sphere of radius RR and charge qq surrounded by a vacuum is U=keq2/2RU=keq2/2R (b) Find the total energy of the system of two spheres in terms of q1,q1, the total charge QQ , and the radii R1R1 and R2R2 . (c) To minimize the energy, differentiate the result to part (b) with respect to q1q1 and set the derivative equal to zero. Solve for q1q1 in terms of QQ and the radii. (d) From the result of to part (c), find the charge q2.q2. (e) Find the potential of each sphere. (f) What is the potential difference between the spheres? • A 30.0 -turn solenoid of length 6.00 cmcm produces a magnetic field of magnitude 2.00 mTmT at its center. Find the current in the solenoid. • A string on a musical instrument is held under tension Tand extends from the point x=0x=0 to the point x=L.x=L. The string is overwound with wire in such a way that its mass per unit length μ(x)μ(x) increases uniformly from μ0μ0 at x=0x=0 to μLμL at x=L.x=L. (a) Find an expression for μ(x)μ(x) as a function of xx over the range 0≤x≤L.0≤x≤L. (b) Find an expression for the time interval required for a transverse pulse to travel the length of the string. • Whenever two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the orbit to be circular and 100 kmkm above the surface of the Moon, where the acceleration due to gravity is 1.52 m/s2m/s2 . The radius of the Moon is 1.70×106m.1.70×106m. Determine (a) the astronaut’s orbital speed and (b) the period of the orbit. • A block of mass M hangs from a rubber cord. The block is supported so that the cord is not stretched. The unstretched length of the cord is L0,L0, and its mass is m,m, much less than M.M. The “spring constant” for the cord is A. The block is released and stops momentarily at the lowest point. ( a) Determine the tension in the string when the block is at this lowest point. (b) What is the length of the cord in this “stretched” position? (c) If the block is held in this lowest position, find the speed of a transverse wave in the cord. • Monochromatic light of wavelength 620 nmnm passes through a very narrow slit SS and then strikes a screen in which are two parallel slits, S1S1 and S2,S2, as shown in Figure P37.71P37.71 on page 1110.1110. Slit S1S1 is directly in line with SS and at a distance of L=1.20mL=1.20m away from SS , whereas S2S2 is displaced a distance dd to one side. The light is detected at point PP on a sec- ond screen, equidistant from S1S1 and S2.S2. When either slit S1S1 or S2S2 is open, equal light intensities are measured at point P.P. When both slits are open, the intensity is three times larger. Find the minimum possible value for the slit separation d.d. • Consider the block–spring collision discussed in Example 8.8. (a) For the situation in part (B), in which the surface exerts a friction force on the block, show that the block never arrives back at x 5 0. (b) What is the maximum value of the coefficient of friction that would allow the block to return to x=0?x=0? • Starting from rest, a 64.0-kg person bungee jumps from a tethered hot-air balloon 65.0 m above the ground. The bungee cord has negligible mass and unstretched length 25.8 m. One end is tied to the basket of the balloon and the other end to a harness around the person’s body. The cord is modeled as a spring that obeys Hooke’s law with a spring constant of 81.0 N/m, and the person’s body is modeled as a particle. The hot-air balloon does not move. (a) Express the gravitational potential energy of the person–Earth system as a function of the person’s variable height y above the ground. (b) Express the elastic potential energy of the cord as a function of y. (c) Express the total potential energy of the person–cord–Earth system as a function of y. (d) Plot a graph of the gravitational, elastic, and total potential energies as functions of y. (e) Assume air resistance is negligible. Determine the minimum height of the person above the ground during his plunge. (f) Does the potential energy graph show any equilibrium position or positions? If so, at what elevations? Are they stable or unstable? (g) Determine the jumper’s maximum speed. • Show that Equation 41.26 is a solution of Equation 41.24 with energy E=12ℏω. • Suppose an ideal (Carnot) heat pump could be constructed for use as an air conditioner. (a) Obtain an expression for the coefficient of performance (COP) for such an air conditioner in terms of ThTh and TcTc (b) Would such an air conditioner operate on a smaller energy input if the difference in the operating temperatures were greater or smaller? (c) Compute the COP for such an air conditioner if the indoor temperature is 20.0∘0∘C and the outdoor temperature is 40.0∘C40.0∘C . • An aluminum rod 0.500 m in length and with a cross- sectional area of 2.50 cm2cm2 is inserted into a thermally insulated vessel containing liquid helium at 4.20 KK . The rod is initially at 300 KK (a) If one-half of the rod is inserted into the helium, how many liters of helium boil off by the time the inserted half cools to 4.20 KK ? Assume the upper half does not yet cool. (b) If the circular surface of the upper end of the rod is maintained at 300 KK , what is the approximate boil-off rate of liquid helium in liters per second after the lower half has reached 4.20 KK ? (Aluminum has thermal conductivity of 3100 W/m⋅KW/m⋅K at 4.20K;4.20K; ignore its temperature variation. The density of liquid helium is 125kg/m3.125kg/m3. ) • Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest. A wooden block of mass MM rests on a table over a large hole as in Figure 9.57 . A bullet of mass mm with an initial velocity of vivi is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of hh . (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Find an expression for the initial velocity of the bullet. • A mercury thermometer is constructed as shown in Figure P19.41P19.41 . The Pyrex glass capillary tube has a diameter of 0.00400cm,0.00400cm, and the bulb has a diameter of 0.250cm.0.250cm. Find the change in height of the mercury column that occurs with a temperature change of 30.0∘0∘C . • Sound with a frequency 650 Hz from a distant source passes through a doorway 1.10 m wide in a sound-absorbing wall. Find (a) the number and (b) the angular directions of the diffraction minima at listening positions along a line parallel to the wall. • Figure P34.13 shows a plane electromagnetic sinusoidal wave propagating in the x direction. Suppose the wavelength is 50.0 m and the electric field vibrates in the xy plane with an amplitude of 22.0 V/m. Calculate (a) the frequency of the wave and (b) the magnetic field ¯BB¯¯¯¯ when the electric field has its maximum value in the negative yy direction. (c) Write an expression for ¯BB¯¯¯¯ with the correct unit vector, with numerical values for Bmax,k,Bmax,k, and ωω , and with its magnitude in the form B=Bmaxcos(kx−ωt)B=Bmaxcos(kx−ωt) • One electron collides elastically with a second electron initially at rest. After the collision, the radii of their trajectories are r1r1 and r2.r2. The trajectories are perpendicular to a uniform magnetic field of magnitude BB . Determine the energy of the incident electron. • A meterstick moving at 0.900c relative to the Earth’s surface approaches an observer at rest with respect to the Earth’s surface. (a) What is the meterstick’s length as measured by the observer? (b) Qualitatively, how would the answer to part (a) change if the observer started running toward the meterstick? • A beam of 541-nm light is incident on a diffraction grating that has 400 grooves/mm. (a) Determine the angle of the second-order ray. (b) What If? If the entire apparatus is immersed in water, what is the new second-order angle of diffraction? (c) Show that the two diffracted rays of parts (a) and (b) are related through the law of refraction. • A photon having energy E0=0.880MeVE0=0.880MeV is scattered by a free electron initially at rest such that the scattering angle of the scattered electron is equal to that of the scattered of the scattered electron is equal to that of the scattered photon as shown in Figure P40.27P40.27 . (a) Determine the scattering angle of the photon and the electron. (b) Determine the energy and momentum of the scattered photon. (c) Determine the kinetic energy and momentum of the scattered electron. • A steplader of negligible weight is constructed as shown in Figure P12.56,P12.56, with AC=BC=ℓ=4.00m.AC=BC=ℓ=4.00m. A painter of mass m=70.0kgm=70.0kg stands on the ladder d=3.00md=3.00m from the bottom. Assuming the floor is frictionless, find (a) the tension in the horizontal bar DEDE connecting the two halves of the ladder, (b) the normal forces at AA and BB , and (c) the components of the reaction exerts on the right half. Suggeslion: Treat the ladder as a single object, but also treat each half of the ladder separately. • The displacement vectors 42.0 cmcm at 15.0∘0∘ and 23.0 cmcm at 65.0∘65.0∘ both start from the origin and form two sides of a parallelogram. Both angles are measured counter clock-wise from the xx axis. (a) Find the area of the parallelogram. (b) Find the length of its longer diagonal. • A light balloon filled with helium of density 0.179 kg/m3kg/m3 is tied to a light string of length L 5 3.00 m. The string is tied to the ground forming an “inverted” simple pendulum (Fig. 15.65a). If the balloon is displaced slightly from equilibrium as in Figure P15.65b and released, (a) show that the motion is simple harmonic and (b) determine the period of the motion. Take the density of air to be 1.20 kg/m3kg/m3. Hint: Use an analogy with the simple pendulum and see Chapter 14. Assume the air applies a buoyant force on the balloon but does not otherwise affect its motion. • A shaft is turning at 65.0 rad/srad/s at time t=0.t=0. Thereafter, its angular acceleration is given by α=−10.0−5.00tα=−10.0−5.00t where αα is in rad/s 22 and tt is in seconds. (a) Find the angular speed of the shaft at t=3.00st=3.00s . (b) Through what angle does it turn between t=0t=0 and t=3.00st=3.00s ? • A 10.0-kg block is released from rest at point in Figure P8.63. The track is frictionless except for the portion between points and , which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2 250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between points and • The refracting telescope at the Yerkes Observatory has a 1.00-m diameter objective lens of focal length 20.0 m. Assume it is used with an eyepiece of focal length 2.50 cm. (a) Determine the magnification of Mars as seen through this telescope. (b) Are the Martian polar caps right side up or upside down? • A child slides across a floor in a pair of rubber-soled shoes. The friction force acting on each foot is 20.0 NN . The foot-print area of each shoe sole is 14.0cm2,14.0cm2, and the thickness of cach sole is 5.00 mmmm . Find the horizontal distance by which the upper and lower surfaces of each sole arfset. The shear modulus of the rubber is 3.00 MN/m2MN/m2 . • One insulated conductor from a household extension cord has a mass per length of 19.0 g/m. A section of this conductor is held under tension between two clamps. A subsection is located in a magnetic field of magnitude 15.3 mT directed perpendicular to the length of the cord. When the cord carries an AC current of 9.00 A at a frequency of 60.0 Hz, it vibrates in resonance in its simplest standing-wave vibration mode. (a) Determine the relationship that must be satisfied between the separation d of the clamps and the tension T in the cord. (b) Determine one possible combination of values for these variables. • Consider an LCLC circuit in which L=500mHL=500mH and C=C= 0.100μFμF . (a) What is the resonance frequency ω0?ω0? (b) If a resistance of 1.00 kΩkΩ is introduced into this circuit, what is the frequency of the damped oscillations? (c) By what percentage does the frequency of the damped oscillations differ from the resonance frequency? • Figure P 3.38 illustrates typical proportions of male (m) and female (f) anatomies. The displacements →d1md→1m and →d1fd→1f from the soles of the feet to the navel have magnitudes of 104 cm and 84.0 cm, respectively. The displacements →d2md→2m and →d2fd→2f from the navel to outstretched fingertips have magnitudes of 100 cm and 86.0 cm, respectively. Find the vector sum of these displacements →d3=→d1+→d2d→3=d→1+d→2 for both people. • In a darkened room, a burning candle is placed 1.50 m from a white wall. A lens is placed between the candle and the wall at a location that causes a larger, inverted image to form on the wall. When the lens is in this position, the object distance is$p_{1}$. When the lens is moved 90.0$\mathrm{cm}$toward the wall, another image of the candle is formed on the wall. From this information, we wish to find$p_{1}$and the focal length of the lens. (a) From the lens equation for the first position of the lens, write an equation relating the focal length$f$of the lens to the object distance$p_{1},$with no other variables in the equation. (b) From the lens equation for the second position of the lens, write another equation relating the focal length$f$of the lens to the object distance$p_{1}$. (c) Solve the equations in parts (a) and (b) simultaneously to find$p_{1} .$(d) Use the value in part (c) to find the focal length$f$of the lens. • Identify the mediators for the two interactions described in the Feynman diagrams shown in Figure P46.62P46.62 . GRAPH CANNOT COPY • Because the Earth rotates about its axis, a point on the equator experiences a centripetal acceleration of 0.0337m/s2,0.0337m/s2, whereas a point at the poles experiences no centripetal acceleration. If a person at the equator has a mass of 75.0kg,75.0kg, calculate (a) the gravitational force (true weight) on the person and (b) the normal force (apparent weight) on the person. (c) Which force is greater? Assume the Earth is a uniform sphere and take g=9.800m/s2.g=9.800m/s2. • A line of charge starts at x=+x0x=+x0 and extends to positive infinity. The linear charge density is λ=λ0x0/x,λ=λ0x0/x, where λ0λ0 is a constant. Determine the electric field at the origin. • The boiling point of liquid hydrogen is 20.3 KK at atmospheric pressure. What is this temperature on (a) the Celsius scale and (b) the Fahrenheit scale? • Show that the kinetic energy of an object rotating about a fixed axis with angular momentum L=IωL=Iω can be written as K=L2/2I.K=L2/2I. • The wave function ψ(x)=Bxe−(mω/2ℏ)x2 is a solution to the simple harmonic oscillator problem. (a) Find the energy of this state. (b) At what position are you least likely to find the particle? (c) At what positions are you most likely to find the particle? (d) Determine the value of B required to normalize the wave function. (e) What If? Determine the classical probability of finding the particle in an interval of small length δ centered at the position x=2(ℏ/mω)1/2. (f) What is the actual probability of finding the particle in this interval? • A periscope (Fig. P36.5) is useful for viewing objects that cannot be seen directly. It can be used in submarines and when watching golf matches or parades from behind a crowd of people. Suppose the object is a distance$p_{1}$from the upper mirror and the centers of the two flat mirrors are separated by a distance$h .$(a) What is the distance of the final image from the lower mirror? (b) Is the final image real or virtual? (c) Is it upright or inverted? (d) What is its magnification? (e) Does it appear to be left-right reversed? • Show that the rms value for the sawtooth voltage shown in Figure P33.63P33.63 is ΔVmax/√3ΔVmax/3–√ • An insulating rod having linear charge density λ=40.0μC/mλ=40.0μC/m and linear mass density μ=0.100kg/mμ=0.100kg/m is released from rest in a uniform electric field E=100V/mE=100V/m directed perpendicular to the rod (Fig. P25.9). (a) Determine the speed of the rod after it has traveled 2.00m.2.00m. (b) What If? How does your answer to part (a) change if the electric field is not perpendicular to the rod? Explain. • A two-dimensional water wave spreads in circular ripples. Show that the amplitude AA at a distance rr from the initial disturbance is proportional to 1/r√/r . Suggestion: Consider the energy carried by one outward-moving ripple. • A 40.0 -kg box initially at rest is pushed 5.00 mm along a rough, horizontal floor with a constant applied horizontal force of 130 NN . The coefficient of friction between box and floor is 0.300 . Find (a) the work done by the applied force, (b) the increase in internal energy in the box-floor system as a result of friction, (c) the work done by the normal force, (d) the work done by the gravitational force, (e) the change in kinetic energy of the box, and (f) the final speed of the box. • A centrifuge in a medical laboratory rotates at an angular speed of 3600 rev/min. When switched off, it rotates through 50.0 revolutions before coming to rest. Find the constant angular acceleration of the centrifuge. • Determine the initial direction of the deflection of charged particles as they enter the magnetic fields shown in Figure P 29.2. • A coil formed by wrapping 50 turns of wire in the shape of a square is positioned in a magnetic field so that the normal to the plane of the coil makes an angle of 30.0∘ with the direction of the field. When the magnetic field is increased uniformly from 200μT to 600μT in 0.400 s , an emf of magnitude 80.0 mV is induced in the coil. What is the total length of the wire in the coil? • Consider a tall building located on the Earth’s equator. As the Earth rotates, a person on the top floor of the building moves faster than someone on the ground with respect to an inertial reference frame because the person on the ground is closer to the Earth’s axis. Consequently, if an object is dropped from the top floor to the ground a distance h below, it lands east of the point vertically below where it was dropped. (a) How far to the east will the object land? Express your answer in terms of h,gh,g and the angular speed ωω of the Earth. Ignore air resistance and assume the free-fall acceleration is constant over this range of heights. (b) Evaluate the eastward displacement for h=50.0mh=50.0m. (c) In your judgment, were we justified in ignoring this aspect of the Coriolis effect in our previous study of free fall? (d) Suppose the angular speed of the Earth were to decrease due to tidal friction with constant angular acceleration. Would the eastward displacement of the dropped object increase or decrease compared with that in part (b)? • At what temperature will aluminum have a resistivity that is three times the resistivity copper has at room temperature? • In Example 2.7, we investigated a jet landing on an aircraft carrier. In a later maneuver, the jet comes in for a landing on solid ground with a speed of 100 m/s, and its acceleration can have a maximum magnitude of 5.00$\mathrm{m} / \mathrm{s}^{2}$as it comes to rest. (a) From the instant the jet touches the runway, what is the minimum time interval needed before it can come to rest? (b) Can this jet land at a small tropical island airport where the runway is 0.800$\mathrm{km}$long? (c) Explain your answer. • The density of gasoline is 730 kg/m3kg/m3 at 0∘0∘C. Its average coefficient of volume expansion is 9.60×10−4(∘C)−19.60×10−4(∘C)−1 . Assume 1.00 gal of gasoline occupies 0.00380m3.0.00380m3. How many extra kilograms of gasoline would you receive if you bought 10.0 gal of gasoline at 0∘C0∘C rather than at 20.0∘C20.0∘C from a pump that is not temperature compensated? • Two glass plates 10.0 cmcm long are in contact at one end and separated at the other end by a thread with a diameter d=d= 0.0500 mmmm (Fig. P37.35). Light containing the two wave- lengths 400 nmnm and 600 nmnm is incident perpendicularly and viewed by reflection. At what distance from the contact point is the next dark fringe? • Using the Maxwell–Boltzmann speed distribution function, verify Equations 21.25 and 21.26 for (a) the rms speed and (b) the average speed of the molecules of a gas at a temperature T. The average value of vn is ¯vn=1N∫∞0vnNvdvvn¯¯¯¯¯=1N∫∞0vnNvdv Use the table of integrals B.6B.6 in Appendix B. • A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east. (a) What is her resultant displacement? (b) What is the total distance she travels? • A billiard ball moving at 5.00 m/s strikes a stationary ball of the same mass. After the collision, the first ball moves at 4.33 m/s at an angle of 30.08 with respect to the original line of motion. Assuming an elastic collision (and ignoring friction and rotational motion), find the struck ball’s velocity after the collision. • A 2.00 -MeV neutron is emitted in a fission reactor. If it loses half its kinetic energy in each collision with a moderator atom, how many collisions does it undergo as it becomes a thermal neutron, with energy 0.039eV?0.039eV? • A large meteoroid enters the Earth’s atmosphere at a speed of 20.0 km/s and is not significantly slowed before entering the ocean. (a) What is the Mach angle of the shock wave from the meteoroid in the lower atmosphere? (b) If we assume the meteoroid survives the impact with the ocean surface, what is the (initial) Mach angle of the shock wave the meteoroid produces in the water? • A strong electromagnet produces a uniform magnetic field of 1.60 T over a cross-sectional area of 0.200m2.0.200m2. A coil having 200 turns and a total resistance of 20.0ΩΩ is placed around the electromagnet. The current in the electromagnet is then smoothly reduced until it reaches zero in 20.0 ms. What is the current induced in the coil? • A uniformly charged rod of length L and total charge Q lies along the x axis as shown in Figure P23.36. (a) Find the components of the electric field at the point P on the y axis a distance d from the origin. (b) What are the approximate values of the field components when d>>d>> L? Explain why you would expect these results. • A rectangular loop of dimensions ℓ and w moves with a constant velocity →v away from a long wire that carries a current I in the plane of the loop (Fig. P31.66). The total resistance of the loop is R. Derive an expression that gives the current in the loop at the instant the near side is a distance r from the wire. • An AC voltage of the form Δv=90.0Δv=90.0 sin 350t,350t, where ΔvΔv is in volts and tt is in seconds, is applied to a series RLCRLC circuit. If R=50.0Ω,C=25.0μF,R=50.0Ω,C=25.0μF, and L=0.200H,L=0.200H, find (a) the impedance of the circuit, (b)(b) the rms current in the circuit, and (c)(c) the average power delivered to the circuit. • A car traveling on a flat (unbanked), circular track accelerates uniformly from rest with a tangential acceleration of 1.70 m/s2m/s2 . The car makes it one quarter of the way around the circle before it skids off the track. From these data, determine the coefficient of static friction between the car and the track. • In the transformer shown in Figure P33.51P33.51 , the load resistance RLRL is 50.0ΩΩ . The turns ratio N1/N2N1/N2 is 2.50 , and the rms source voltage is ΔVs=80.0VΔVs=80.0V . If a voltmeter across the load resistance measures an rms voltage of 25.0V,25.0V, what is the source resistance Rs?Rs? • A block of mass m=2.00kgm=2.00kg is released from rest at h=0.500mh=0.500m above the surface of a table, at the top of a θ=30.0∘θ=30.0∘ incline as shown in Figure P5.75.P5.75. The frictionless incline is fixed on a table of height H=2.00m.H=2.00m. (a) Determine the acceleration of the block as it slides down the incline. (b) What is the velocity of the block as it leaves the incline? (c) How far from the table will the block hit the floor? (d) What time interval elapses between when the block is released and when it hits the floor? (e) Does the mass of the block affect any of the above calculations? • Prove that the first term in the Schrödinger equation, −(ℏ2/2m)(d2ψ/dx2), reduces to the kinetic energy of the quantum particle multiplied by the wave function (a) for a freely moving particle, with the wave function given by Equation 41.4, and (b) for a particle in a box, with the wave function given by Equation 41.13 . • Example 26.1 explored a cylindrical capacitor of length ℓℓ with radii aa and bb for the two conductors. In the What If? section of that example, it was claimed that increasing ℓℓ by 10%% is more effective in terms of increasing the capacitance than increasing aa by 10%% if b>2.85ab>2.85a . Verify this claim mathematically. • A sample of a solid substance has a mass mm and a density ρ0ρ0 at a temperature T0T0 . (a) Find the density of the substance if its temperature is increased by an amount ΔTΔT in terms of the coefficient of volume expansion ββ . (b) What is the mass of the sample if the temperature is raised by an amount ΔT?ΔT? • A cube of edge length ℓ=2.50cmℓ=2.50cm is positioned as shown in Figure P30.45.P30.45. A uniform magnetic field given by →B=(5ˆi+4ˆj+3ˆk)TB→=(5i^+4j^+3k^)T exists throughout the region. (a) Calculate the magnetic flux through the shaded face. (b) What is the total flux through the six faces? • In about 1657 , Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres (Fig. Pl4.54). Two teams of eight horses each could pull the hemispheres apart only on some trials and then “with greatest difficulty,” with the resulting sound likened to a cannon firing. Find the force$F$required to pull the thin- walled evacuated hemispheres apart in terms of$R,$the radius of the hemispheres;$P$, the pressure inside the hemispheres; and atmospheric pressure$P_{0}$. • An inclined plane of angle θ=20.0∘ has a spring of force constant k=500N/m fastened securely at the bottom so that the spring is parallel in Figure P7.63. A block of mass m=2.50kg is placed on the plane at a distance d=0.300m from the spring. From this position, the block is projected downward toward the spring with speed v=0.750m/s . By what distance is the spring compressed when the block momentarily comes to rest? • Calculate the net torque (magnitude and direction) on the beam in Figure P 11.5 about (a) an axis through OO perpendicular to the page and (b) an axis through CC perpendicular to the page. • A zoom lens system is a combination of lenses that produces a variable magnification of a fixed object as it maintains a fixed image position. The magnification is varied by moving one or more lenses along the axis. Multiple lenses are used in practice, but the effect of zooming in on an object can be demonstrated with a simple two-lens system. An object, two converging lenses, and a screen are mounted on an optical bench. Lens$1,$which is to the right of the object, has a focal length of$f_{1}=5.00 \mathrm{cm},$and lens$2,$which is to the right of the first lens, has a focal length of$f_{2}=10.0 \mathrm{cm} .$The screen is to the right of lens$2 .$Initially, an object is situated at a distance of 7.50$\mathrm{cm}$to the left of lens$1,$and the image formed on the screen has a magnification of$+1.00$. (a) Find the distance between the object and the screen. (b) Both lenses are now moved along their common axis while the object and the screen maintain fixed positions until the image formed on the screen has a magnification of$+3.00$. Find the displacement of each lens from its initial position in part (a). (c) Can the lenses be displaced in more than one way? • A laboratory electromagnet produces a magnetic field of magnitude 1.50 T. A proton moves through this field with a speed of 6.00×106m/s6.00×106m/s . (a) Find the magnitude of the maximum magnetic force that could be exerted on the proton. (b) What is the magnitude of the maximum acceleration of the proton? (c) Would the field exert the same magnetic force on an electron moving through the field with the same speed? (d) Would the electron experience the same acceleration? Explain. • A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N. As the elevator later stops, the scale reading is 391 N. Assuming the magnitude of the acceleration is the same during starting and stopping, determine (a) the weight of the person, (b) the person’s mass, and (c) the acceleration of the elevator. • A river flows with a steady speed vv . A student swims upstream a distance dd and then back to the starting point. The student can swim at speed cc in still water. (a) In terms of d,v,d,v, and cc , what time interval is required for the round trip? (b) What time interval would be required if the water were still? (c) Which time interval is larger? Explain whether it is always larger. • Why is the following situation impossible? A meteoroid strikes the Earth directly on the equator. At the time it lands, it is traveling exactly vertical and downward. Due to the impact, the time for the Earth to rotate once increases by 0.5 s, so the day is 0.5 s longer, undetectable to laypersons. After the impact, people on the Earth ignore the extra half-second each day and life goes on as normal. (Assume the density of the Earth is uniform.) • A homeowner has a solar water heater installed on the roof of his house (Fig. P34.61). The heater is a flat, closed box with excellent thermal insulation. Its interior is painted black, and its front face is made of insulating glass. Its emissivity for visible light is 0.900, and its emissivity for infrared light is 0.700. Light from the noontime Sun is incident perpendicular to the glass with an intensity of 1000W/m2,1000W/m2, and no water enters or leaves the box. Find the steady-state temperature of the box’s interior. (b) What If? The homeowner builds an identical box with no water tubes. It lies flat on the ground in front of the house. He uses it as a cold frame, where he plants seeds in early spring. Assuming the same noontime Sun is at an elevation angle of 50.0∘0∘ , find the steady-state temperature of the interior of the box when its ventilation slots are tightly closed. • The Fermi energy of copper at 300 K is 7.05 eV. (a) What is the average energy of a conduction electron in copper at 300 K? (b) At what temperature would the average translational energy of a molecule in an ideal gas be equal to the energy calculated in part (a)? • A synchronous satellite, which always remains above the same point on a planet’s equator, is put in orbit around Jupiter to study that planet’s famous red spot. Jupiter rotates once every 9.84 h. Use the data of Table 13.2 to find the altitude of the satellite above the surface of the planet. • A certain element has its outermost electron in a 3pp sub- shell. It has valence +3+3 because it has three more electrons than a certain noble gas. What element is it? • A cyclotron (Fig. 29.16) designed to accelerate protons has an outer radius of 0.350 m. The protons are emitted nearly at rest from a source at the center and are accelerated through 600 V each time they cross the gap between the dees. The dees are between the poles of an electromagnet where the field is 0.800 T. (a) Find the cyclotron frequency for the protons in this cyclotron. Find (b) the speed at which protons exit the cyclotron and (c) their maximum kinetic energy. (d) How many revolutions does a proton make in the cyclotron? (e) For what time interval does the proton accelerate? • A car of mass mm moving at a speed v1v1 collides and couples with the back of a truck of mass 2 mm moving initially in the same direction as the car at a lower speed v2.v2. (a) What is the speed vfvf of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the car-truck system in the collision? • Let A→=60.0cmA→=60.0cm at 270∘270∘ measured from the horizontal. Let B→=80.0cmB→=80.0cm at some angle θ.θ. (a) Find the magnitude of A→+B→A→+B→ as a function of θ.θ. (b) From the answer to part (a), for what value of θθ does |A→+B→||A→+B→| take on its maximum value? What is this maximum value? (c) From the answer to part (a),(a), for what value of θθ does |A→+B→||A→+B→| take on its minimum value? What is this minimum value? (d) Without reference to the answer to part (a), argue that the answers to each of parts (b) and (c) do or do not make sense. • A person going for a walk follows the path shown in Figure P 3.47. The total trip consists of four straight-line paths. At the end of the walk, what is the person’s resultant displacement measured from the starting point? • When a gymnast performing on the rings executes the iron cross, he maintains the position at rest shown in Figure P12.53aP12.53a . In this maneuver, the gymnast’s feet (not shown) are off the floor. The primary muscles involved in supporting this position are the latissimus dorsi (“lats”) and the pectoralis major (“pecs”). One of the rings exerts an upward force F→hF→h on a hand as shown in Figure P12.53bP12.53b . The force F→sF→s is exerted by the shoulder joint on the arm. The latissimus dorsi and pectoralis major muscles exert a total force F→mF→m on the arm. (a) Using the information in the figure, find the magnitude of the force F→mF→m for an athlete of weight 750 NN . (b) Suppose an athlete in training cannot perform the iron cross but can hold a position similar to the figure in which the arms make a 45∘45∘ angle with the horizontal rather than being horizontal. Why is this position easier for the athlete? • Iridescent peacock feathers are shown in Figure P 38.58a. The surface of one microscopic barbule is composed of transparent keratin that supports rods of dark brown melanin in a regular lattice, represented in Figure P 38.58b. (Your fingernails are made of keratin, and melanin is the dark pigment giving color to human skin.) In a portion of the feather that can appear turquoise (blue-green), assume the melanin rods are uniformly separated by 0.25μm,0.25μm, with air between them. (a) Explain how this structure can appear turquoise when it contains no blue or green pigment. (b) Explain how it can also appear violet if light falls on it in a different direction. (c) Explain how it can present different colors to your two eyes simultaneously, which is a characteristic of iridescence. (d) A compact disc can appear to be any color of the rainbow. Explain why the portion of the feather in Figure P 38.58b cannot appear yellow or red. (e) What could be different about the array of melanin rods in a portion of the feather that does appear to be red? • Four identical particles, each having charge qq and mass mm, are released from rest at the vertices of a square of side LL. How fast is each particle moving when their distance from the center of the square doubles? • Two light sources are used in a photoelectric experiment to determine the work function for a particular metal surface. When green light from a mercury lamp (λ=546.1nm)(λ=546.1nm) is used, a stopping potential of 0.376 VV reduces the photocurrent to zero. (a) Based on this measurement, what is the work function for this metal? (b) What stopping potential would be observed when using the yellow light from a helium discharge tube (λ=587.5nm)(λ=587.5nm) ? • Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest. Why is the following situation impossible? An astronaut, together with the equipment he carries, has a mass of 150 kgkg . He is taking a space walk outside his spacecraft, which is drifting through space with a constant velocity. The astronaut accidentally pushes against the spacecraft and begins moving away at 20.0 m/sm/s , relative to the spacecraft, without a tether. To return, he takes equipment off his space suit and throws it in the direction away from the spacecraft. Because of his bulky space suit, he can throw equipment at a maximum speed of 5.00 m/sm/s relative to himself. After throwing enough equipment, he starts moving back to the spacecraft and is able to grab onto it and climb inside. • A spherical steel ball bearing has a diameter of 2.540 cmcm at 25.00∘00∘C . (a) What is its diameter when its temperature is raised to 100.0∘C100.0∘C ? (b) What temperature change is required to increase its volume by 1.000%% ? • Two long wires hang vertically. Wire 1 carries an upward current of 1.50 A. Wire 2, 20.0 cm to the right of wire 1, carries a downward current of 4.00 A. A third wire, wire 3, is to be hung vertically and located such that when it carries a certain current, each wire experiences no net force. (a) Is this situation possible? Is it possible in more than one way? Describe (b) the position of wire 3 and (c) the magnitude and direction of the current in wire 3. • A series RLC circuit is operating at 2.00×103Hz . At this frequency, XL=XC=1884Ω. The resistance of the circuit is 40.0Ω. (a) Prepare a table showing the values of XL,XC, and Z for f=300,600,800,1.00×103,1.50×103,2.00× 103,3.00×103,4.00×103,6.00×103, and 1.00×104Hz (b) Plot on the same set of axes XL,XC, and Z as a function of lnf. • The accompanying table shows measurements of the Hall voltage and corresponding magnetic field for a probe used to measure magnetic fields. (a) Plot these data and deduce a relationship between the two variables. (b) If the measurements were taken with a current of 0.200 A and the sample is made from a material having a charge- carrier density of 1.00×10261.00×1026 carriers /m3,/m3, what is the thickness of the sample? ΔVH(μV)0111928425061687990102B(T)0.000.100.200.200.300.400.500.800.800.901.00ΔVH(μV)B(T)00.00110.10190.20280.20420.30500.40610.50680.80790.80900.901021.00 • Calculate the equivalent capacitance between points aa and bb in Figure P26.73P26.73 . Notice that this system is not a simple series or parallel combination. Suggestion: Assume a potential difference ΔVΔV between points aa and b.b. Write expressions for ΔVabΔVab in terms of the charges and capacitances for the various possible pathways from aa to bb and require conservation of charge for those capacitor plates that are connected to each other. • Assume you are agile enough to run across a horizontal surface at 8.50 m/s, independently of the value of the gravitational field. What would be (a) the radius and (b) the mass of an airless spherical asteroid of uniform density 1.10×103kg/m31.10×103kg/m3 on which you could launch yourself into orbit by running? (c) What would be your period? (d) Would your running significantly affect the rotation of the asteroid? Explain. • Astronomers often take photographs with the objective lens or mirror of a telescope alone, without an eyepiece. (a) Show that the image size$h^{\prime}$for such a telescope is given by$h^{\prime}=f h /(f-p),$where$f$is the objective focal length,$h$is the object size, and$p$is the object distance. (b) What If? Simplify the expression in part (a) for the case in which the object distance is much greater than objective focal length. (c) The “wingspan” of the International Space Station is$108.6 \mathrm{m},$the overall width of its solar panel configuration. When the station is orbiting at an altitude of$407 \mathrm{km},$find the width of the image formed by a telescope objective of focal length$4.00 \mathrm{m} .$• A vertical electric field of magnitude 2.00×104N/C2.00×104N/C exists above the Earth’s surface on a day when a thunderstorm is brewing. A car with a rectangular size of 6.00 m by 3.00 m is traveling along a dry gravel roadway sloping downward at 10.0∘0∘. Determine the electric flux through the bottom of the car. • Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest. A student performs a ballistic pendulum experiment using an apparatus similar to that discussed in Example 9.6 and shown in Figure P9.56. She obtains the following average data: h=8.68cm, projectile mass m1= 68.8 g , and pendulum mass m2=263g . (a) Determine the initial speed v1A of the projectile. (b) The second part of her experiment is to obtain v1A by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its final horizontal position x and distance of fall y (Fig. P9.70). What numerical value does she obtain for v1A based on her measured values of x=257cm and y=85.3cm2 (c) What factors might account for the difference in this value compared with that obtained in part (a)? • A series AC circuit contains a resistor, an inductor of 150mH,150mH, a capacitor of 5.00μFμF , and a source with ΔVmax=ΔVmax= 240 VV operating at 50.0 HzHz . The maximum current in the circuit is 100 mAmA . Calculate (a) the inductive reactance, (b) the capacitive reactance, (c) the impedance, (d) the resistance in the circuit, and (e) the phase angle between the current and the source voltage. • A metal rod of mass mm carrying a current II glides on two horizontal rails a distance dd apart. If the coefficient of kinetic friction between the rod and rails is μ,μ, what vertical magnetic field is required to keep the rod moving at a constant speed? • The probability of a nuclear reaction increases dramatically when the incident particle is given energy above the “Coulomb barrier,” which is the electric potential energy of the two nuclei when their surfaces barely touch. Compute the Coulomb barrier for the absorption of an alpha particle by a gold nucleus. • The K series of the discrete x-ray spectrum of tungsten contains wavelengths of 0.0185nm,0.0209nm,0.0185nm,0.0209nm, and 0.0215nm.0.0215nm. The K-shell ionization energy is 69.5 keVkeV . (a) Determine the ionization energies of the L,M,L,M, and NN shells. (b) Draw a diagram of the transitions. • A simple pendulum has a length of 1.00 mm and a mass of 1.00 kgkg . The maximum horizontal displacement of the pendulum bob from equilibrium is 3.00cm.3.00cm. Calculate the quantum number nn for the pendulum. • A particle having charge q=+2.00μCq=+2.00μC and mass m=m= 0.0100 kgkg is connected to a string that is L=1.50mL=1.50m long and tied to the pivot point PP in Figure P25.7.P25.7. The particle, string, and pivot point all lie on a frictionless, horizontal table. The particle is released from rest when the string makes an angle θ=60.0∘θ=60.0∘ with a uniform electric field of magnitude E=300V/mE=300V/m . Determine the speed of the particle when the string is parallel to the electric field. • A plano-concave lens having index of refraction 1.50 is placed on a flat glass plate as shown in Figure P37.61. Its curved surface, with radius of curvature 8.00 m, is on the bottom. The lens is illuminated from above with yellow sodium light of wavelength 589 nm, and a series of concentric bright and dark rings is observed by reflection. The interference pattern has a dark spot at the center that is surrounded by 50 dark rings, the largest of which is at the outer edge of the lens. (a) What is the thickness of the air layer at the center of the interference pattern? (b) Calculate the radius of the outermost dark ring. (c) Find the focal length of the lens. • A puck of mass m=50.0gm=50.0g is attached to a taut cord passing through a small hole in a friction less, horizontal surface (Fig. P 11.52). The puck is initially orbiting with speed vi=1.50m/svi=1.50m/s in a circle of radius ri=0.300m.ri=0.300m. The cord is then slowly pulled from below, decreasing the radius of the circle to r=0.100m.r=0.100m. (a) What is the puck’s speed at the smaller radius? (b) Find the tension in the cord at the smaller radius. (c) How much work is done by the hand in pulling the cord so that the radius of the puck’s motion changes from 0.300 m to 0.100 m? • Assume an object of mass M is suspended from the bottom of the rope of mass mm and length LL in Problem 58.58. (a) Show that the time interval for a transverse pulse to travel the length of the rope is Δt=2Lmg−−−√(M+m−−−−−−√−M−−√)Δt=2Lmg(M+m−M) (b) What If? Show that the expression in part (a) reduces to the result of Problem 58 when M=0M=0 . (c) Show that for m<<M,m<<M, the expression in part (a) reduces to Δt=mLMg−−−−√Δt=mLMg • Consider a nucleus at rest, which then spontaneously splits into two fragments of masses m1m1 and m2m2 . (a) Show that the fraction of the total kinetic energy carried by fragment m1m1 is K1Ktot=m2m1+m2K1Ktot=m2m1+m2 and the fraction carried by m2m2 is K2Ktot=m1m1+m2 assuming relativistic corrections can be ignored. A stationary 236 U nucleus fissions spontaneously into two primary fragments, 8733Br and 14957La . (b) Calculate the disintegration energy. The required atomic masses are 86.920711 u for 8735Br,148.934370ufor 14957La, and 236.045562 u for 23692U (c) How is the disintegration energy split between the two primary fragments? (d) Calculate the speed of each fragment immediately after the fission. • An air-core solenoid with 68 turns is 8.00 cm long and has a diameter of 1.20 cm. When the solenoid carries a current of 0.770 A, how much energy is stored in its magnetic field? • Find (a) the equivalent capacitance of the capacitors in Figure P26.18, (b) the charge on each capacitor, and (c) the potential difference across each capacitor. • Show that the minimum period for a satellite in orbit around a spherical planet of uniform density ρρ is Tmin=3πGρ−−−√Tmin=3πGρ independent of the planet’s radius. • For each of the following systems and time intervals, write the appropriate expanded version of Equation 8.2, the conservation of energy equation. (a) the heating coils in your toaster during the first five seconds after you turn the toaster on (b) your automobile from just before you fill it with gasoline until you pull away from the gas station at speed v (c) your body while you sit quietly and eat a peanut butter and jelly sandwich for lunch (d) your home during five minutes of a sunny afternoon while the temperature in the home remains fixed • Two boys are sliding toward each other on a friction less, ice-covered parking lot. Jacob, mass 45.0 kg, is gliding to the right at 8.00 m/s, and Ethan, mass 31.0 kg, is gliding to the left at 11.0 m/s along the same line. When they meet, they grab each other and hang on. (a) What is their velocity immediately thereafter? (b) What fraction of their original kinetic energy is still mechanical energy after their collision? That was so much fun that the boys repeat the collision with the same original velocities, this time moving along parallel lines 1.20 m apart. At closest approach, they lock arms and start rotating about their common center of mass. Model the boys as particles and their arms as a cord that does not stretch. (c) Find the velocity of their center of mass. (d) Find their angular speed. (e) What fraction of their original kinetic energy is still mechanical energy after they link arms? (f) Why are the answers to parts (b) and (e) so different? • A quantum particle is described by the wave function ψ(x)={Acos(2πxL)for−L4≤x≤L40elsewhere (a) Determine the normalization constant A. (b) What is the probability that the particle will be found between x=0 and x=L/8 if its position is measured? • Compute an order-of-magnitude estimate for the frequency of an electromagnetic wave with wavelength equal to (a) your height and (b) the thickness of a sheet of paper. How is each wave classified on the electromagnetic spectrum? • A rock sample contains traces of 238U,235U,232Th,208Pb,238U,235U,232Th,208Pb, 207Pb,207Pb, and 206 Pb . Analysis shows that the ratio of the amount of 238U to 206Pb is 1.164. (a) Assuming the rock originally contained no lead, determine the age of the rock. (b) What should be the ratios of 235 U to 207 Pb and of 232 Th to 208 Pb so that they would yield the same age for the rock? Ignore the minute amounts of the intermediate decay products in the decay chains. Note: This form of multiple dating gives reliable geological dates. • A 2.00-H inductor carries a steady current of 0.500 A. When the switch in the circuit is opened, the current is effectively A 2.00-H inductor carries a steady current of 0.500 A. When the switch in the circuit is opened, the current is effectively • Two nuclei having atomic numbers Z1Z1 and Z2Z2 approach each other with a total energy E.E. (a) When they are far apart, they interact only by electric repulsion. If they approach to a distance of 1.00×10−14m,1.00×10−14m, the nuclear force suddenly takes over to make them fuse. Find the minimum value of E,E, in terms of Z1Z1 and Z2,Z2, required to produce fusion. (b) State how EE depends on the atomic numbers. (c) If Z1+Z2Z1+Z2 is to have a certain target value such as 60,60, would it be energetically favorable to take Z1=1Z1=1 and Z2=59,Z2=59, or Z1=Z2=30,Z1=Z2=30, or some other choice? Explain your answer. (d) Evaluate from your expression the minimum energy for fusion for the D−DD−D and D−D− T reactions (the first and third reactions in Eq. 45.4).45.4). • A potential energy function for a system in which a two-dimensional force acts is of the form U=3x3y−7x . Find the force that acts at the point (x,y). • A solid sphere of brass (bulk modulus of$14.0 \times10^{10} \mathrm{N} / \mathrm{m}^{2} )$with a diameter of 3.00$\mathrm{m}$is thrown into the ocean. By how much does the diameter of the sphere decrease as it sinks to a depth of 1.0$\mathrm{km}$? • An object is placed 50.0 cm from a concave spherical mirror with focal length of magnitude 20.0 cm. (a) Find the location of the image. (b) What is the magnification of the image? (c) Is the image real or virtual? (d) Is the image upright or inverted • A uniform rod of mass 300 g and length 50.0 cm rotates in a horizontal plane about a fixed, friction less, vertical pin through its center. Two small, dense beads, each of mass m,m, are mounted on the rod so that they can slide without friction along its length. Initially, the beads are held by catches at positions 10.0 cm on each side of the center and the system is rotating at an angular speed of 36.0 rad/s. The catches are released simultaneously, and the beads slide outward along the rod. (a) Find an expression for the angular speed ωfωf of the system at the instant the beads slide off the ends of the rod as it depends on m.m. (b) What are the maximum and the minimum possible values for ωfωf and the values of mm to which they correspond? • A child of mass mm starts from rest and slides without friction from a height hh along a slide next to a pool (Fig. P8.27). She is launched from a height h/5h/5 into the air over the pool. We wish to find the maximum height she reaches above the water in her projectile motion. (a) Is the child-Earth system isolated or nonisolated? Why? (b) Is there a nonconservative force acting within the system? (c) Define the configuration of the system when the child is at the water level as having zero gravitational potential energy. Express the total energy of the system when the child is at the top of the water slide. (d) Express the total energy of the system when the child is at the launching point. (e) Express the total energy of the system when the child is at the highest point in her projectile motion. (f) From parts (c) and (d), determine her initial speed vivi at the launch point in terms of gg and h.h. (g) From parts (d), (e),(e), and (f),(f), determine her maximum airborne height ymax in terms ofhand the launch angleθ.(h) Would yourymax in terms ofhand the launch angleθ.(h) Would your answers be the same if the waterslide were not frictionless? Explain. • After a ball rolls off the edge of a horizontal table at time t=0,t=0, its velocity as a function of time is given by v→=1.2i^−9.8tjv→=1.2i^−9.8tj where v→v→ is in meters per second and tt is in seconds. The ball’s displacement away from the edge of the table, during the time interval of 0.380 s for which the ball is in flight, is given by Δr→=∫0.380s0v→dtΔr→=∫00.380sv→dt To perform the integral, you can use the calculus theorem ∫[A+Bf(x)]dx=∫Adx+B∫f(x)dx∫[A+Bf(x)]dx=∫Adx+B∫f(x)dx You can think of the units and unit vectors as constants, represented by AA and B.B. Perform the integration to calculate the displacement of the ball from the edge of the table at 0.380 s. • A snowmobile is originally at the point with position vector 29.0 m at 95.0∘ counterclockwise from the x axis, moving with velocity 4.50 m/s at 40.0∘. It moves with constant acceleration 1.90 m/s2 at 200∘. After 5.00 s have elapsed, find (a) its velocity and (b) its position vector. • Two converging lenses having focal lengths of$f_{1}=10.0 \mathrm{cm}$and$f_{2}=20.0 \mathrm{cm}$are placed a distance$d=50.0 \mathrm{cm}$apart as shown in Figure$\mathrm{P} 36.64$. The image due to light passing through both lenses is to be located between the lenses at the position$x=31.0 \mathrm{cm}$indicated. (a) At what value of$p$p should the object be positioned to the left of the first lens? (b) What is the magnification of the final image? (c) Is the final image upright or inverted? (d) Is the final image real or virtual? • A Σ0Σ0 particle traveling through matter strikes a proton; then a Σ+Σ+ and a gamma ray as well as a third particle Use the quark model of each to determine the identity of the third particle. • An electric field in a region of space is parallel to the xx axis. The electric potential varies with position as shown in Figure P25.36P25.36 . Graph the xx component of the electric field versus position in this region of space. • Two infinitely long solenoids (seen in cross section) pass through a circuit as shown in Figure P31.71. The magnitude of →B inside each is the same and is increasing at the rate of 100 T/s . What is the current in each resistor? • A shaft is turning at angular speed ωω at time t=0t=0 Thereafter, its angular acceleration is given by α=A+Btα=A+Bt (a) Find the angular speed of the shaft at time tt (b) Through what angle does it turn between t=0t=0 and t?t? • Show that the rate of change of the free-fall acceleration with vertical position near the Earth’s surface is dgdr=−2GMER3Edgdr=−2GMERE3 This rate of change with position is called a gradient. (b) Assuming hh is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between two points separated by vertical distance hh is |Δg|=2GMEhR3E|Δg|=2GMEhRE3 (c) Evaluate this difference for h=6.00m,h=6.00m, a typical height for a two-story building. • A block slides down a frictionless plane having an inclination of θ=15.0∘.θ=15.0∘. The block starts from rest at the top, and the length of the incline is 2.00 mm . (a) Draw a free-body diagram of the block. Find (b) the acceleration of the block and (c)(c) its speed when it reaches the bottom of the incline. • An enemy ship is on the east side of a mountain island as shown in Figure P4.71P4.71 . The enemy ship has maneuvered to within 2500 mm of the 1800 -m-high mountain peak and can shoot projectiles with an initial speed of 250 m/sm/s . If the western shoreline is horizontally 300 mm from the peak, what are the distances from the western shore at which a ship can be safe from the bombardment of the enemy ship? • The wave function for a traveling wave on a taut string is (in SI units) y(x,t)=0.350sin(10πt−3πx+π4)y(x,t)=0.350sin⁡(10πt−3πx+π4) (a) What are the speed and direction of travel of the wave? (b) What is the vertical position of an element of the string at t=0,x=0.100mt=0,x=0.100m ? What are (c)(c) the wavelength and (d) the frequency of the wave? (e) What is the maximum transverse speed of an element of the string? • In an experiment designed to measure the Earth’s magnetic field using the Hall effect, a copper bar 0.500 cm thick is positioned along an east–west direction. Assume n=8.46×1028n=8.46×1028 electrons /m3/m3 and the plane of the bar is rotated to be perpendicular to the direction of →BB→ . If a current of 8.00 AA in the conductor results in a Hall voltage of 5.10×10−12V,5.10×10−12V, what is the magnitude of the Earth’s magnetic field at this location? • A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as in Figure P15.23. A horizontal force F S causes the spring to stretch a distance of 5.46 cm from its equilibrium position. (a) Find the magnitude of F S. (b) What is the total energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block just after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, would your answer to part (d) be larger or smaller? (f) What other information would you need to know to find the actual answer to part (d) in this case? (g) What is the largest value of the coefficient of friction that would allow the block to reach the equilibrium position? • A small ball of mass M is attached to the end of a uniform rod of equal mass M and length L that is pivoted at the top (Fig. P15.51). Determine the tensions in the rod (a) at the pivot and (b) at the point P when the system is stationary. (c) Calculate the period of oscillation for small displacements from equilibrium and (d) determine this period for L 5 2.00 m. • The circuit shown in Figure P28.72P28.72 is set up in the laboratory to measure an unknown capacitance CinCin series with a resistance R=10.0MΩR=10.0MΩ powered by a battery whose emf is 6.19 VV . The data given in the table are the measured volt- ages across the capacitor as a function of time, where t=t= 0 represents the instant at which the switch is thrown to position bb . (a) Construct a graph of ln(E/ΔV)ln⁡(E/ΔV) versus tt and perform a linear least-squares fit to the data. (b) From the slope of your graph, obtain a value for the time constant of the circuit and a value for the capacitance. • A proton is at rest at the plane boundary of a region containing a uniform magnetic field BB (Fig. P 29.59). An alpha particle moving horizontally makes a head-on elastic collision with the proton. Immediately after the collision, both particles enter the magnetic field, moving perpendicular to the direction of the field. The radius of the proton’s trajectory is R.R. The mass of the alpha particle is four times that of the proton, and its charge is twice that of the proton. Find the radius of the alpha particle’s trajectory. • For the circuit shown in Figure P33.10,ΔVmax=80.0V,ω=P33.10,ΔVmax=80.0V,ω= 0πrad/s,65.0πrad/s, and L=70.0mH.L=70.0mH. Calculate the current in the inductor at t=15.5mst=15.5ms • The cost of energy delivered to residences by electrical transmission varies from$0.070/kWh$0.070/kWh to$0.258/kWh$0.258/kWh throughout the United States;$0.110/kWh$0.110/kWh is the average value. At this average price, calculate the cost of (a) leaving a 40.0−W40.0−W porch light on for two weeks while you are on vacation, (b) making a piece of dark toast in 3.00 minmin with a 970−W970−W toaster, and (c)(c) drying a load of clothes in 40.0 minmin in a 5.20×103−W5.20×103−W dryer. • A deep-space vehicle moves away from the Earth with a speed of 0.800 cc . An astronaut on the vehicle measures a time interval of 3.00 s to rotate her body through 1.00 rev as she floats in the vehicle. What time interval is required for this rotation according to an observer on the Earth? • A concave spherical mirror has a radius of curvature of magnitude 20.0 cm. (a) Find the location of the image for object distances of (i) 40.0 cm, (ii) 20.0 cm, and (iii) 10.0 cm. For each case, state whether the image is (b) real or virtual and (c) upright or inverted. (d) Find the magnification in each case. • A steel beam being used in the construction of a skyscraper has a length of 35.000 mm when delivered on a cold day at temperature of 15.000∘000∘F . What is the length of the beam when it is being installed later on a warm day when the temperature is 90.000∘F90.000∘F ? • Consider a black body of surface area 20.0 cm2cm2 and temperature 5000 KK (a) How much power does it radiate? (b) At what wavelength does it radiate most intensely? Find the spectral power per wavelength interval at (c) this wavelength and at wavelengths of (d) 1.00 nmnm (an xx – or gamma ray), (e) 5.00 nmnm (ultraviolet light or an x-ray), (f) 400 nmnm (at the boundary between UV and visible light), (g) 700 nmnm (at the boundary between visible and infrared light), (h) 1.00 mmmm (infrared light or a microwave), and (i) 10.0 cmcm (a microwave or radio wave). (j) Approximately how much power does the object radiate as visible light? • A 200 -kg load is hung on a wire of length 4.00m,4.00m, cross-sectional area 0.200×10−4m2,0.200×10−4m2, and Young’s modulus 8.00×1010N/m2.8.00×1010N/m2. What is its increase in length? • A cooking vessel on a slow burner contains 10.0 kgkg of water and an unknown mass of ice in equilibrium at 0∘C0∘C at time t=0.t=0. The temperature of the mixture is measured at various times, and the result is plotted in Figure P20.63P20.63 . During the first 50.0 min, the mixture remains at 0∘C0∘C . From 50.0 min to 60.0 minmin , the temperature increases to 2.00∘00∘C . Ignoring the heat capacity of the vessel, determine the initial mass of the ice. • The index of refraction for violet light in silica flint glass is nV,nV, and that for red light is nRnR . What is the angular spread of visible light passing through a prism of apex angle ΦΦ if the angle of incidence is θθ ? See Figure P35.33P35.33 . • Light traveling in a medium of index of refraction n1n1 is incident at an angle θθ on the surface of a medium of index n2.n2. The angle between reflected and refracted rays is β.β. Show that tanθ=n2sinβn1−n2cosβtan⁡θ=n2sin⁡βn1−n2cos⁡β (b) What If? Show that this expression for tan θθ reduces to Brewster’s law when β=90∘.β=90∘. • A light source recedes from an observer with a speed vsvs that is small compared with cc (a) Show that the fractional shift in the measured wavelength is given by the approximate expression Δλλ≈vscΔλλ≈vsc This phenomenon is known as the redshift because the visible light is shifted toward the red. (b) Spectroscopic measurements of light at λ=397λ=397 nm coming from a galaxy in Ursa Major reveal a redshift of 20.0 nmnm . What is the recessional speed of the galaxy? • A map suggests that Atlanta is 730 miles in a direction of 5.00° north of east from Dallas. The same map shows that Chicago is 560 miles in a direction of 21.0° west of north from Atlanta. Figure P 3.22 shows the locations of these three cities. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago. • A 12.0-V battery is connected into a series circuit containing a 10.0−Ω10.0−Ω resistor and a 2.00−H2.00−H inductor. In what time interval will the current reach (a) 50.0%% and (b) 90.0%% of its final value? • Unpolarized light passes through two ideal Polaroid sheets. The axis of the first is vertical, and the axis of the second is at 30.0° to the vertical. What fraction of the incident light is transmitted? • Potassium iodide (KI) has the same crystalline structure as NaCl, with atomic planes separated by 0.353 nm. A monochromatic x-ray beam shows a first-order diffraction maximum when the grazing angle is 7.60°. Calculate the x-ray wavelength. • The danger to the body from a high dose of gamma rays is not due to the amount of energy absorbed; rather, it is due to the ionizing nature of the radiation. As an illustration, calculate the rise in body temperature that results if a “lethal” dose of 1 000 rad is absorbed strictly as internal energy. Take the specific heat of living tissue as 4186 J/kg⋅∘CJ/kg⋅∘ • An object is placed 20.0 cm from a concave spherical mirror having a focal length of magnitude 40.0 cm. (a) Use graph paper to construct an accurate ray diagram for this situation. (b) From your ray diagram, determine the location of the image. (c) What is the magnification of the image? (d) Check your answers to parts (b) and (c) using the mirror equation. • A student holds a laser that emits light of wavelength 632.8 nmnm . The laser beam passes though a pair of slits separated by 0.300 mmmm , in a glass plate attached to the front of the laser. The beam then falls perpendicularly on a screen, creating an interference pattern on it. The student begins to walk directly toward the screen at 3.00 m/sm/s . The central maximum on the screen is stationary. Find the speed of the 50 th-order maxima on the screen. • In a choir practice room, two parallel walls are 5.30$\mathrm{m}$apart. The singers stand against the north wall. The organist faces the south wall, sitting 0.800$\mathrm{m}$away from it. To enable her to see the choir, a flat mirror 0.600$\mathrm{m}$wide is mounted on the south wall, straight in front of her. What width of the north wall can the organist see? Suggestion: Draw a top-view diagram to justify your answer. • In Figure P28.29,P28.29, find (a)(a) the current in each resistor and (b) the power delivered to each resistor. • Why is the following situation impossible? A photon of wave- length 88.0 nmnm strikes a clean aluminum surface, ejecting a photoelectron. The photoelectron then strikes a hydrogen atom in its ground state, transferring energy to it and exciting the atom to a higher quantum state. • Residential building codes typically require the use of 12-gauge copper wire (diameter 0.205 cm) for wiring receptacles. Such circuits carry currents as large as 20.0 A. If a wire of smaller diameter (with a higher gauge number) carried that much current, the wire could rise to a high temperature and cause a fire. (a) Calculate the rate at which internal energy is produced in 1.00 mm of 12 -gauge copper wire carrying 20.0 AA . (b) What If? Repeat the calculation for a 12 -gauge aluminum wire. (c) Explain whether a 12 -gauge aluminum wire would be as safe as a copper wire. • A quantum particle is in the n=1 state of an infinitely deep square well with walls at x=0 and x=L. Let ℓ be an arbitrary value of x between x=0 and x=L. (a) Find an expression for the probability, as a function of ℓ, that the particle will be found between x=0 and x=ℓ. (b) Sketch the probability as a function of the variable ℓ/L. Choose values of ℓ/L ranging from 0 to 1.00 in steps of 0.100. (c) Explain why the probability function must have particular values at ℓ/L=0 and at ℓ/L=1. (d) Find the value of ℓ for which the probability of finding the particle between x=0 and x=ℓ is twice the probability of finding the particle between x=ℓ and x=L. Suggestion: Solve the transcendental equation for ℓ/L numerically. • Assume three identical uncharged particles of mass mm and spin12spin⁡12 are contained in a one-dimensional box of length L. What is the ground-state energy of this system? • In a Young’s interference experiment, the two slits are separated by 0.150 mmmm and the incident light includes two wavelengths: λ1=540nmλ1=540nm (green) and λ2=450nmλ2=450nm (blue). The overlapping interference patterns are observed on a screen 1.40 mm from the slits. Calculate the minimum distance from the center of the screen to a point where a bright fringe of the green light coincides with a bright fringe of the blue light. • The current in a 4.00 mHmH -inductor varies in time as shown in Figure P32.8.P32.8. Construct a graph of the self-induced emf across the inductor over the time interval t=0t=0 to t=t= 12.0ms.12.0ms. • A long, straight metal rod has a radius of 5.00 cm and a charge per unit length of 30.0 nC/m. Find the electric field (a) 3.00 cm, (b) 10.0 cm, and (c) 100 cm from the axis of the rod, where distances are measured perpendicular to the rod’s axis. • A 4.00 -kg particle moves along the xx axis. Its position varies with time according to x=t+2.0t3,x=t+2.0t3, where xx is in meters and tt is in seconds. Find (a)(a) the kinetic energy of the particle at any time t,t, (b) the acceleration of the particle and the force acting on it at time t,(c)t,(c) the power being delivered to the particle at time t,t, and (d)(d) the work done on the particle in the interval t=0t=0 to t=2.00st=2.00s . • Two charged particles, Q1=Q1= +5.00nC+5.00nC and Q2=−3.00nC,Q2=−3.00nC, are separated by 35.0cm.35.0cm. (a) What is the electric potential energy of the pair? Explain the significance of the algebraic sign of your answer. (b) What is the electric potential at a point midway between the charged particles? • The intensity of a sound wave at a fixed distance from a speaker vibrating at 1.00 kHzkHz is 0.600W/m2.0.600W/m2. (a) Determine the intensity that results if the frequency is increased to 2.50 kHzkHz while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to 0.500 kHzkHz and the displacement amplitude is doubled. • For copper at 300 K, calculate the probability that a state with an energy equal to 99.0% of the Fermi energy is occupied. • A particle of mass mm moves in the xyxy plane with a velocity of →v=vxˆi+vyˆj.v→=vxi^+vyj^. Determine the angular momentum of the particle about the origin when its position vector is →r=xi+yj.r→=xi+yj. • Calculate the momentum of an electron moving with a speed of (a) 0.0100c,0.0100c, (b) 0.500c,0.500c, and (c)0.900c(c)0.900c • Taking R=1.00kΩR=1.00kΩ and ε=250Vε=250V in Figure P28.27P28.27 , determine the direction and magnitude of the current in the horizontal wire between aa and e.e. • Because of the Earth’s rotation, a plumb bob does not hang exactly along a line directed to the center of the Earth. How much does the plumb bob deviate from a radial line at 35.0∘0∘ north latitude? Assume the Earth is spherical. • Three displacements are →A=200mA→=200m due south, →B=250mB→=250m due west, and →C=150mC→=150m at 30.0∘0∘ east of north. (a) Construct a separate diagram for each of the following possible ways of adding these vectors: ¯R1=→A+→B+→CR¯¯¯¯1=A→+B→+C→ ; →R2=→B+→C+→A;→R3=→C+→B+→A.R→2=B→+C→+A→;R→3=C→+B→+A→. (b) Explain what you can conclude from comparing the diagrams. • Use Kepler’s third law to determine how many days it takes a spacecraft to travel in an elliptical orbit from a point 6 670 km from the Earth’s center to the Moon, 385 000 km from the Earth’s center. • X-ray pulses from Cygnus X-1, the first black hole to be identified and a celestial x-ray source, have been recorded during high-altitude rocket flights. The signals can be interpreted as originating when a blob of ionized matter orbits a black hole with a period of 5.0 ms. If the blob is in a circular orbit about a black hole whose mass is 20MSun,20MSun, what is the orbit radius? • One method of producing neutrons for experimental use is bombardment of light nuclei with alpha particles. In the method used by James Chadwick in 1932, alpha particles emitted by polonium are incident on beryllium nuclei: 42He+94Be→126C+10n What is the Q value of this reaction? (b) Neutrons are also often produced by small-particle accelerators. In one design, deuterons accelerated in a Van de Graaff generator bombard other deuterium nuclei and cause the reaction 21H+21H→32He+10n Calculate the Q value of the reaction. (c) Is the reaction in part (b) exothermic or endothermic? • Natural gold has only one isotope, 197 Au . If natural gold is irradiated by a flux of slow neutrons, electrons are emitted. (a) Write the reaction equation. (b) Calculate the maximum energy of the emitted electrons. • A capacitor in a series LC circuit has an initial charge Q and is being discharged. When the charge on the capacitor is Q/2,Q/2, find the flux through each of the NN turns in the coil of the inductor in terms of Q,N,L,Q,N,L, and C.C. • A horizontal laser beam of wavelength 632.8 nm has a circular cross section 2.00 mm in diameter. A rectangular aperture is to be placed in the center of the beam so that when the light falls perpendicularly on a wall 4.50 m away, the central maximum fills a rectangle 110 mm wide and 6.00 mm high. The dimensions are measured between the minima bracketing the central maximum. Find the required (a) width and (b) height of the aperture. (c) Is the longer dimension of the central bright patch in the diffraction pattern horizontal or vertical? (d) Is the longer dimension of the aperture horizontal or vertical? (e) Explain the relationship between these two rectangles, using a diagram. • A small object of mass m carries a charge q and is suspended by a thread between the vertical plates of a parallel-plate capacitor. The plate separation is d. If the thread makes an angle θθ with the vertical, what is the potential difference between the plates? • Use Stefan’s law to find the intensity of the cosmic background radiation emitted by the fireball of the big bang at a temperature of 2.73 KK . • A uniformly charged ring of radius 10.0 cm has a total charge of 75.0μCμC . Find the electric field on the axis of the ring at (a) 1.00cm,1.00cm, (b) 5.00cm,(c)30.0cm,5.00cm,(c)30.0cm, and (d)100cm(d)100cm from the center of the ring. • A flat, circular coil does not actually produce a uniform magnetic field in the area it encloses. Nevertheless, estimate the inductance of a flat, compact, circular coil with radius RR and NN turns by assuming the field at its center is uniform over its area. (b) A circuit on a laboratory table consists of a 1.50 -volt battery, a 270−Ω270−Ω resistor, a switch, and three 30.0 -cm-long patch connecting them. Suppose the circuit is arranged to be circular. Think of it as a flat coil with one turn. Compute the order of magnitude of its inductance and (c)(c) of the time constant describing how fast the current increases when you close the switch. • A coil of 15 turns and radius 10.0 cmcm surrounds a long solenoid of radius 2.00 cmcm and 1.00×1031.00×103 turns/meter (Fig. P31.12). The current in the solenoid changes as I=I= 5.00 sin 120t,120t, where II is in amperes and tt is in seconds. Find the induced emf in the 15 -turn coil as a function of time. • Model a penny as 3.10 gg of pure copper. Consider an anti-penny minted from 3.10 gg of copper anti-atoms, each with 29 positrons in orbit around a nucleus comprising 29 anti- protons and 34 or 36 antineutrons. (a) Find the energy released if the two coins collide. (b) Find the value of this energy at the unit price of$0.11/kWh,$0.11/kWh, a representative retail rate for energy from the electric company. • Consider a conical pendulum (Fig. P6.8) with a bob of mass m=80.0kgm=80.0kg on a string of length L=10.0mL=10.0m that makes an angle of θ=5.00∘θ=5.00∘ with the vertical. Determine (a) the horizontal and vertical components of the force exerted by the string on the pendulum and (b) the radial acceleration of the bob. • A crate of mass 10.0 kgkg is pulled up a rough incline with an initial speed of 1.50 m/sm/s . The pulling force is 100 NN parallel to the incline, which makes an angle of 20.0∘0∘ with the horizontal. The coefficient of kinetic friction is 0.400 , and the crate is pulled 5.00m.5.00m. (a) How much work is done by the gravitational force on the crate? (b) Determine the increase in internal energy of the crate-incline system owing to friction. (c) How much work is done by the 100−N100−N force on the crate? (d) What is the change in kinetic energy of the crate? (e) What is the speed of the crate after being pulled 5.00 mm ? • Draw motion diagrams for (a) an object moving to the right at constant speed, (b) an object moving to the right and speeding up at a constant rate, (c) an object moving to the right and slowing down at a constant rate, (d) an object moving to the left and speeding up at a constant rate, and (e) an object moving to the left and slowing down at a constant rate. (f) How would your drawings change if the changes in speed were not uniform, that is, if the speed were not changing at a constant rate? • A sphere of radius 2aa is made of a nonconducting material that has a uniform volume charge density ρ.ρ. Assume the material does not affect the electric field. A spherical cavity of radius aa is now removed from the sphere as shown in Figure P 24.60. Show that the electric field within the cavity is uniform and is given by Ex=0Ex=0 and Ey=ρa/3ϵ0.Ey=ρa/3ϵ0. • Shannon observes two light pulses to be emitted from the same location, but separated in time by 3.00μμ s. Kimmie observes the emission of the same two pulses to be separated in time by 9.00μμ s. (a) How fast is Kimmie moving relative to Shannon? (b) According to Kimmie, what is the separation in space of the two pulses? • Show that baryon number and charge are conserved in the following reactions of a pion with a proton: (1)π++p→K++Σ+(2)π++p→π++Σ+(b) The first reaction is observed, but the second neveroccurs. Explain.(1)π++p→K++Σ+(2)π++p→π++Σ+(b) The first reaction is observed, but the second neveroccurs. Explain. • Figure P2.35 represents part of the performance data of a car owned by a proud physics student. (a) Calculate the total distance traveled by computing the area under the red-brown graph line. (b) What distance does the car travel between the times$t=10 \mathrm{s}$and$t=40 \mathrm{s}$? (c) Draw a graph of its acceleration versus time between$t=$0 and$t=50 \mathrm{s}$. (d) Write an equation for$x$as a function of time for each phase of the motion, represented by the segments$0 a, a b,$and$b c$(e) What is the average velocity of the car between$t=0$and$t=50 \mathrm{s}$? • A 1.00 -mol sample of hydrogen gas is heated at constant pressure from 300 KK to 420 KK . Calculate (a) the energy transferred to the gas by heat, (b) the increase in its internal energy, and (c) the work done on the gas. • The force on a magnetic moment μzμz in a nonuniform magnetic field BzBz is given by Fz=μz(dBz/dz).Fz=μz(dBz/dz). If a beam of silver atoms travels a horizontal distance of 1.00 mm through such a field and each atom has a speed of 100 m/sm/s , how strong must be the field gradient dBz/dzdBz/dz to deflect the beam 1.00 mmmm ? • The number N of atoms in a particular state is called the population of that state. This number depends on the energy of that state and the temperature. In thermal equilibrium, the population of atoms in a state of energy En is given by a Boltzmann distribution expression N=Nge−(En−Eg)/kBTN=Nge−(En−Eg)/kBT where NgNg is the population of the ground state of energy Eg,kBEg,kB is Boltzmann’s constant, and TT is the absolute temperature. For simplicity, assume each energy level has only one quantum state associated with it. (a) Before the power is switched on, the neon atoms in a laser are in thermal equilibrium at 27.0∘0∘C . Find the equilibrium ratio of the populations of the states E4∗E4∗ and E3E3 shown for the red transition in Figure P42.54P42.54 . Lasers operate by a clever artificial production of a “population inversion” between the upper and lower atomic energy states involved in the lasing transition. This term means that more atoms are in the upper excited state than in the lower one. Consider the E∗4−E3E∗4−E3 transition in Figure P42.54P42.54 . Assume 2%% more atoms occur in the upper state than in the lower. (b) To demonstrate how unnatural such a situation is, find the temperature for which the Boltzmann distribution describes a 2.00%% population inversion. (c) Why does such a situation not occur naturally? • Determine the minimum height of a vertical flat mirror in which a person 178$\mathrm{cm}$tall can see his or her full image. Suggestion: Drawing a ray diagram would be helpful. • A certain uniform string is held under constant tension. (a) Draw a side-view snapshot of a sinusoidal wave on a string as shown in diagrams in the text. (b) Immediately below diagram (a), draw the same wave at a moment later by one-quarter of the period of the wave. (c) Then, draw a wave with an amplitude 1.5 times larger than the wave in diagram (a). (d) Next, draw a wave differing from the one in your diagram (a) just by having a wavelength 1.5 times larger. (e) Finally, draw a wave differing from that in diagram (a) just by having a frequency 1.5 times larger. • A proton travels with a speed of 5.02×106m/s5.02×106m/s in a direction that makes an angle of 60.0∘0∘ with the direction of a magnetic field of magnitude 0.180 TT in the positive xx direction. What are the magnitudes of (a) the magnetic force on the proton and (b) the proton’s acceleration? • A 42.0 -kg boy uses a solid block of Styrofoam as a raft while fishing on a pond. The Styrofoam has an area of 1.00$\mathrm{m}^{2}$and is 0.0500$\mathrm{m}$thick. While sitting on the surface of the raft, the boy finds that the raft just supports him so that the top of the raft is at the level of the pond. Determine the density of the Styrofoam. • Our discussion of the techniques for determing constructive and destructive interference by reflection from a thin film in air has been confined to rays striking the film at nearly normal incidence. What If? Assume a ray is incident at an angle of 30.0∘0∘ (relative to the normal) on a film with index of refraction 1.38 surrounded by vacuum. Calculate the minimum thickness for constructive interference of sodium light with a wavelength of 590 nmnm . • Two blocks are free to slide along the frictionless, wooden track shown in Figure P9.27P9.27 . The block of mass m1=m1= 5.00 kgkg is released from the position shown, at height h=h= 5.00 mm above the flat part of the track. Protruding from its front end is the north pole of a strong magnet, which repels the north pole of an identical magnet embedded in the back end of the block of mass m2=10.0kgm2=10.0kg , initially at rest. The two blocks never touch. Calculate the maximum height to which m1m1 rises after the elastic collision. • An ambulance moving at 42 m/s sounds its siren whose frequency is 450 Hz. A car is moving in the same direction as the ambulance at 25 m/s. What frequency does a person in the car hear (a) as the ambulance approaches the car? (b) After the ambulance passes the car? • An athlete whose mass is 70.0 kg drinks 16.0 ounces (454 g) of refrigerated water. The water is at a temperature of 35.0∘0∘F . ( a) Ignoring the temperature change of the body that results from the water intake (so that the body is regarded as a reservoir always at 98.6∘F)98.6∘F) , find the entropy increase of the entire system. (b) What If? Assume the entire body is cooled by the drink and the average specific heat of a person is equal to the specific heat of liquid water. Ignoring any other energy transfers by heat and any metabolic energy release, find the athlete’s temperature after she drinks the cold water given an initial body temperature of 98.6aF.98.6aF. (c) Under these assumptions, what is the entropy increase of the entire system? (d) State how this result compares with the one you obtained in part (a). • Why is the following situation impossible? Manny Ramírez hits a home run so that the baseball just clears the top row of bleachers, 24.0 mm high, located 130 mm from home plate. The ball is hit at 41.7 m/sm/s at an angle of 35.0∘0∘ to the horizontal, and air resistance is negligible. • Three metal rods are located relative to each other as shown in Figure P17.62, where L1+L1+ L2=L3.L2=L3. The speed of sound in a rod is given by v=√Y/ρ,v=Y/ρ−−−−√, where YY is Young’s modulus for the rod and ρρ is the density. Values of density and Young’s modulus for the three materials are ρ1=2.70×103kg/m3,Y1=7.00×1010N/m2,ρ2=11.3×ρ1=2.70×103kg/m3,Y1=7.00×1010N/m2,ρ2=11.3× 103kg/m3,Y2=1.60×1010N/m2,ρ3=8.80×103kg/m3103kg/m3,Y2=1.60×1010N/m2,ρ3=8.80×103kg/m3 Y3=11.0×1010N/m2.Y3=11.0×1010N/m2. If L3=1.50m,L3=1.50m, what must the ratio L1/L2L1/L2 be if a sound wave is to travel the length of rods 1 and 2 in the same time interval required for the wave to travel the length of rod 3?? • A quantum particle of mass m is placed in a one-dimensional box of length L. Assume the box is so small that the particle’s motion is relativistic and K=p2/2m is not valid. (a) Derive an expression for the kinetic energy levels of the particle. (b) Assume the particle is an electron in a box of length L=1.00×10−12m . Find its lowest possible kinetic energy. (c) By what percent is the nonrelativistic equation in error? Suggestion: See Equation 39.23 . • An airplane of mass 1.50×104kg1.50×104kg is in level flight, initially moving at 60.0 m/sm/s . The resistive force exerted by air on the airplane has a magnitude of 4.0×104N.4.0×104N. By Newton’s third law, if the engines exert a force on the exhaust gases to expel them out of the back of the engine, the exhaust gases exert a force on the engines in the direction of the airplane’s travel. This force is called thrust, and the value of the thrust in this situation is 7.50×104N7.50×104N . (a) Is the work done by the exhaust gases on the airplane during some time interval equal to the change in the airplane’s kinetic energy? Explain. (b) Find the speed of the airplane after it has traveled 5.0×102m.5.0×102m. • On a single, light, vertical cable that does not stretch, a crane is lifting a 1 207-kg Ferrari and, below it, a 1 461-kg BMW Z8. The Ferrari is moving upward with speed 3.50 m/s and acceleration 1.25 m/s2m/s2 . (a) How do the velocity and acceleration of the BMW compare with those of the Ferrari? (b) Find the tension in the cable between the BMW and the Ferrari. (c) Find the tension in the cable above the Ferrari. • In a Rutherford scattering experiment, alpha particles having kinetic energy of 7.70 MeV are fired toward a gold nucleus that remains at rest during the collision. The alpha particles come as close as 29.5 fm to the gold nucleus before turning around. (a) Calculate the de Broglie wavelength for the 7.70-MeV alpha particle and compare it with the distance of closest approach, 29.5 fm. (b) Based on this comparison, why is it proper to treat the alpha particle as a particle and not as a wave in the Rutherford scattering experiment? • Protons having a kinetic energy of 5.00 MeV(1eV=1.60×10−19J)MeV(1eV=1.60×10−19J) are moving in the positive xx direction and enter a magnetic field B→=0.0500k^B→=0.0500k^ T directed out of the plane of the page and extending from x=0x=0 to x=1.00mx=1.00m as shown in Figure P 29.74. (a) Ignoring relativistic effects, find the angle a between the initial velocity vector of the proton beam and the velocity vector after the beam emerges from the field. (b) Calculate the yy component of the protons’ momenta as they leave the magnetic field. • A certain telescope has an objective mirror with an aperture diameter of 200 mm and a focal length of 2 000 mm. It captures the image of a nebula on photographic film at its prime focus with an exposure time of 1.50 min. To produce the same light energy per unit area on the film, what is the required exposure time to photograph the same nebula with a smaller telescope that has an objective with a 60.0-mm diameter and a 900-mm focal length? • A 65.0-kg boy and his 40.0-kg sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with velocity 2.90 m/s toward the west. Ignore friction. (a) Describe the subsequent motion of the girl. (b) How much potential energy in the girl’s body is converted into mechanical energy of the boy–girl system? (c) Is the momentum of the boy–girl system conserved in the pushing-apart process? If so, explain how that is possible considering (d) there are large forces acting and (e) there is no motion beforehand and plenty of motion afterward. • A magnetized sewing needle has a magnetic moment of 9.70mA⋅9.70mA⋅m2. At its location, the Earth’s magnetic field is 55.0μTμT northward at 48.0∘48.0∘ below the horizontal. Identify the orientations of the needle that represent (a) the minimum potential energy and (b) the maximum potential energy of the needle-field system. (c) How much work must be done on the system to move the needle from the minimum to the maximum potential energy orientation? • A supersonic jet traveling at Mach 3.00 at an altitude of h=20000mh=20000m is directly over a person at time t=0t=0 as shown in Figure P17.45.P17.45. Assume the average speed of sound in air is 335 m/sm/s over the path of the sound. (a) At what time will the person encounter the shock wave due to the sound emitted at t=0?t=0? (b) Where will the plane be when this shock wave is heard? • In Example 5.7, we pushed on two blocks on a table. Suppose three blocks are in contact with one another on a frictionless, horizontal surface as shown in Figure P5.59.P5.59. A horizontal force →FF→ is applied to m1.m1. Take m1=2.00kgm1=2.00kg , m2=3.00kg,m3=4.00kg,m2=3.00kg,m3=4.00kg, and F=18.0N.F=18.0N. (a) Draw a separate free-body diagram for each block. (b) Determine the acceleration of the blocks. (c) Find the resultant force on each block. (d) Find the magnitudes of the contact forces between the blocks. (e) You are working on a construction project. A coworker is nailing up plasterboard on one side of a light partition, and you are on the opposite side, providing “backing” by leaning against the wall with your back pushing on it. Every hammer blow makes your back sting. The supervisor helps you put a heavy block of wood between the wall and your back. Using the situation analyzed in parts (a) through (d) as a model, explain how this change works to make your job more comfortable. • If a loop of chain is spun at high speed, it can roll along the ground like a circular hoop without collapsing. Consider a chain of uniform linear mass density μμ whose center of mass travels to the right at a high speed v0v0 as shown in Figure P16.65P16.65 . (a) Determine the tension in the chain in terms of μμ and v0v0 . Assume the weight of an individual link is negligible compared to the tension. (b) If the loop rolls over a small bump, the resulting deformation of the chain causes two transverse pulses to propagate along the chain, one moving clockwise and one moving counterclockwise. What is the speed of the pulses traveling along the chain? (c) Through what angle does each pulse travel during the time interval over which the loop makes one revolution? • A boy throws a stone horizontally from the top of a cliff of height hh toward the ocean below. The stone strikes the ocean at distance dd from the base of the cliff. In terms of h,d,h,d, and g,g, find expressions for (a) the time tt at which the stone lands in the ocean, (b) the initial speed of the stone, (c) the speed of the stone immediately before it reaches the ocean, and (d) the direction of the stone’s velocity immediately before it reaches the ocean. • In 1911, Ernest Rutherford and his assistants Geiger and Marsden conducted an experiment in which they scattered alpha particles (nuclei of helium atoms) from thin sheets of gold. An alpha particle, having charge +2e+2e and mass 6.64×10−27kg6.64×10−27kg , is a product of certain radioactive decays. The results of the experiment led Rutherford to the idea that most of an atom’s mass is in a very small nucleus, with electrons in orbit around it. (This is the planetary model of the atom, which we’ll study in Chapter 42.)42.) Assume an alpha particle, initially very far from a stationary gold nucleus, is fired with a velocity of 2.00×107m/s2.00×107m/s directly toward the nucleus (charge +79e).+79e). What is the smallest distance between the alpha particle and the nucleus before the alpha particle reverses direction? Assume the gold nucleus remains stationary. • A spherical conductor has a radius of 14.0 cmcm and a charge of 26.0μCμC . Calculate the electric field and the electric potential at (a)r=10.0cm,(b)r=20.0cm,(a)r=10.0cm,(b)r=20.0cm, and (c)r=(c)r= 14.0 cmcm from the center. • A 30.0 -kg hammer, moving with speed 20.0 m/sm/s , strikes a steel spike 2.30 cmcm in diameter. The hammer rebounds with speed 10.0 m/sm/s after 0.110 s. What is the average strain in the spike during the impact? • Figure P37.57P37.57 shows a radio-wave transmitter and a receiver separated by a distance dd and both a distance hh above the ground. The receiver can receive signals both directly from the transmitter and indirectly from signals that reflect from the ground. Assume the ground is level between the transmitter and receiver and a 180∘180∘ phase shift occurs upon reflection. Determine the longest wavelengths that interfere (a) constructively and (b) destructively. • At t=0t=0 , the open switch in Figure P32.66P32.66 is thrown closed. We wish to find a symbolic expression for the current in the inductor for time t>0.t>0. Let this current be called II and choose it to be downward in the inductor in Figure P32.66P32.66 . Identify I1I1 as the current to the right through R1R1 and I2I2 as the current downward through R2R2 . (a) Use Kirchhoff’s junction rule to find a relation among the three currents. (b) Use Kirchhoff’s loop rule around the left loop to find another relationship. (c) Use Kirchhoff’s loop rule around the outer loop to find a third relationship. (d) Eliminate IandI2IandI2 among the three equations to find an equation involving only the current II . (e) Compare the equation in part (d) with Equation 32.6 in the text. Use this comparison to rewrite Equation 32.7 in the text for the situation in this problem and show that I(t)=εR1[1−e−(R/I)t]whereR′=R1R2/(R1+R2)I(t)=εR1[1−e−(R/I)t]whereR′=R1R2/(R1+R2) • A taut rope has a mass of 0.180 kgkg and a length of 3.60 mm . What power must be supplied to the rope so as to generate sinusoidal waves having an amplitude of 0.100 mm and a wavelength of 0.500 mm and traveling with a speed of 30.0 m/sm/s ? • The rectangle shown in Figure P 3.54 has sides parallel to the xx and yy axes. The position vectors of two corners are A→=10.0mA→=10.0m at 50.0∘0∘ and B→=B→= 12.0 mm at 30.0∘.30.0∘. (a) Find the perimeter of the rectangle. (b) Find the magnitude and direction of the vector from the origin to the upper-right corner of the rectangle. • On the PV diagram for an ideal gas, one isothermal curve and one adiabatic curve pass through each point as shown in Figure P21.66 . Prove that the slope of the adiabatic curve is steeper than the slope of the isotherm at that point by the factor γ • Suppose your portable DVD player draws a current of 350 mAmA at 6.00 VV . How much power does the player require? • An inquisitive physics student and mountain climber climbs a 50.0-m-high cliff that overhangs a calm pool of water. He throws two stones vertically downward, 1.00 s apart, and observes that they cause a single splash. The first stone has an initial speed of 2.00$\mathrm{m} / \mathrm{s}$. (a) How long after release of the first stone do the two stones hit the water? (b) What initial velocity must the second stone have if the two stones are to hit the water simultaneously? (c) What is the speed of each stone at the instant the two stones hit the water? • For any two vectors →AA→ and →B→ show that →A⋅→B=AxBx+ AyBy+AzBz. Suggestions: Write →A and →B in unit-vector form and use Equations 7.4 and 7.5. • A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find (a) the period of its motion, (b) the frequency in hertz, and (c) the angular frequency in radians per second. • Steam at 100°C is added to ice at 0°C. (a) Find the amount of ice melted and the final temperature when the mass of steam is 10.0 g and the mass of ice is 50.0 g. (b) What If? Repeat when the mass of steam is 1.00 g and the mass of ice is 50.0 g. • A fisherman sets out upstream on a river. His small boat, powered by an outboard motor, travels at a constant speed vv in still water. The water flows at a lower constant speed vwvw . The fisherman has traveled upstream for 2.00 kmkm when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another 15.0 minmin . At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as he returns to his starting point. How fast is the river flowing? Solve this problem in two ways. (a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed v−vwv−vw and downstream at v+vw∗v+vw∗ (b) AA second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets. • The angular momentum vector of a precessing gyroscope sweeps out a cone as shown in Figure P 11.43. The angular speed of the tip of the angular momentum vector, called its precessional frequency, is given by ωp=τ/L,ωp=τ/L, where ττ is the magnitude of the torque on the gyroscope and LL is the magnitude of its angular momentum. In the motion called precession of the equinoxes, the Earth’s axis of rotation precesses about the perpendicular to its orbital plane with a period of 2.58×1042.58×104 yr. Model the Earth as a uniform sphere and calculate the torque on the Earth that is causing this precession. • A potter’s wheel-a thick stone disk of radius 0.500 mm and mass 100kg−100kg− is freely rotating at 50.0 rev/min. The potter can stop the wheel in 6.00 s by pressing a wet rag against the rim and exerting a radially inward force of 70.0 NN . Find the effective coefficient of kinetic friction between wheel and rag. • A parallel-plate capacitor with vacuum between its horizontal plates has a capacitance of 25.0μFμF . A nonconducting liquid with dielectric constant 6.50 is poured into the space between the plates, filling up a fraction ff of its volume. (a) Find the new capacitance as a function of ff (b) What should you expect the capacitance to be when f=0f=0 ? Does your expression from part (a) agree with your answer? (c) What capacitance should you expect when f=1?f=1? Does the expression from part (a) agree with your answer? • A straight, cylindrical wire lying along the xx axis has a length of 0.500 mm and a diameter of 0.200mm.0.200mm. It is made of a material described by Ohm’s law with a resistivity of ρ=4.00×10−8Ω⋅ρ=4.00×10−8Ω⋅m. Assume a potential of 4.00 VV is maintained at the left end of the wire at x=0.x=0. Also assume V=0V=0 at x=0.500mx=0.500m . Find (a)(a) the magnitude and direction of the electric field in the wire, (b) the resistance of the wire, (c) the magnitude and direction of the electric current in the wire, and (d) the current density in the wire. (e) Show that E=ρJE=ρJ. • Consider the circuit shown in Figure P26.24,P26.24, where C1=C1= 6.00μF,C2=3.00μF,6.00μF,C2=3.00μF, and ΔV=20.0VΔV=20.0V . Capacitor C1C1 is first charged by closing switch S1S1 . Switch S1S1 is then opened, and the charged capacitor is connected to the uncharged capacitor by closing S2S2 . Calculate (a) the initial charge acquired by C1C1 and (b)(b) the final charge on each capacitor. • Consider an electron near the Earth’s equator. In which direction does it tend to deflect if its velocity is (a) directed downward? (b) Directed northward? (c) Directed west-ward? (d) Directed southeastward? • Earthquakes at fault lines in the Earth’s crust create seismic waves, which are longitudinal (P waves) or transverse (S waves). The PP waves have a speed of about 7 km/skm/s . Estimate the average bulk modulus of the Earth’s crust given that the density of rock is about 2500 kg/m3kg/m3 . • In Figure P31.58 (page 924), the rolling axle, 1.50 m long, is pushed along horizontal rails at a constant speed v=3.00m/s. A resistor R=0.400Ω is connected to the rails at points a and b, directly opposite each other. The wheels make good electrical contact with the rails, so the axle, rails, and R form a closed-loop circuit. The only significant resistance in the circuit is R . A uniform magnetic field B=0.0800T is vertically downward. (a) Find the induced current I in the resistor. (b) What horizontal force F is required to keep the axle rolling at constant speed? (c) Which end of the resistor, a or b, is at the higher electric potential? (d) What If? After the axle rolls past the resistor, does the current in R reverse direction? Explain your answer. • A convex spherical mirror has a radius of curvature of magnitude 40.0 cm. Determine the position of the virtual image and the magnification for object distances of (a) 30.0 cm and (b) 60.0 cm. (c) Are the images in parts (a) and (b) upright or inverted? • Neutrons traveling at 0.400 m/sm/s are directed through a pair of slits separated by 1.00 mmmm . An array of detectors is placed 10.0 mm from the slits. (a) What is the de Broglie wavelength of the neutrons? (b) How far off axis is the first zero-intensity point on the detector array? (c) When a neutron reaches a detector, can we say which slit the neutron passed through? Explain. • A submarine is 300 m horizontally from the shore of a freshwater lake and 100 m beneath the surface of the water. A laser beam is sent from the submarine so that the beam strikes the surface of the water 210 m from the shore. A building stands on the shore, and the laser beam hits a target at the top of the building. The goal is to find the height of the target above sea level. (a) Draw a diagram of the situation, identifying the two triangles that are important in finding the solution. (b) Find the angle of incidence of the beam striking the water–air interface. (c) Find the angle of refraction. (d) What angle does the refracted beam make with the horizontal? (e) Find the height of the target above sea level. • Radio transmitter A operating at 60.0 MHzMHz is 10.0 mm from another similar transmitter BB that is 180∘180∘ out of phase with A. How far must an observer move from A toward B along the line connecting the two transmitters to reach the nearest point where the two beams are in phase? • A 0.001 60-nm photon scatters from a free electron. For what (photon) scattering angle does the recoiling electron have kinetic energy equal to the energy of the scattered photon? • A Carnot heat engine operates between temperatures ThTh and Tc∗Tc∗ (a) If Th=500KTh=500K and Tc=350K,Tc=350K, what is the efficiency of the engine? (b) What is the change in its efficiency for each degree of increase in ThTh above 500 KK ? (c) What is the change in its efficiency for each degree of change in Tc%Tc% (d) Does the answer to part (c) depend on T2cT2c Explain. • A concave spherical mirror forms an inverted image 4.00 times larger than the object. Assuming the distance between object and image is 0.600$\mathrm{m}, find the focal length of the mirror. (b) What If? Suppose the mirror is convex. The distance between the image and the object is the same as in part (a), but the image is 0.500 the size of the object. Determine the focal length of the mirror. • A certain lightbulb has a tungsten filament with a resistance of 19.0ΩΩ when at 20.0∘0∘C and 140ΩΩ when hot. Assume the resistivity of tungsten varies linearly with temperature even over the large temperature range involved here. Find the temperature of the hot filament. • Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest. A small block of mass m1=0.500kg is released from rest at the top of a frictionless, curve-shaped wedge of mass m2= 3.00 kg , which sits on a frictionless, horizontal surface as shown in Figure P9.66a . When the block leaves the wedge, its velocity is measured to be 4.00 m/s to the right as shown in Figure P9.66b . (a) What is the velocity of the wedge after the block reaches the horizontal surface? (b) What is the height h of the wedge? • Consider the hemispherical closed surface in Figure P30.46. The hemisphere is in a uniform magnetic field that makes an angle θθ with the vertical. Calculate the magnetic flux through (a)(a) the flat surface S1S1 and (b)(b) the hemispherical surface S2.S2. • Two light pulses are emitted simultaneously from a source. Both pulses travel through the same total length of air to a detector, but mirrors shunt one pulse along a path that carries it through an extra length of 6.20 mm of ice along the way. Determine the difference in the pulses’ times of arrival at the detector. • An ice tray contains 500 g of liquid water at 0∘C0∘C . Calculate the change in entropy of the water as it freezes slowly and completely at 0∘0∘C. • Figure P 11.17 represents a small, flat puck with mass m=2.40kgm=2.40kg sliding on a friction less, horizontal surface. It is held in a circular orbit about a fixed axis by a rod with negligible mass and length R=1.50m,R=1.50m, pivoted at one end. Initially, the puck has a speed of v=5.00m/sv=5.00m/s . A 1.30−kg1.30−kg ball of putty is dropped vertically onto the puck from a small distance above it and immediately sticks to the puck. (a) What is the new period of rotation? (b) Is the angular momentum of the puck–putty system about the axis of rotation constant in this process? (c) Is the momentum of the system constant in the process of the putty sticking to the puck? (d) Is the mechanical energy of the system constant in the process? • When a battery is connected to the plates of a 3.00−μF3.00−μF capacitor, it stores a charge of 27.0μCμC . What is the voltage of the battery? (b) If the same capacitor is connected to another battery and 36.0μCμC of charge is stored on the capacitor, what is the voltage of the battery? • In the circuit of Figure P28.30,P28.30, the current I1=I1= 3.00 AA and the values of εε for the ideal battery and RR are unknown. What are the currents (a) I2I2 and (b)I3?(c)(b)I3?(c) Can you find the values of εε and RR ? If so, find their values. If not, explain. • A string is wound around a uniform disk of radius RR and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P10.73). Show that (a) the tension in the string is one third of the weight of the disk, (b) the magnitude of the acceleration of the center of mass is 2g/3,2g/3, and (4gh/3)1/2(4gh/3)1/2 after of the center of mass is (4gh/3)1/2(4gh/3)1/2 after the disk has descended through distance h.h. (d) Verify your answer to part (c) using the energy approach. • The wavelength of red helium-neon laser light in air is 632.8 nmnm . (a) What is its frequency? (b) What is its wave-length in glass that has an index of refraction of 1.50 ? (c) What is its speed in the glass? • Plaskett’s binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (Fig. P13.17). Assume the orbital speed of • A 1.00 -kg object is attached to a horizontal spring. The spring is initially stretched by 0.100 mm , and the object is released from rest there. It proceeds to move without friction. The next time the speed of the object is zero is 0.500 ss later. What is the maximum speed of the object? • A molecule of DNA (deoxyribonucleic acid) is 2.17μm2.17μm long. The ends of the molecule become singly ionized: negative on one end, positive on the other. The • A generator produces 24.0 V when turning at 900 rev/min . What emf does it produce when turning at 500 rev/min ? • Trucks carrying garbage to the town dump form a nearly steady procession on a country road, all traveling at 19.7 m/s in the same direction. Two trucks arrive at the dump every 3 min. A bicyclist is also traveling toward the dump, at 4.47 m/s. (a) With what frequency do the trucks pass the cyclist? (b) What If? A hill does not slow down the trucks, but makes the out-of-shape cyclist’s speed drop to 1.56 m/s. How often do the trucks whiz past the cyclist now? • A sphere of radius R has a uniform volume charge density ρ.ρ. When the sphere rotates as a rigid object with angular speed ωω about an axis through its center (Fig. P30.74), determine (a) the magnetic field at the center of the sphere and (b) the magnetic moment of the sphere. • An electromagnetic wave is called ionizing radiation if its photon energy is larger than, say, 10.0 eV so that a single photon has enough energy to break apart an atom. With reference to Figure P40.34, explain what region or regions of the electromagnetic spectrum fit this definition of ionizing radiation and what do not. (If you wish to consult a larger version of Fig. P40.34, see Fig. 34.13.) • Most of the mass of an atom is in its nucleus. Model the mass distribution in a diatomic molecule as two spheres of uniform density, each of radius 2.00×10−15m2.00×10−15m and mass 1.00×10−26kg,1.00×10−26kg, located at points along the yy axis as in Active Figure 43.5a,43.5a, and separated by 2.00×10−10m2.00×10−10m . Rotation about the axis joining the nuclei in the diatomic molecule is ordinarily ignored because the first excited state would have an energy that is too high to access. To see why, calculate the ratio of the energy of the first excited state for rotation about the yy axis to the energy of the first excited state for rotation about the xx axis. • One electron collides elastically with a second electron initially at rest. After the collision, the radii of their trajectories are 1.00 cm and 2.40 cm. The trajectories are perpendicular to a uniform magnetic field of magnitude 0.044 0 T. Determine the energy (in keV) of the incident electron. • The longest pipe on a certain organ is 4.88 m. What is the fundamental frequency (at 0.00°C) if the pipe is (a) closed at one end and (b) open at each end? (c) What will be the frequencies at 20.0°C? • A Styrofoam cup holding 125 gg of hot water at 100∘C100∘C cools to room temperature, 20.0∘0∘C . What is the change in entropy of the room? Neglect the specific heat of the cup and any change in temperature of the room. • A 90.0-kg fullback running east with a speed of 5.00 m/s is tackled by a 95.0-kg opponent running north with a speed of 3.00 m/s. (a) Explain why the successful tackle constitutes a perfectly inelastic collision. (b) Calculate the velocity of the players immediately after the tackle. (c) Determine the mechanical energy that disappears as a result of the collision. Account for the missing energy. • The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kgkg . The air outside is at 10.0∘C10.0∘C and 101 kPakPa . The volume of the balloon is 400 m3m3 . To what temperature must the air in the balloon be warmed before the balloon will lift off? (Air density at 10.0∘C10.0∘C is 1.244kg/m3.)1.244kg/m3.) • A student drives a moped along a straight road as described by the velocity-versus-time graph in Figure P2.48. Sketch this graph in the middle of a sheet of graph paper. (a) Directly above your graph, sketch a graph of the position versus time, aligning the time coordinates of the two graphs. (b) Sketch a graph of the acceleration versus time directly below the velocity-versus-time graph, again aligning the time coordinates. On each graph, show the numerical values ofx$and$a_{x}$for all points of inflection. (c) What is the acceleration at$t=6.00$s? (d) Find the position (relative to the starting point) at$t=$6.00 s. (e) What is the moped’s final position at$t=9.00 \mathrm{s}$? • If the intensity of sunlight at the Earth’s surface under a fairly clear sky is 1000W/m2,1000W/m2, how much electromagnetic energy per cubic meter is contained in sunlight? • In a control system, an accelerometer consists of a 4.70−g object sliding on a calibrated horizontal rail. A low-mass spring attaches the object to a flange at one end of the rail. Grease on the rail makes static friction negligible, but rapidly damps out vibrations of the sliding object. When subject to a steady acceleration of 0.800 g, the object should be at a location 0.500 cm away from its equilibrium position. Find the force constant of the spring required for the calibration to be correct. • On a printed circuit board, a relatively long, straight conductor and a conducting rectangular loop lie in the same plane as shown in Figure P32.39P32.39 . Taking h=0.400mm,w=h=0.400mm,w= 1.30mm,1.30mm, and ℓ=2.70mm,ℓ=2.70mm, find their mutual inductance. • An 80.0-kg skydiver jumps out of a balloon at an altitude of 1 000 m and opens his parachute at an altitude of 200 m. (a) Assuming the total retarding force on the skydiver is constant at 50.0 N with the parachute closed and constant at 3 600 N with the parachute open, find the speed of the skydiver when he lands on the ground. (b) Do you think the skydiver will be injured? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s? (d) How realistic is the assumption that the total retarding force is constant? Explain. • A thin rod of mass 0.630 kgkg and length 1.24 mm is at rest, hanging vertically from a strong, fixed hinge at its top end. Suddenly, a horizontal impulsive force 14.7 iNiN is applied to it. (a) Suppose the force acts at the bottom end of the rod. Find the acceleration of its center of mass and (b) the horizontal force the hinge exerts. (c) Suppose the force acts at the midpoint of the rod. Find the acceleration of this point and (d) the horizontal hinge reaction force. (e) Where can the impulse be applied so that the hinge will exert no horizontal force? This point is called the center of percussion. • A sinusoidal wave of wavelength 2.00 mm and amplitude 0.100 mm travels on a string with a speed of 1.00 m/sm/s to the right. At t=0t=0 , the left end of the string is at the origin. For this wave, find (a) the frequency, (b) the angular frequency, (c) the angular wave number, and (d) the wave function in SI units. Determine the equation of motion in SI units for (e) the left end of the string and (f) the point on the string at x=1.50mx=1.50m to the right of the left end. (g) What is the maximum speed of any element of the string? • Photons of wavelength λλ are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius RR by a magnetic field having a magnitude BB . What is the work function of the metal? • The neutron has a mass of 1.67×10−27kg.1.67×10−27kg. Neutrons emitted in nuclear reactions can be slowed down by collisions with matter. They are referred to as thermal neutrons after they come into thermal equilibrium with the environment. The average kinetic energy (32kBT)(32kBT) of a thermal neutron is approximately 0.04 eV. (a) Calculate the de Broglie wavelength of a neutron with a kinetic energy of 0.0400 eV. (b) How does your answer compare with the characteristic atomic spacing in a crystal? (c) Explain whether you expect thermal neutrons to exhibit diffraction effects when scattered by a crystal. • In the liquid-drop model of nuclear structure, why does the surface-effect term −C2A2/3−C2A2/3 have a negative sign? (b) What If? The binding energy of the nucleus increases as the volume-to-surface area ratio increases. Calculate this ratio for both spherical and cubical shapes and explain which is more plausible for nuclei. • In the circuit of Figure P28.39P28.39 , the switch SS has been open for a long time. It is then suddenly closed. Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at t=0.t=0. Determine the current in the switch as a function of time. • An electric dipole is located along the yy axis as shown in Figure P25. 71 . The magnitude of its electric dipole moment is defined as p=2aqp=2aq . (a) At a point P,P, which is far from the dipole (r>>a),(r>>a), show that the electric potential is V=kepcosθr2V=kepcos⁡θr2 (b) Calculate the radial component ErEr and the perpendicular component EθEθ of the associated electric field. Note that Eθ=−(1/r)(∂V/∂θ).Eθ=−(1/r)(∂V/∂θ). Do these results seem reasonable for (c)θ=90∘(c)θ=90∘ and 0∘?0∘? (d) For r=0r=0 ? (e) For the dipole arrangement shown in Figure P25.71, express VV in terms of Cartesian coordinates using r=(x2+y2)1/2r=(x2+y2)1/2 and cosθ=y(x2+y2)1/2cos⁡θ=y(x2+y2)1/2 (f) Using these results and again taking r>>r>> calculate the field components ExEx and EyEy . • Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest. A 0.400−kg0.400−kg blue bead slides on a frictionless, curved wire, starting from rest at point A in Figure P9.63, where h= 1.50m. At point B , the blue bead collides elastically with a 0.600−kg green bead at rest. Find the maximum height the green bead rises as it moves up the wire. • An electron confined to a box absorbs a photon with wavelength λ.λ. As a result, the electron makes a transition from the n=1n=1 state to the n=3n=3 state. (a) Find the length of the box. (b) What is the wavelength λ′ of the photon emitted when the electron makes a transition from the n= 3 state to the n=2 state? • In the arrangement shown in Figure P 18.24, an object can be hung from a string (with linear mass density μ=0.00200kg/mμ=0.00200kg/m ) that passes over a light pulley. The string is connected to a vibrator (of constant frequency f),f), and the length of the string between point PP and the pulley is L=2.00m.L=2.00m. When the mass mm of the object is either 16.0 kg or 25.0 kg, standing waves are observed; no standing waves are observed with any mass between these values, however. (a) What is the frequency of the vibrator? Note: The greater the tension in the string, the smaller the number of nodes in the standing wave. (b) What is the largest object mass for which standing waves could be observed? • Consider a small, spherical particle of radius rr located in space a distance R=3.75×1011mR=3.75×1011m from the Sun. Assume in the particle has a perfectly absorbing surface and a mass density of ρ=1.50g/cm3.ρ=1.50g/cm3. Use S=214W/m2S=214W/m2 as the value of the solar intensity at the location of the particle. Calcu- late the value of rr for which the particle is in equilibrium between the gravitational force and the force exerted by solar radiation. • In introductory physics laboratories, a typical Caven-dish balance for measuring the gravitational constant G uses lead spheres with masses of 1.50 kg and 15.0 g whose centers are separated by about 4.50 cm. Calculate the gravitational force between these spheres, treating each as a particle located at the sphere’s center. • At$t=0,$one athlete in a race running on a long, straight track with a constant speed$v_{1}$is a distance$d_{1}$behind a second athlete running with a constant speed$v_{2} .$(a) Under what circumstances is the first athlete able to overtake the second athlete? (b) Find the time$t$at which the first athlete overtakes the second athlete, in terms of$d_{1}, v_{1},$and$v_{2} .$(c) At what minimum distance$d_{2}$from the leading athlete must the finish line be located so that the trailing athlete can at least tie for first place? Express$d_{2}$in terms of$d_{1}, v_{1},$and$v_{2}$by using the result of part$(\mathrm{b})$• A 1 000-kg satellite orbits the Earth at a constant altitude of 100 km. (a) How much energy must be added to the system to move the satellite into a circular orbit with altitude 200 km? What are the changes in the system’s (b) kinetic energy and (c) potential energy? • An object of mass m1=9.00kgm1=9.00kg is in equilibrium when connected to a light spring of constant k=100N/mk=100N/m that is fastened to a wall as shown in Figure P15.73aP15.73a . A second object, m2=7.00kg,m2=7.00kg, is slown pushed up against m1,m1, compressing the spring by the amount A=0.200mA=0.200m (see Fig. Pl5.73b). The system is then released, and both objects start moving to the right on the frictionless surface. (a) When m1m1 reaches the equilibrium point, m2m2 loses contact with m1m1 (see Fig. P15.73 c) and moves to the right with speed v.v. Determine the value of vv . (b) How far apart are the objects when the spring is fully stretched for the first time (the distance DD in Fig. P15.48d)?P15.48d)? • At time ti,ti, the kinetic energy of a particle is 30.0 JJ and the potential energy of the system to which it belongs is 10.0 JJ . At some later time tf,tf, the kinetic energy of the par- ticle is 18.0 JJ . (a) If only conservative forces act on the particle, what are the potential energy and the the total energy of the system at time tf?tf? (b) If the potential energy of the system at time tftf is 5.00 JJ , are any nonconservative forces acting on the particle? (c) Explain your answer to part (b).(b). • The force acting on a particle is Fx=(8x−16), where F is in newtons and x is in meters. (a) Make a plot of this force versus x from x=0 to x=3.00m. (b) From your graph, find the net work done by this force on the particle as it moves from x=0 to x=3.00m. • Three point charges lie along a circle of radius r at angles of 30∘,150∘,30∘,150∘, and 270∘270∘ as shown in Figure P23. 26 . Find a symbolic expression for the resultant electric field at the center of the circle. • Sand from a stationary hopper falls onto a moving conveyor belt at the rate of 5.00 kg/s as shown in Figure P9.78. The conveyor belt is supported by frictionless rollers and moves at a constant speed of v=0.750m/s under the action of a constant horizontal external force →Fext Supplied by the motor that drives the belt. Find (a) the sand’s rate of change of momentum in the horizontal direction, (b) the force of friction exerted by the belt on the sand, (c) the external force →Fext,(d) the work done by →Fext in 1s, and (e) the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion. (f) Why are the answers to parts (d) and (e) different? • Mirror M1M1 in Active Figure 37.13 is moved through a displacement ΔL.ΔL. During this displacement, 250 fringe reversals (formation of successive dark or bright bands) are counted. The light being used has a wavelength of 632.8nm.632.8nm. Calculate the displacement ΔLΔL . • The magnetic field created by a large current passing through plasma (ionized gas) can force current-carrying particles together. This pinch effect has been used in designing fusion reactors. It can be demonstrated by making an empty aluminum can carry a large current parallel to its axis. Let R represent the radius of the can and I the current, uniformly distributed over the can’s curved wall. Determine the magnetic field (a) just inside the wall and (b) just outside. (c) Determine the pressure on the wall. • Why is the following situation impossible? A softball pitcher has a strange technique: she begins with her hand at rest at the highest point she can reach and then quickly rotates her arm backward so that the ball moves through a half-circle path. She releases the ball when her hand reaches the bot- tom of the path. The pitcher maintains a component of force on the 0.180-kg ball of constant magnitude 12.0 N in the direction of motion around the complete path. As the ball arrives at the bottom of the path, it leaves her hand with a speed of 25.0 m/s. • You put a diode in a microelectronic circuit to protect the system in case an untrained person installs the battery backward. In the correct forward-bias situation, the current is 200 mA with a potential difference of 100 mV across the diode at room temperature (300 K). If the battery were reversed, so that the potential difference across the diode is still 100 mV but with the opposite sign, what would be the magnitude of the current in the diode? • Four 1.50−V1.50−V AA batteries in series are used to power a small If the batteries can move a charge of 240C,240C, how long will they last if the radio has a resistance of 200Ω?Ω? • A ball whirls around in a vertical circle at the end of a string. The other end of the string is fixed at the center of the circle. Assuming the total energy of the ball–Earth system remains constant, show that the tension in the string at the bottom is greater than the tension at the top by six times the ball’s weight. • A room contains air in which the speed of sound is 343 m/sm/s . The walls of the room are made of concrete in which the speed of sound is 1850 m/sm/s . ( a) Find the critical angle for total internal reflection of sound at the concrete- air boundary. (b) In which medium must the sound be initially traveling if it is to undergo total internal reflection? (c) “A bare concrete wall is a highly efficient mirror for sound.” Give evidence for or against this statement. • A 5.50-kg black cat and her four black kittens, each with mass 0.800 kg, sleep snuggled together on a mat on a cool night, with their bodies forming a hemisphere. Assume the hemisphere has a surface temperature of 31.0∘C,31.0∘C, an emissivity of 0.970,0.970, and a uniform density of 990kg/m3.990kg/m3. Find (a)(a) the radius of the hemisphere, (b)(b) the area of its curved surface, (c)(c) the radiated power emitted by the cats at their curved surface and, (d) the intensity of radiation at this surface. You may think of the emitted electromagnetic wave as having a single predominant frequency. Find (e) the amplitude of the electric field in the electromagnetic wave just outside the surface of the cozy pile and (f) the amplitude of the magnetic field. (g) What If? The next night, the kittens all sleep alone, curling up into separate hemispheres like their mother. Find the total radiated power of the family. (For simplicity, ignore the cats’ absorption of radiation from the environment.) • To measure her speed, a skydiver carries a buzzer emitting a steady tone at 1 800 Hz. A friend on the ground at the landing site directly below listens to the amplified sound he receives. Assume the air is calm and the speed of sound is independent of altitude. While the skydiver is falling at terminal speed, her friend on the ground receives waves of frequency 2 150 Hz. (a) What is the skydiver’s speed of descent? (b) What If? Suppose the skydiver can hear the sound of the buzzer reflected from the ground. What frequency does she receive? • Three sheets of plastic have unknown indices of refraction. Sheet 1 is placed on top of sheet 2,2, and a laser beam is directed onto the sheets from above. The laser beam enters sheet 1 and then strikes the interface between sheet 1 and sheet 2 at an angle of 26.5∘5∘ with the normal. The refracted beam in sheet 2 makes an angle of 31.7∘31.7∘ with the normal. The experiment is repeated with sheet 3 on top of sheet 2 , and, with the same angle of incidence on the sheet 3 -sheet 2 interface, the refracted beam makes an angle of 36.7∘36.7∘ with the normal. If the experiment is repeated again with sheet 1 on top of sheet 3,3, with that same angle of incidence on the sheet 1 -sheet 3 interface, what is the expected angle of refraction in sheet 3 ? • If A battery with emf εε and no internal resistance supplies current to the circuit shown in Figure P28.11P28.11 . When the double-throw switch SS is open as shown in the figure, position a,a, the current in the battery is IaIa . When the switch is closed in position bb , the current in the battery is IbIb . Find the resistances (a) R1,R1, (b) R2,R2, and (c)R3(c)R3 . • Besides the gravitational force, a 2.80−kg object is subjected to one other constant force. The object starts from rest and in 1.20 s experiences a displacement of (4.20ˆi−3.30ˆj)m, where the direction of ˆj is the upward vertical direction. Determine the other force. • The half-life of tritium is 12.3 yr. (a) If the TFTR fusion reactor contained 50.0 m3 of tritium at a density equal to 2.00×1014 ions/cm ^{3} , \text { how many curies of tritium } were in the plasma? (b) State how this value compares with a fission inventory (the estimated supply of fissionable material) of 4.00×1010Ci. • What is the fractional change in the resistance of an iron filament when its temperature changes from 25.0∘0∘C to 50,0∘C50,0∘C? • A heat pump used for heating shown in Figure P22.31P22.31 is essentially an air conditioner installed backward. It extracts energy from colder air outside and deposits it in a warmer room. Suppose the ratio of the actual energy entering the room to the work done by the device’s motor is 10.0%% of the theoretical maximum ratio. Determine the energy entering the room per joule of work done by the motor given that the inside temperature is 20.0∘0∘C and the outside temperature is −5.00∘C−5.00∘C . • A flat coil of wire has an inductance of 40.0 mHmH and a resistance of 5.00ΩΩ . It is connected to a 22.0−V22.0−V battery at the instant t=0.t=0. Consider the moment when the current is 3.00 A. (a) At what rate is energy being delivered by the battery? (b) What is the power being delivered to the resistance of the coil? (c) At what rate is energy being stored in the magnetic field of the coil? (d) What is the relationship among these three power values? (e) Is the relationship described in part (d) true at other instants as well? (f) Explain the relationship at the moment immediately after t=0t=0 and at a moment several seconds later. • A certain ideal gas has a molar specific heat of CV=72R.CV=72R. A 2.00 -mol sample of the gas always starts at pressure 1.00×1051.00×105 Pa and temperature 300 KK . For each of the following processes, determine (a) the final pressure, (b) the final volume, (c) the final temperature, (d) the change in internal energy of the gas, (e) the energy added to the gas by heat, and (f) the work done on the gas. (i) The gas is heated at constant pressure to 400 KK . (ii) The gas is heated at constant volume to 400 KK . (iii) The gas is compressed at constant temperature to 1.20×105Pa1.20×105Pa (iv) The gas is compressed adiabatically to 1.20×105Pa1.20×105Pa . • A global positioning system (GPS) satellite moves in a circular orbit with period 11 h 58 min. (a) Determine the radius of its orbit. (b) Determine its speed. (c) The nonmilitary GPS signal is broadcast at a frequency of 1 575.42 MHz in the reference frame of the satellite. When it is received on the Earth’s surface by a GPS receiver (Fig. P39.55), what is the fractional change in this frequency due to time dilation as described by special relativity? (d) The gravitational “blueshift” of the frequency according to general relativity is a separate effect. It is called a blueshift to indicate a change to a higher frequency.The magnitude of that fractional change is given by Δff=ΔUgmc2Δff=ΔUgmc2 where UgUg is the change in gravitational potential energy of an object-Earth system when the object of mass mm is moved between the two points where the signal is observed. Calculate this fractional change in frequency due to the change in position of the satellite from the Earth’s surface to its orbital position. (e) What is the overall fractional change in frequency due to both time dilation and gravitational blueshift? • Two blocks, each of mass m,m, are hung from the ceiling of an elevator as in Figure P5.31. The elevator has an upward acceleration aa . The strings have negligible mass. (a) Find the tensions T1T1 and T2T2 in the upper and lower strings in terms of m,am,a and gg . (b) Compare the two tensions and determine which string would break first if aa is made sufficiently large. (c) What are the tensions if the cable supporting the elevator breaks? • A puck of mass m1=80.0gm1=80.0g and radius r1=4.00cmr1=4.00cm glides across an air table at a speed of →v=1.50m/sv→=1.50m/s as shown in Figure P11.36a. It makes a glancing collision with a second puck of radius r2=6.00cmr2=6.00cm and mass m2=120gm2=120g (initially at rest) such that their rims just touch. Because their rims are coated with instant-acting glue, the pucks stick together and rotate after the collision (Fig. P 11.36 b). (a) What is the angular momentum of the system relative to the center of mass? (b) What is the angular speed about the center of mass? • We wish to show that the most probable radial position for an electron in the 2 s state of hydrogen is r= 236a0 . (a) Use Equations 42.24 and 42.26 to find the radial probability density for the 2 s state of hydrogen. (b) Calculate the derivative of the radial probability density with respect to r (c) Set the derivative in part (b) equal to zero and identify three values of r that represent minima in the function. (d) Find two values of r that represent maxima in the function. (e) Identify which of the values in part (c) represents the highest probability. • A 0.600-kg particle has a speed of 2.00 m/s at point and kinetic energy of 7.50 J at point . What is (a) its kinetic energy at , (b) its speed at , and (c) the net work done on the particle by external forces as it moves from to ? • A large, flat sheet carries a uniformly distributed electric current with current per unit width JsJs . This current creates a magnetic field on both sides of the sheet, parallel to the sheet and perpendicular to the current, with magnitude B=12μ0J.B=12μ0J. If the current is in the yy direction and oscillates in time according to Jmax(cosωt)ˆj=Jmax[cos(−ωt)]ˆjJmax(cosωt)j^=Jmax[cos(−ωt)]j^ the sheet radiates an electromagnetic wave. Figure P34.43 on page 1006 shows such a wave emitted from one point on the sheet chosen to be the origin. Such electromagnetic waves are emitted from all points on the sheet. The magnetic field of the wave to the right of the sheet is described by the wave function →B=12μ0Jmax[cos(kx−ωt)]ˆkB→=12μ0Jmax[cos(kx−ωt)]k^ (a) Find the wave function for the electric field of the wave to the right of the sheet. (b) Find the Poynting vector as a function of xx and t.t. (c) Find the intensity of the wave. (d) What If the sheet is to emit radiation in each direction (normal to the plane of the sheet) with intensity 570 W/m2W/m2 , what maximum value of sinusoidal current density is required? • Air (a diatomic ideal gas) at 27.08C and atmospheric pressure is drawn into a bicycle pump (see the chapter-opening photo on page 599) that has a cylinder with an inner diameter of 2.50 cm and length 50.0 cm. The down stroke adiabatically compresses the air, which reaches a gauge pressure of 8.00 3 105 Pa before entering the tire. We wish to investigate the temperature increase of the pump. (a) What is the initial volume of the air in the pump? (b) What is the number of moles of air in the pump? (c) What is the absolute pressure of the compressed air? (d) What is the volume of the compressed air? (e) What is the temperature of the compressed air? (f) What is the increase in internal energy of the gas during the compression? What If? The pump is made of steel that is 2.00 mm thick. Assume 4.00 cm of the cylinder’s length is allowed to come to thermal equilibrium with the air. (g) What is the volume of steel in this 4.00-cm length? (h) What is the mass of steel in this 4.00-cm length? (i) Assume the pump is compressed once. After the adiabatic expansion, conduction results in the energy increase in part (f) being shared between the gas and the 4.00-cm length of steel. What will be the increase in temperature of the steel after one compression? • The eye is most sensitive to light having a frequency of 5.45×1014Hz,5.45×1014Hz, which is in the green-yellow region of the visible electromagnetic spectrum. What is the wavelength of this light? • Use Bohr’s model of the hydrogen atom to show that when the electron moves from the nn state to the n−1n−1 state, the frequency of the emitted light is f=(2π2mek2ee4h3)2n−1n2(n−1)2.f=(2π2meke2e4h3)2n−1n2(n−1)2. (b) Bohr’s correspondence principle claims that quantum results should reduce to classical results in the limit of large quantum numbers. Show that as n→∞,n→∞, this expression varies as 1/n31/n3 and reduces to the classical frequency one expects the atom to emit. Suggestion: To calculate the classical frequency, note that the frequency of revolution is v/2πr,v/2πr, where vv is the speed of the electron and rr is given by Equation 42.10. • If you roll two dice, what is the total number of ways in which you can obtain (a) a 12 and (b) a 7 ? • You are given a certain volume of copper from which you can make copper wire. To insulate the wire, you can have as much enamel as you like. You will use the wire to make a tightly wound solenoid 20 cm long having the greatest possible magnetic field at the center and using a power supply that can deliver a current of 5 A. The solenoid can be wrapped with wire in one or more layers. (a) Should you make the wire long and thin or shorter and thick? Explain. (b) Should you make the radius of the solenoid small or large? Explain. • Consider the block–spring–surface system in part (B) of Example 8.6. (a) Using an energy approach, find the position x of the block at which its speed is a maximum. (b) In the What If? section of this example, we explored the effects of an increased friction force of 10.0 N. At what position of the block does its maximum speed occur in this situation? • For a metal at temperature TT , calculate the probability that a state with an energy equal to βEFβEF is occupied where ββ is a fraction between 0 and 1. • A 50.0 -g object connected to a spring with a force constant of 35.0 N/mN/m oscillates with an amplitude of 4.00 cmcm on a frictionless, horizontal surface. Find (a) the total energy of the system and (b) the speed of the object when its position is 1.00cm.1.00cm. Find (c)(c) the kinetic energy and (d) the potential energy when its position is 3.00cm.3.00cm. • A 3.00-g copper coin at 25.0°C drops 50.0 m to the ground. (a) Assuming 60.0% of the change in gravitational potential energy of the coin–Earth system goes into increasing the internal energy of the coin, determine the coin’s final temperature. (b) What If? Does the result depend on the mass of the coin? Explain. • A student throws a set of keys vertically upward to her sorority sister, who is in a window 4.00 m above. The second student catches the keys 1.50 s later. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught? • Calculate the work that must be done on charges brought from infinity to charge a spherical shell of radius RR to a total charge QQ • A bicycle tire is inflated to a gauge pressure of 2.50 atm when the temperature is 15.0∘0∘C . While a man rides the bicycle, the temperature of the tire rises to 45.0∘C45.0∘C . Assuming the volume of the tire does not change, find the gauge pressure in the tire at the higher temperature. • A train slows down as it rounds a sharp horizontal turn, going from 90.0 km/hkm/h to 50.0 km/hkm/h in the 15.0 ss it takes to round the bend. The radius of the curve is 150m.150m. Compute the acceleration at the moment the train speed reaches 50.0 km/hkm/h . Assume the train continues to slow down at this time at the same rate. • Two thin rods are fastened to the inside of a circular ring as shown in Figure P2.69. One rod of length$D$is vertical, and the other of length$L$makes an angle$\theta$with the horizontal. The two rods and the ring lie in a vertical plane. Two small beads are free to slide without friction along the rods. (a) If the two beads are released from rest simultaneously from the positions shown, use your intuition and guess which bead reaches the bottom first. (b) Find an expression for the time interval required for the red bead to fall from point$@$to point$\mathbb{C}$in terms of$g$and$D .$(c) Find an expression for the time interval required for the blue bead to slide from point$\mathbb{B}$to point$\mathbb{C}$in terms of$g, L,$and$\theta .$(d) Show that the two time intervals found in parts$(b)$and$(c)$are equal. Hint: What is the angle between the chords of the circle$\mathbb{Q}$and$\mathbb{C}$(e) Do these results surprise you? Was your intuitive guess in part (a) correct? This problem was inspired by an article by Thomas B. Greenslade, Jr., “Galileo’s Paradox,” Phys. Teach. 46, 294 (May 2008 ). • The primary coil of a transformer has N1=350N1=350 turns, and the secondary coil has N2=2000N2=2000 turns. If the input voltage across the primary coil is Δv=170cosωt,Δv=170cosωt, where ΔvΔv is in volts and tt is in seconds, what rms voltage is developed across the secondary coil? • The motion of a transparent medium influences the speed of light. This effect was first observed by Fizeau in 1851. Consider a light beam in water. The water moves with speed v in a horizontal pipe. Assume the light travels in the same direction as the water moves. The speed of light with respect to the water is c/n,c/n, where n=1.33n=1.33 is the index of refraction of water. (a) Use the velocity transformation equation to show that the speed of the light measured in the laboratory frame is u=cn(1+nv/c1+v/nc)u=cn(1+nv/c1+v/nc) (b) Show that for v<<c,v<<c, the expression from part (a) becomes, to a good approximation, u≈cn+v−vn2u≈cn+v−vn2 (c) Argue for or against the view that we should expect the result to be u=(c/n)+vu=(c/n)+v according to the Galilean transformation and that the presence of the term −v/n2−v/n2 represents a relativistic effect appearing even at “nonrelativistic” speeds. (d) Evaluate uu in the limit as the speed of the water approaches cc . • In an electron microscope, there is an electron gun that contains two charged metallic plates 2.80 cm apart. An electric force accelerates each electron in the beam from rest to 9.60% of the speed of light over this distance. (a) Determine the kinetic energy of the electron as it leaves the electron gun. Electrons carry this energy to a phosphorescent viewing screen where the microscope’s image is formed, making it glow. For an electron passing between the plates in the electron gun, determine (b) the magnitude of the constant electric force acting on the electron, (c) the acceleration of the electron, and (d) the time interval the electron spends between the plates. • There are (one can say) three coequal theories of motion for a single particle: Newton’s second law, stating that the total force on the particle causes its acceleration; the work–kinetic energy theorem, stating that the total work on the particle causes its change in kinetic energy; and the impulse-momentum theorem, stating that the total impulse on the particle causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A 3.00−kg3.00−kg object has velocity 7.00ˆjm/sj^m/s . Then, a constant net force 12.0 iN acts on the object for 5.00 s. (a) Calculate the object’s final velocity, using the impulse-momentum theorem. (b) Calculate its acceleration from →a=(→vf→vi)/Δt . (c) Calculate its acceleration from →a=ΣF/m . (d) Find the object’s vector displacement from Δr=→vit+12→at2. (e) Find the work done on the object from W=→F⋅Δ→r. (f) Find the final kinetic energy from 12mv2f=12m→vf⋅→vf. (g) Find the final kinetic energy from 12mv2i+W. (h) State the result of comparing the answers to parts (b) and (c), and the answers to parts (f) and (g). • The ammeter shown in Figure P28.23P28.23 reads 2.00 A. Find (a) I1,I1, (b) I2,I2, and (c)ε(c)ε • The magnetic field inside a superconducting solenoid is 4.50 T. The solenoid has an inner diameter of 6.20 cmcm and a length of 26.0cm.26.0cm. Determine (a) the magnetic energy density in the field and (b) the energy stored in the magnetic field within the solenoid. • When a phosphorus atom is substituted for a silicon atom in a crystal, four of the phosphorus valence electrons form bonds with neighboring atoms and the remaining electron is much more loosely bound. You can model the electron as free to move through the crystal lattice. The phosphorus nucleus has one more positive charge than does the silicon nucleus, however, so the extra electron provided by the phosphorus atom is attracted to this single nuclear charge +e.+e. The energy levels of the extra electron are similar to those of the electron in the Bohr hydrogen atom with two important exceptions. First, the Coulomb attraction between the electron and the positive charge on the phosphorus nucleus is reduced by a factor of 1/K/K from what it would be in free space (see Eq. 26.21 ), where κκ is the dielectric constant of the crystal. As a result, the orbit radii are greatly increased over those of the hydrogen atom. Second, the influence of the periodic electric potential of the lattice causes the electron to move as if it had an effective mass m∗,m∗, which is quite different from the mass meme of a free electron. You can use the Bohr as if it had an effective mass m∗,m∗, which is quite different from the mass meme of a free electron. You can use the Bohr important role in semiconductor devices. Assume κ=11.7κ=11.7 for silicon and m∗=0.220mcm∗=0.220mc (a) Find a symbolic expression for the smallest radius of the electron orbit in terms of a0,a0, the Bohr radius. (b) Substitute numerical values to find the numerical value of the smallest radius. (c) Find a symbolic expression for the energy levels E′nEn′ of the electron in the Bohr orbits around the donor atom in terms of me,m∗,me,m∗, κ,κ, and En,En, the energy of the hydrogen atom in the Bohr model. (d) Find the numerical value of the energy for the ground state of the electron. • A quantum particle has a wave function ψ(x)={√2ae−x/aforx>00forx<0 (a) Find and sketch the probability density. (b) Find the probability that the particle will be at any point where x<0. (c) Show that ψ is normalized and then (d) find the probability of finding the particle between x=0 and x=a . • A nylon string has mass 5.50 g and length L=86.0cm.L=86.0cm. The lower end is tied to the floor, and the upper end is tied to a small set of wheels through a slot in a track on which the wheels move (Fig. P 18.64). The wheels have a mass that is negligible compared with that of the string, and they roll without friction on the track so that the upper end of the string is essentially free. At equilibrium, the string is vertical and motionless. When it is carrying a small-amplitude wave, you may assume the string is always under uniform tension 1.30 N. (a) Find the speed of transverse waves on the string. (b) The string’s vibration possibilities are a set of standing-wave states, each with a node at the fixed bottom end and an antinode at the free top end. Find the node–antinode distances for each of the three simplest states. (c) Find the frequency of each of these states. • You use a sequence of ideal polarizing filters, each with its axis making the same angle with the axis of the previous filter, to rotate the plane of polarization of a polarized light beam by a total of 45.0°. You wish to have an intensity reduction no larger than 10.0%. (a) How many polarizers do you need to achieve your goal? (b) What is the angle between adjacent polarizers? • An aluminum ring of radius r1r1 and resistance RR is placed around one end of a long air-core solenoid with nn turns per meter and smaller radius r2r2 as shown in Figure P31.9. Assume the axial component of the field produced by the solenoid over the area of the end of the solenoid is one-half as strong as at the center of the solenoid. Also assume the solenoid produces negligible field outside it so cross-sectional area. The current in the solenoid is increasing at a rate of ΔI/ΔtΔI/Δt (a) What is the induced current in the ring? (b) At the center of the ring, what is the magnetic field produced by the induced current in the ring? (c) What is the direction of this field? • A parallel-plate capacitor has a charge Q and plates of area A. What force acts on one plate to attract it toward the other plate? Because the electric field between the plates is E=Q/Aϵ0,E=Q/Aϵ0, you might think the force is F=QE=Q2/Aϵ0F=QE=Q2/Aϵ0 This conclusion is wrong because the field EE includes contributions from both plates, and the field created by the positive plate cannot exert any force on the positive plate. Show that the force exerted on each plate is actually F=F= Q2/2Aϵ0.Q2/2Aϵ0. Suggestion: Let C=ϵ0A/xC=ϵ0A/x for an arbitrary plate separation xx and note that the work done in separating the two charged plates is W=∫Fdx.W=∫Fdx. • A rod of length LL (Fig. P25.42 lies along the xx axis with its left end at the origin. It has a nonuniform charge density λ=αx,λ=αx, where αα is a positive constant. (a) What are the units of αα ? (b) Calculate the electric potential at A.A. • A bar on a hinge starts from rest and rotates with an angular acceleration α=10+6t,α=10+6t, where αα is in rad/s 22 and tt is in seconds. Determine the angle in radians through which the bar turns in the first 4.00 s. • To destroy a cancerous tumor, a dose of gamma radiation with a total energy of 2.12 J is to be delivered in 30.0 days from implanted sealed capsules containing palladium-103. Assume this isotope has a half-life of 17.0 d and emits gamma rays of energy 21.0 keV, which are entirely absorbed within the tumor. (a) Find the initial activity of the set of capsules. (b) Find the total mass of radioactive palladium these “seeds” should contain. • A very long, thin strip of metal of width w carries a current II along its length as shown in Figure P30.53P30.53 . The cur- rent is distributed uniformly across the width of the strip. Find the magnetic field at point PP in the diagram. Point PP is in the plane of the strip at distance bb away from its edge. • Two identical hard spheres, each of mass m and radius r, are released from rest in otherwise empty space with their centers separated by the distance R. They are allowed to collide under the influence of their gravitational attraction. (a) Show that the magnitude of the impulse received by each sphere before they make contact is given by [Gm3(1/2r−1/R)]1/2.[Gm3(1/2r−1/R)]1/2. (b) What If? Find the magnitude of the impulse each receives during their contact if they collide elastically. • A spherical mirror is to be used to form an image 5.00 times the size of an object on a screen located 5.00 m from the object. (a) Is the mirror required concave or convex? (b) What is the required radius of curvature of the mirror? (c) Where should the mirror be positioned relative to the object? • A driver travels northbound on a highway at a speed of 25.0 m/s. A police car, traveling southbound at a speed of 40.0 m/s, approaches with its siren producing sound at a frequency of 2 500 Hz. (a) What frequency does the driver observe as the police car approaches? (b) What frequency does the driver detect after the police car passes him? (c) Repeat parts (a) and (b) for the case when the police car is behind the driver and travels northbound. • The speed of sound in air (in meters per second) depends on temperature according to the approximate expression v=331.5+0.607TCv=331.5+0.607TC where TCTC is the Celsius temperature. In dry air, the temperature decreases about 1∘C1∘C for every 150 −m−m rise in altitude. (a) Assume this change is constant up to an altitude of 9000 mm . What time interval is required for the sound from an airplane flying at 9000 mm to reach the ground on a day when the ground temperature is 30∘C30∘C ? (b) What If? Compare your answer with the time interval required if the air were uniformly at 30∘30∘C. Which time interval is longer? • A sled of mass mm is given a kick on a frozen pond. The kick imparts to the sled an initial speed of 2.00 m/sm/s . The coefficient of kinetic friction between sled and ice is 0.100.0.100. Use energy considerations to find the distance the sled moves before it stops. • A counterweight of mass m=4.00kgm=4.00kg is attached to a light cord that is wound around a pulley as in Figure P 11.18. The pulley is a thin hoop of radius R=8.00cmR=8.00cm and mass M=2.00kgM=2.00kg . The spokes have negligible mass. (a) What is the magnitude of the net torque on the system about the axle of the pulley? (b) When the counterweight has a speed vv the pulley has an angular speed ω=v/Rω=v/R . Determine the magnitude of the total angular momentum of the system about the axle of the pulley. (c) Using your result from part (b) and →τ=d→L/dt,τ⃗=dL→/dt, calculate the acceleration of the counterweight. • For a neutral atom of element 110,110, what would be the prob- able ground-state electronic configuration? • Figure$\mathrm{P} 14.57$shows a valve separating a reservoir from a water tank. If this valve is opened, what is the maximum height above point$B$attained by the water stream coming out of the right side of the tank? Assume$h=10.0 \mathrm{m}, L=2.00 \mathrm{m},$and$\theta=30.0^{\circ}$, and assume the cross- sectional area at$A$is very large compared with that at$B$. • An object is placed 12.0 cm to the left of a diverging lens of focal length$-6.00 \mathrm{cm} .$A converging lens of focal length 12.0$\mathrm{cm}$is placed a distance$d$to the right of the diverging lens. Find the distance$d$so that the final image is infinitely far away to the right. • A variable air capacitor used in a radio tuning circuit is made of N semicircular plates, each of radius R and positioned a distance d from its neighbors, to which it is electrically connected. As shown in Figure P26.10, a second identical set of plates is enmeshed with the first set. Each plate in the second set is halfway between two plates of the first set. The second set can rotate as a unit. Determine the capacitance as a function of the angle of rotation θθ where θ=0θ=0 corresponds to the maximum capacitance. • The following fission reaction is typical of those occurring in a nuclear electric generating station: 10n+25592U→141Ba+9236Kr+3(10)10n+25592U→141Ba+9236Kr+3(10) a) Find the energy released in the reaction. The masses of the products are 140.914 411 u for 141Ba141Ba and 91.926 156 u for 92 36 KrKr (b) What fraction of the initial rest energy of the system is transformed to other forms? • If the spacing between planes of atoms in a NaCl crystal is 0.281 nm, what is the predicted angle at which 0.140-nm x-rays are diffracted in a first-order maximum? • Suppose you wish to fabricate a uniform wire from a mass mm of a metal with density ρmρm and resistivity ρ.ρ. If the wire is to have a resistance of RR and all the metal is to be used, what must be (a) the length and (b) the diameter of this wire? • The ground-state wave function for the electron in a hydro- gen atom is ψ1s(r)=1√πa30e−r/a0ψ1s(r)=1πa30−−−√e−r/a0 where rr is the radial coordinate of the electron and a0a0 is the Bohr radius. (a) Show that the wave function as givenis normalized. (b) Find the probability of locating the electron between r1=a0/2r1=a0/2 and r2=3a0/2r2=3a0/2 • Two identical parallel-plate capacitors, each with capacitance 10.0μFμF , are charged to potential difference 50.0 VV and then disconnected from the battery. They are then connected to each other in parallel with plates of like sign connected. Finally, the plate separation in one of the capacitors is doubled. (a) Find the total energy of the system of two capacitors before the plate separation is doubled. (b) Find the potential difference across each capacitor after the plate separation is doubled. (c) Find the total energy of the system after the plate separation is doubled. (d) Reconcile the difference in the answers to parts (a) and (c) with the law of conservation of energy. • In the circuit of Figure P28.28P28.28 , determine (a) the current in each resistor and (b)(b) the potential difference across the 200−Ω200−Ω resistor. • A boy stands on a diving board and tosses a stone into a swimming pool. The stone is thrown from a height of 2.50 mm above the water surface with a velocity of 4.00 m/sm/s at an angle of 60.0∘0∘ above the horizontal. As the stone strikes the water surface, it immediately slown to exactly half the speed it had when it struck the water and maintains that speed while in the water. After the stone enters the water, it moves in a straight line in the direction of the velocity it had when it struck the water. If the pool is 3.00 mm deep, how much time elapses between when the stone is thrown and when it strikes the bottom of the pool? • Consider the two circuits shown in Figure P28.8 in which the batteries are identical. The resistance of each light-bulb is R. Neglect the internal resistances of the batteries. (a) Find expressions for the currents in each light-bulb. (b) How does the brightness of B compare with that of C? Explain. (c) How does the brightness of A compare with that of B and of C? Explain. • A sinusoidal wave on a string is described by the wave function y=0.15sin(0.80x−50t)y=0.15sin⁡(0.80x−50t) where xx and yy are in meters and tt is in seconds. The mass per unit length of this string is 12.0 g/mg/m . Determine (a) the speed of the wave, (b) the wavelength, (c) the frequency, and (d) the power transmitted by the wave. • A series RLRL circuit with L=3.00HL=3.00H and a series RCRC circuit with C=3.00μFC=3.00μF have equal time constants. If the two circuits contain the same resistance R,R, (a) what is the value of R?(b) What is the time constant?R?(b) What is the time constant? • A right circular cone can theoretically be balanced on a horizontal surface in three different ways. Sketch these three equilibrium configurations and identify them as positions of stable, unstable, or neutral equilibrium. • Two identical conducting small spheres are placed with their centers 0.300 mm apart. One is given a charge of 12.0 nCnC and the other a charge of −18.0nC−18.0nC . (a) Find the electric force exerted by one sphere on the other. (b) What If? The spheres are connected by a conducting wire. Find the electric force each exerts on the other after they have come to equilibrium. • In Figure P5.75,P5.75, the incline has mass MM and is fastened to the stationary horizontal tabletop. The block of mass mm is placed near the bottom of the incline and is released with a quick push that sets it sliding upward. The block stops near the top of the incline as shown in the figure and then slides down again, always without friction. Find the force that the tabletop exerts on the incline throughout this motion in terms of m,M,g,m,M,g, and θ.θ. • Two handheld radio transceivers with dipole antennas are separated by a large, fixed distance. If the transmitting antenna is vertical, what fraction of the maximum received power will appear in the receiving antenna when it is inclined from the vertical (a) by 15.0∘?(b) By45.0∘? (c) By15.0∘?(b) By45.0∘? (c) By 90.0∘0∘ ? • A 120−V120−V motor has mechanical power output of 2.50 hphp . It is 90.0%% efficient in converting power that it takes in by electrical transmission into mechanical power. (a) Find the current in the motor. (b) Find the energy delivered to the motor by electrical transmission in 3.00 hh of operation. (c) If the electric company charges$0.110/kWh,$0.110/kWh, what does it cost to run the motor for 3.00 hh ? • A bag of cement whose weight is FgFg hangs in equilibrium from three wires as shown in Figure 24. Two of the wires make angles θ1θ1 and θ2θ2 with the horizontal. Assuming the system is in equilibrium, show that the tension in the left-hand wire is T1=Fgcosθ2sin(θ1+θ2)T1=Fgcosθ2sin(θ1+θ2) • A photon of initial energy E0E0 undergoes Compton scattering at an angle θθ from a free electron (mass me)me) initially at rest. Derive the following relationship for the final energy E′E′ of the scattered photon: E′=E01+(E0mec2)(1−cosθ)E′=E01+(E0mec2)(1−cos⁡θ) • In x-ray production, electrons are accelerated through a high voltage and then decelerated by striking a (a) To make possible the production of x-rays of wavelength l, what is the minimum potential difference DV through which the electrons must be accelerated? (b) State in words how the required potential difference depends on the wavelength. (c) Explain whether your result predicts the correct minimum wavelength in Figure 42.22. (d) Does the relationship from part (a) apply to other kinds of electromagnetic radiation besides x-rays? (e) What does the potential difference approach as l goes to zero? (f) What does the potential difference approach as l increases with-out limit? • A bar of mass m and resistance R slides without friction in a horizontal plane, moving on parallel rails as shown in Figure P31.72 . The rails are separated by a distance d. A battery that maintains a constant emf E is connected between the rails, and a constant magnetic field →B is directed perpendicularly out of the page. Assuming the bar starts from rest at time t=0, show that at time t it moves with a speed v=EBd(1−e−B2d2t/mR) • A neodymium-yttrium-aluminum garnet laser used in eye surgery emits a 3.00 -mJ pulse in 1.00 nsns , focused to a spot 30.0μmμm in diameter on the retina. (a) Find (in SI units) the power per unit area at the retina. (In the optics industry, this quantity is called the irradiance.) (b) What energy is delivered by the pulse to an area of molecular size, taken as a circular area 0.600 nmnm in diameter? • The magnetic coils of a tokamak fusion reactor are in the shape of a toroid having an inner radius of 0.700 mm and an outer radius of 1.30 mm . The toroid has 900 turns of large-diameter wire, each of which carries a current of 14.0 kAkA . Find the magnitude of the magnetic field inside the toroid along (a) the inner radius and (b) the outer radius. • Why is the following situation impossible? An illuminated object is placed a distance$d=2.00 \mathrm{m}$from a screen. By placing a converging lens of focal length$f=60.0 \mathrm{cm}$at two locations between the object and the screen, a sharp, real image of the object can be formed on the screen. In one location of the lens, the image is larger than the object, and in the other, the image is smaller. • A very light rigid rod of length 0.500 m extends straight out from one end of a meterstick. The combination is suspended from a pivot at the upper end of the rod as shown in Figure P15.34. The combination is then pulled out by a small angle and released. (a) Determine the period of oscillation of the system. (b) By what percentage does the period differ from the period of a simple pendulum 1.00 m long? • The rectangular plate shown in Figure P19.49 has an area AiAi equal to ℓw.ℓw. If the temperature increases by ΔT,ΔT, each dimension increases according to Equation 19.4,19.4, where αα is the average coefficient of linear expansion. (a) Show that the increase in area is ΔA=2αAiΔT.ΔA=2αAiΔT. (b) What approximation does this expression assume? • A community plans to build a facility to convert solar radiation to electrical power. The community requires 1.00 MW of power, and the system to be installed has an efficiency of 30.0% (that is, 30.0% of the solar energy incident on the surface is converted to useful energy that can power the community). Assuming sunlight has a constant intensity of 1000W/m2,1000W/m2, what must be the effective area of a perfectly absorbing surface used in such an installation? • Transverse waves travel with a speed of 20.0 m/sm/s on a string under a tension of 6.00 NN . What tension is required for a wave speed of 30.0 m/sm/s on the same string? • A particle with charge Q=5.00μCQ=5.00μC is located at the center of a cube of edge L=0.100m.L=0.100m. In addition, six other identical charged particles having q=−1.00μCq=−1.00μC are positioned symmetrically around QQ as shown in Figure P 24.19. Determine the electric flux through one face of the cube. • One end of a uniform 4.00 -m-long rod of weight FgFg is supported by a cable at an angle of θ=37∘θ=37∘ with the rod. The other end rests against the wall, where it is held by friction as shown in Figure P12.23. The coefficient of static friction between the wall and the rod is μs=0.500μs=0.500 . Determine the minimum distance xx from point AA at which an additional object, also with the same weight Fg,Fg, can be hung without causing the rod to slip at point A.A. • A tennis player receives a shot with the ball (0.0600kg)(0.0600kg) traveling horizontally at 50.0 m/sm/s and returns the shot with the ball traveling horizontally at 40.0 m/sm/s in the opposite direction. (a) What is the impulse delivered to the ball by the tennis racquet? (b) What work does the racquet do on the ball? • A golf ball is hit off a tee at the edge of a cliff. Its xx and yy coordinates as functions of time are given by x=18,0tx=18,0t and y=4.00t−4.90t2,y=4.00t−4.90t2, where xx and yy are in meters and tt is in seconds. (a) Write a vector expression for the ball’s position as a function of time, using the unit vectors i and ˆjj^ . tBy taking derivatives, obtain expressions for (b) the velocity vector →vv→ as a function of time and (c) the acceleration vector →aa→ as a function of time. (d) Next use unit-vector notation to write expressions for the position, the velocity, and the acceleration of the golf ball at t=3.00s . • Slit 1 of a double slit is wider than slit 2 so that the light from slit 1 has an amplitude 3.00 times that of the light from slit 2.2. Show that Equation 37.13 is replaced by the equation I=Imax(1+3cos2ϕ/2)I=Imax(1+3cos2⁡ϕ/2) for this situation. • A sealed capsule containing the radiopharmaceutical phosphorus- 32, an e− emitter, is implanted into a patient’s tumor. The average kinetic energy of the beta particles is 700 keV. The initial activity is 5.22 MBq . Assume the beta particles are completely absorbed in 100 g of tissue. Determine the absorbed dose during a 10.0 -day period. • The average speed of a nitrogen molecule in air is about 6.70×102m/s6.70×102m/s , and its mass is 4.68×10−26kg4.68×10−26kg . (a) If it takes 3.00×10−13s3.00×10−13s for a nitrogen molecule to hit a wall and rebound with the same speed but moving in the opposite direction, what is the average acceleration of the molecule during this time interval? (b) What average force does the molecule exert on the wall? • Raise your hand and hold it flat. Think of the space between your index finger and your middle finger as one slit and think of the space between middle finger and ring finger as a second slit. (a) Consider the interference resulting from sending coherent visible light perpendicularly through this pair of openings. Compute an order-of- magnitude estimate for the angle between adjacent zones of constructive interference. (b) To make the angles in the interference pattern easy to measure with a plastic protractor, you should use an electromagnetic wave with frequency of what order of magnitude? (c) How is this wave classified on the electromagnetic spectrum? • Four long, parallel conductors carry equal currents of I=5.00I=5.00 A. Figure P30.32P30.32 is an end view of the conductors. The current direction is into the page at points AA and BB and out of the page at points CC and D.D. Calculate (a) the magnitude and (b)(b) the direction of the magnetic field at point P,P, located at the center of the square of edge length ℓ=0.200m.ℓ=0.200m. • In a single-slit diffraction pattern, assuming each side maximum is halfway between the adjacent minima, find the ratio of the intensity of (a) the first-order side maximum and (b) the second-order side maximum to the intensity of the central maximum. • As the people sing in church, the sound level everywhere inside is 101 dBdB . No sound is transmitted through the massive walls, but all the windows and doors are open on a summer morning. Their total area is 22.0m2.22.0m2. (a) How much sound energy is radiated through the windows and doors in 20.0 minmin ? (b) Suppose the ground is a good reflector and sound radiates from the church uniformly in all horizontal and upward directions. Find the sound level 1.00 kmkm away. • Find the energy released in the fission reaction 10n+23592U→9840Zr+13552Te+3(10)10n+23592U→9840Zr+13552Te+3(10) The atomic masses of the fission products are 97.912735 uu for 9840Zr9840Zr and 134.916450 u for 13552Te13552Te • Find the speed of light in (a) flint glass, (b) water, and (c) cubic zirconia. • A village maintains a large tank with an open top, containing water for emergencies. The water can drain from the tank through a hose of diameter$6.60 \mathrm{cm} .$The hose ends with a nozzle of diameter$2.20 \mathrm{cm} .$A rubber stopper is inserted into the nozzle. The water level in the tank is kept 7.50$\mathrm{m}$above the nozzle. (a) Calculate the friction force exerted on the stopper by the nozzle. (b) The stopper is removed. What mass of water flows from the nozzle in 2.00$\mathrm{h}$? (c) Calculate the gauge pressure of the flowing water in the hose just behind the nozzle. • When a 4.25-kg object is placed on top of a vertical spring, the spring compresses a distance of 2.62 cm. What is the force constant of the spring? • People who ride motorcycles and bicycles learn to look out for bumps in the road and especially for wash-boarding, a condition in which many equally spaced ridges are worn into the road. What is so bad about wash boarding? A motorcycle has several springs and shock absorbers in its suspension, but you can model it as a single spring supporting a block. You can estimate the force constant by thinking about how far the spring compresses when a heavy rider sits on the seat. A motorcyclist traveling at highway speed must be particularly careful of washboard bumps that are a certain distance apart. What is the order of magnitude of their separation distance? • A muon formed high in the Earth’s atmosphere is measured by an observer on the Earth’s surface to travel at speed v=v= 0.990 cfor a distance of 4.60 kmkm before it decays into an electron, a neutrino, and an antineutrino (μ−→e−+ν+¯ν)(μ−→e−+ν+ν¯¯¯). (a) For what time interval does the muon live as measured in its reference frame? (b) How far does the Earth travel as measured in the frame of the muon? • The record distance in the sport of throwing cowpats is 81.1 mm . This record toss was set by Steve Uren of the United States in 1981.1981. Assuming the initial launch angle was 45∘45∘ and neglecting air resistance, determine (a) the initial speed of the projectile and (b) the total time interval the projectile was in flight. (c) How would the answers change if the range were the same but the launch angle were greater than 45∘?45∘? Explain. • A particular radioactive source produces 100 mrad of 2.00-MeV gamma rays per hour at a distance of 1.00 m from the source. (a) How long could a person stand at this distance before accumulating an intolerable dose of 1.00 rem? (b) What If? Assuming the radioactive source is a point source, at what distance would a person receive a dose of 10.0 mrad/h? • A conductor consists of a circular loop of radius R=R= 15.0 cmcm and two long, straight sections as shown in Figure P30.7. The wire lies in the plane of the paper and carries a current I=1.00I=1.00 A. Find the magnetic field at the center of the loop. • Use the semiempirical binding-energy formula (Eq. 44.3) to compute the binding energy for 5626Fe5626Fe . (b) What percentage is contributed to the binding energy by each of the four terms? • Two parallel plates having charges of equal magnitude but opposite sign are separated by 12.0cm.12.0cm. Each plate has a surface charge density of 36.0nC/m2.36.0nC/m2. A proton is released from rest at the positive plate. Determine (a) the magnitude of the electric field between the plates from the charge density, (b) the potential difference between the plates, (c) the kinetic energy of the proton when it reaches the negative plate, (d) the speed of the proton just before it strikes the negative plate, (e) the acceleration of the proton, and (f) the force on the proton. (g) From the force, find the magnitude of the electric field. (h) How does your value of the electric field compare with that found in part (a)? • A man drops a rock into a well. (a) The man hears the sound of the splash 2.40$\mathrm{s}$after he releases the rock from rest. The speed of sound in air (at the ambient temperature) is 336$\mathrm{m} / \mathrm{s}$. How far below the top of the well is the surface of the water? (b) What If? If the travel time for the sound is ignored, what percentage error is introduced when the depth of the well is calculated? • Three point charges are arranged as shown in Figure P23.9. Find (a) the magnitude and (b) the direction of the electric force on the particle at the origin. • A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency of 0.450 HzHz . The pendulum has a mass of 2.20 kgkg , and the pivot is. located 0.350 mm from the center of mass. Determine the moment of inertia of the pendulum about the pivot point. • A hydrogen atom is in its fifth excited state, with principal quantum number 6.6. The atom emits a photon with a wavelength of 1090 nmnm . Determine the maximum possible magnitude of the orbital angular momentum of the atom after emission. • A 2.00 -mol sample of a diatomic ideal gas expands slowly and adiabatically from a pressure of 5.00 atm and a volume of 12.0 LL to a final volume of 30.0 LL (a) What is the final pressure of the gas? (b) What are the initial and final temperatures? Find (c) Q,Q, (d) ΔEintΔEint , and (e) WW for the gas during this process. • Assume a 150-W loudspeaker broadcasts sound equally in all directions and produces sound with a level of 103 dB at a distance of 1.60 m from its center. (a) Find its sound power output. If a salesman claims the speaker is rated at 150 W, he is referring to the maximum electrical power input to the speaker. (b) Find the efficiency of the speaker, that is, the fraction of input power that is converted into useful output power. • A helium-neon laser produces a beam of diameter 1.75mm,1.75mm, delivering 2.00×10182.00×1018 photons/s. Each photon has a wavelength of 633 nmnm . Calculate the amplitudes of (a) the electric fields and (b) the magnetic fields inside the beam. (c) If the beam shines perpendicularly onto a perfectly reflecting surface, what force does it exert on the surface? (d) If the beam is absorbed by a block of ice at 0∘C0∘C for 1.50 hh , what mass of ice is melted? • Identify the unknown particle on the left side of the following reaction: ?+p→n+μ+?+p→n+μ+ • In the Atwood machine discussed in Example 5.9 and shown in Active Figure 5.14a,m1=2.00kg and m2= 7.00 kg . The masses of the pulley and string are negligible by comparison. The pulley turns without friction, and the string does not stretch. The lighter object is released with a sharp push that sets it into motion at vi=2.40m/s downward. (a) How far will m1 descend below its initial level? (b) Find the velocity of m1 after 1.80 s . • Two gases in a mixture diffuse through a filter at rates proportional to their rms speeds. (a) Find the ratio of speeds for the two isotopes of chlorine, 35 ClCl and 37 ClCl , as they diffuse through the air. (b) Which isotope moves faster? • The electric potential immediately outside a charged conducting sphere is 200 V, and 10.0 cm farther from the center of the sphere the potential is 150 V. Determine (a) the radius of the sphere and (b) the charge on it. The electric potential immediately outside another charged conducting sphere is 210 V, and 10.0 cm farther from the center the magnitude of the electric field is 400 V/m. Determine (c) the radius of the sphere and (d) its charge on it. (e) Are the answers to parts (c) and (d) unique? • Why is the following situation impossible? A shopper pushing a cart through a market follows directions to the canned goods and moves through a displacement 8.00 ˆii^ m down one aisle. He then makes a 90.0° turn and moves 3.00 m along the y axis. He then makes another 90.0° turn and moves 4.00 m along the x axis. Every shopper who follows these directions correctly ends up 5.00 m from the starting point. • Why is the following situation impossible? Figure P 29.68 shows an experimental technique for altering the direction of travel for a charged particle. A particle of charge q=1.00μC and mass m=2.00×10−13kg enters the bottom of the region of uniform magnetic field at speed v=2.00×105m/s, with a velocity vector perpendicular to the field lines. The magnetic force on the particle causes its direction of travel to change so that it leaves the region of the magnetic field at the top traveling at an angle from its original direction. The magnetic field has magnitude B=0.400T and is directed out of the page. The length h of the magnetic field region is 0.110 m. An experimenter performs the technique and measures the angle θ at which the particles exit the top of the field. She finds that the angles of deviation are exactly as predicted. • Two blocks, each of mass m=m= 3.50kg,3.50kg, are hung from the ceiling of an elevator as in Figure P5.31.P5.31. (a) If the elevator moves with an upward acceleration →aa→ of magnitude 1.60 m/s2 , find the tensions T1 and T2 in the upper and lower strings. (b) If the strings can withstand a maximum tension of 85.0 N , what maximum acceleration can the elevator have before a string breaks? • In a constant-volume process, 209 JJ of energy is transferred by heat to 1.00 molmol of an ideal monatomic gas initially at 300 KK . Find (a) the work done on the gas, (b)(b) the increase in internal energy of the gas, and (c)(c) its final temperature. • A baby bounces up and down in her crib. Her mass is 12.5 kg, and the crib mattress can be modeled as a light spring with force constant 700 N/m. (a) The baby soon learns to bounce with maximum amplitude and minimum effort by bending her knees at what frequency? (b) If she were to use the mattress as a trampoline—losing contact with it for part of each cycle—what minimum amplitude of oscillation does she require? • Assume a deuteron and a triton are at rest when they fuse according to the reaction 21H+31H→42He+10n Determine the kinetic energy acquired by the neutron. • Colonel John P. Stapp, USAF, participated in studying whether a jet pilot could survive emergency ejection. On March 19, 1954, he rode a rocket-propelled sled that moved down a track at a speed of 632$\mathrm{mi} / \mathrm{h}$. He and the sled were safely brought to rest in 1.40$\mathrm{s}$(Fig. P2.25). Determine (a) the negative acceleration he experienced and (b) the distance he traveled during this negative acceleration. • Why is the following situation impossible? Starting from rest, a disk rotates around a fixed axis through an angle of 50.0 rad in a time interval of 10.0 s. The angular acceleration of the disk is constant during the entire motion, and its final angular speed is 8.00 rad/srad/s . • A lens made of glass (ng=1.52)(ng=1.52) is coated with a thin film of MgF2(ns=1.38)MgF2(ns=1.38) of thickness t.t. Visible light is incident normally on the coated lens as in Figure P37.38P37.38 . (a) For what minimum value of tt will the reflected light of wavelength 540 nmnm (in air) be missing? (b) Are there • The pair of capacitors in Figure P28.59P28.59 are fully charged by a 12.0−V12.0−V battery. The battery is disconnected, and the switch is then closed. After 1.00 msms has elapsed, (a) how much charge remains on the 3.00−μF3.00−μF capacitor? (b) How much charge remains on the 2.00−μF2.00−μF capacitor? (c) What is the current in the resistor at this time? • Why is the following situation impossible? Figure P14.12 shows Superman attempting to drink cold water through a straw of length$\ell=12.0 \mathrm{m} .$The walls of the tubular straw are very strong and do not collapse. With his great strength, he achieves maximum possible suction and enjoys drinking the cold water. • The true weight of an object can be measured in a vacuum, where buoyant forces are absent. A measurement in air, however, is disturbed by buoyant forces. An object of volume V is weighed in air on an equal-arm balance with the use of counterweights of density$\rho .$Representing the density of air as$\rho_{\text { air }}$and the balance reading as$F_{g}^{\prime},$show that the true weight$F_{g}$is Fg=F′g+(V−F′gρg)ρairgFg=Fg′+(V−Fg′ρg)ρairg • Why is the following situation impossible? An engineer working on nuclear power makes a breakthrough so that he is able to control what daughter nuclei are created in a fission reaction. By carefully controlling the process, he is able to restrict the fission reactions to just this single possibility: the uranium-235 nucleus absorbs a slow neutron and splits into lanthanum-141 and bromine-94. Using this break- through, he is able to design and build a successful nuclear reactor in which only this single process occurs. • The surface of the Sun has a temperature of about 5800 K . The radius of the Sun is 6.96×108m. Calculate the total energy radiated by the Sun each second. Assume the emissivity of the Sun is 0.986 . • Why is the following situation impossible? A group of campers arises at 8:30 a.m. and uses a solar cooker, which consists of a curved, reflecting surface that concentrates sunlight onto the object to be warmed (Fig. P20.60). During the day, the maximum solar intensity reaching the Earth’s surface at the cooker’s location is T=600W/m2.T=600W/m2. The cooker faces the Sun and has a face diameter of d=0.600m.d=0.600m. Assume a fraction ff of 40.0%% of the incident energy is transferred to 1.50 LL of water in an open container, initially at 20.0∘0∘C . The water comes to a boil, and the campers enjoy hot coffee for breakfast before hiking ten miles and returning by noon for lunch. • A 2.00-mol sample of helium gas initially at 300 K, and 0.400 atm is compressed isothermally to 1.20 atm. Noting that the helium behaves as an ideal gas, find (a) the final volume of the gas, (b) the work done on the gas, and (c) the energy transferred by heat. • A wire of length L,L, Young’s modulus Y,Y, and cross-sectional area AA is stretched elastically by an amount AL. By Hooke’s law, the restoring force is −kΔL.−kΔL. (a) Show that k=YA/L.k=YA/L. (b) Show that the work done in stretching the wire by an amount ΔLΔL is W=12YA(ΔL)2/LW=12YA(ΔL)2/L. • The position of a particle moving along the$x$axis varies in time according to the expression$x=3 t^{2},$where$x$is in meters and$t$is in seconds. Evaluate its position (a) at$t=$00$\mathrm{s}$and$(\mathrm{b})$at$3.00 \mathrm{s}+\Delta t .$(c) Evaluate the limit of$\Delta x / \Delta t$as$\Delta t$approaches zero to find the velocity at$t=3.00 \mathrm{s}$. • Make an order-of-magnitude estimate of the cost of one person’s routine use of a handheld hair dryer for 1 year. If you do not use a hair dryer yourself, observe or interview someone who does. State the quantities you estimate and their values. • A proton moving in the plane of the page has a kinetic energy of 6.00 MeV. A magnetic field of magnitude B=1.00B=1.00 TT is directed into the page. The proton enters the magnetic field with its velocity vector at an angle θ=45.0∘θ=45.0∘ to the linear boundary of the field as shown in the linear boundary of the field as shown in Figure P 29.76 . (a) Find x,x, the distance from the point of entry to where the proton will leave the field. (b) Determine θθ , the angle between the boundary and the proton’s velocity vector as it leaves the field. • An air-traffic controller observes two aircraft on his radar screen. The first is at altitude 800 m, horizontal distance 19.2 km, and 25.0° south of west. The second aircraft is at altitude 1 100 m, horizontal distance 17.6 km, and 20.0° south of west. What is the distance between the two aircraft? (Place the x axis west, the yy axis south, and the zz axis vertical.) • In the Bohr model of the hydrogen atom, an electron moves in a circular path around a proton. The speed of the electron is approximately 2.20×106m/s2.20×106m/s . Find (a) the force acting on the electron as it revolves in a circular orbit of radius 0.530×10−10m0.530×10−10m and (b) the centripetal acceleration of the electron. • A certain vibrating string on a piano has a length of 74.0 cm and forms a standing wave having two antinodes. (a) Which harmonic does this wave represent? (b) Determine the wavelength of this wave. (c) How many nodes are there in the wave pattern? • Consider the popgun in Example 8.3. Suppose the projectile mass, compression distance, and spring constant remain the same as given or calculated in the example. Suppose, however, there is a friction force of magnitude 2.00 N acting on the projectile as it rubs against the interior of the barrel. The vertical length from point to the end of the barrel is 0.600 m. (a) After the spring is compressed and the popgun fired, to what height does the projectile rise above point ? (b) Draw four energy bar charts for this situation, analogous to those in Figures 8.6c-d. • Design an incandescent lamp filament. A tungsten wire radiates electromagnetic waves with power 75.0 WW when its ends are connected across a 120−V120−V power supply. Assume its constant operating temperature is 2900 KK and its emissivity is 0.450 . Also assume it takes in energy only by electric transmission and emits energy only by electromagnetic radiation. You may take the resistivity of tungsten at 2900 KK as 7.13×10−7Ω⋅m.7.13×10−7Ω⋅m. Specify (a) the radius and (b) the length of the filament. • A capacitor, a coil, and two resistors of equal resistance are arranged in an AC circuit as shown in Figure P33.66. An AC source provides an emf of ΔVrms=20.0VΔVrms=20.0V at a frequency of 60.0 HzHz . When the double-throw switch SS is open as shown in the figure, the rms current is 183 mAmA . When the switch is closed in position a,a, the rms current is 298 mAmA . When the switch is closed in position b,b, the rms current is 137 mAmA . Determine the values of (a)R,(b)C,(a)R,(b)C, and (c)L.(c)L. (d) Is more than one set of values possible? Explain. • The energy required to excite an atom is on the order of 1 eV. As the temperature of the Universe dropped below a threshold, neutral atoms could form from plasma and the Universe became transparent. Use the Boltzmann distribution function e−E/k0Te−E/k0T to find the order of magnitude of the threshold temperature at which 1.00%% of a population of photons has energy greater than 1.00 eV. • A golfer tees off from a location precisely at ϕi=ϕi= 35.0∘0∘ north latitude. He hits the ball due south, with range 285m.285m. The ball’s initial velocity is at 48.0∘48.0∘ above the horizontal. Suppose air resistance is negligible for the golf ball. (a) For how long is the ball in flight? The cup is due south of the golfer’s location, and the golfer would have a hole-in-one if the Earth were not rotating. The Earth’s rotation makes the tee move in a circle of radius REcosϕi=(6.37×106m)cos35.0∘REcos⁡ϕi=(6.37×106m)cos⁡35.0∘ as shown in Figure P6.67P6.67 . The tee completes one revolution each day. (b) Find the eastward speed of the tee relative to the stars. The hole is also moving east, but it is 285 mm farther south and thus at a slightly lower latitude ϕf.ϕf. Because the hole moves in a slightly larger circle, its speed must be greater than that of the tee. (c) By how much does the hole’s speed exceed that of the tee? During the time interval the ball is in flight, it moves upward and downward as well as south-ward with the projectile motion you studied in Chapter 4, but it also moves eastward with the speed you found in part (b). The hole moves to the east at a faster speed, however, pulling ahead of the ball with the relative speed you found in part (c). (d) How far to the west of the hole does the ball land? • When a star has exhausted its hydrogen fuel, it may fuse other nuclear fuels. At temperatures above 1.00×108K1.00×108K , helium fusion can occur. Consider the following processes. (a) Two alpha particles fuse to produce a nucleus AA and a gamma ray. What is nucleus AA ? (b) Nucleus AA from part (a) absorbs an alpha particle to produce nucleus BB and a gamma ray. What is nucleus BB ? (c) Find the total energy released in the sequence of reactions given in parts (a) and (b). • How much energy is required to change a 40.0 -g ice cube from ice at −10.0∘C−10.0∘C to steam at 110∘C110∘C ? • The Bernoulli effect can have important consequences for the design of buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure. The higher atmospheric pressure in the still air inside the buildings can cause windows to pop out. As originally constructed, the John Hancock Building in Boston popped windowpanes that fell many stories to the sidewalk below. (a) Suppose a horizontal wind blows with a speed of 11.2$\mathrm{m} / \mathrm{s}$outside a large pane of plate glass with dimensions$4.00 \mathrm{m} \times 1.50 \mathrm{m}$. Assume the density of the air to be constant at 1.20$\mathrm{kg} / \mathrm{m}^{3}$. The air inside the building is at atmospheric pressure. What is the total force exerted by air on the windowpane? (b) What If? If a second skyscraper is built nearby, the airspeed can be especially high where wind passes through the narrow separation between the buildings. Solve part (a) again with a wind speed of 22.4$\mathrm{m} / \mathrm{s}$, twice as high. • For hydrogen in the 1 s state, what is the probability of finding the electron farther than βa0βa0 from the nucleus, where ββ is an arbitrary number? • In the circuit shown in Figure P32.16,P32.16, let L=7.00H,R=L=7.00H,R= 9.00Ω,9.00Ω, and E=120VE=120V . What is the self-induced emf 0.200 ss after the switch is closed? • A small object with electric dipole moment p→p→ is placed in a nonuniform electric field E→=E(x)i^E→=E(x)i^ . That is, the field is in the xx direction, and its magnitude depends only on the coordinate xx . Let θθ represent the angle between the dipole moment and the xx direction. Prove that the net force on the dipole is F=p(dEdx)cosθF=p(dEdx)cos⁡θ acting in the direction of increasing field. • A wire 2.80 m in length carries a current of 5.00 A in a region where a uniform magnetic field has a magnitude of 0.390 T. Calculate the magnitude of the magnetic force on the wire assuming the angle between the magnetic field and the current is (a) 60.0∘,60.0∘, (b) 90.0∘,90.0∘, and (c) 120∘.120∘. • Two concrete spans that form a bridge of length LL are placed end to end so that no room is allowed for expansion (Fig. P19.55a).P19.55a). If a temperature increase of ΔΔ Toccurs, what is the height yy to which the spans rise when they buckle (Fig. Pl9.55b)? • Light of wavelength 500 nm is incident normally on a dif- fraction grating. If the third-order maximum of the diffraction pattern is observed at 32.0°, (a) what is the number of rulings per centimeter for the grating? (b) Determine the total number of primary maxima that can be observed in this situation. • A satellite in Earth orbit has a mass of 100 kgkg and is at an altitude of 2.00×106m.2.00×106m. (a) What is the potential energy of the satellite-Earth system? (b) What is the magnitude of the gravitational force exerted by the Earth on the satellite? (c) What If? What force, if any, does the satellite exert on the Earth? • A pair of narrow, parallel slits separated by 0.250 mm are illuminated by green light (λ=546.1nm)(λ=546.1nm). The interference pattern is observed on a screen 1.20 m away from the plane of the parallel slits. Calculate the distance (a) from the central maximum to the first bright region on either side of the central maximum and (b) between the first and second dark bands in the interference pattern. • A particle of mass mm moves along a straight line with constant velocity v→0v→0 in the xx direction, a distance bb from the xx axis (Fig. P13.16). (a) Does the particle possess any angular momentum about the origin? (b) Explain why the amount of its angular momentum should change or should stay constant. (c) Show that Kepler’s second law is satisfied by showing that the two shaded triangles in the figure have the same area when tD−tC=tB−tAtD−tC=tB−tA. • The daughter nucleus formed in radioactive decay is often radioactive. Let N10N10 represent the number of parent nuclei at time t=0,N1(t)t=0,N1(t) the number of parent nuclei at time t,t, and λ1λ1 the decay constant of the parent. Suppose the number of daughter nuclei at time t=0t=0 is Let N2(t)N2(t) be the number of daughter nuclei at time tt and let λ2λ2 be the decay constant of the daughter. Show that N2(t)N2(t) satisfies the differential equation dN2dt=λ1N1−λ2N2dN2dt=λ1N1−λ2N2 (b) Verify by substitution that this differential equation has the solution N2(t)=N10λ1λ1−λ2(e−λ2t−e−λ1t)N2(t)=N10λ1λ1−λ2(e−λ2t−e−λ1t) This equation is the law of successive radioactive decays. (c) 218 Po decays into 214 Pb with a half-life of 3.10 minmin , and 214 Pb decays into 214 Bi with a half-life of 26.8 min. On the same axes, plot graphs of N1(t)N1(t) for 218218 Po and N2(t)N2(t) for 214 Pb. Let N10=1000N10=1000 nuclei and choose values of tt from 0 to 36 minmin in 2 -min intervals. (d) The curve for 219 PbPb obtained in part (c)(c) at first rises to a maximum and then starts to decay. At what instant tmtm is the number of 214 PbPb nuclei a maximum? (e) By applying the condition for a maximum dN2/dt=0,dN2/dt=0, derive a symbolic equation for tmtm in terms of λ1λ1 and λ2.λ2. (f) Explain whether the value obtained in part (c) agrees with this equation. • Interpret the graph in Figure 6.16(b), which describes the results for falling coffee filters discussed in Example 6.10. Proceed as follows. (a) Find the slope of the straight line, including its units. (b) From Equation 6.6,R=12DρAv2,6.6,R=12DρAv2, identify the theoretical slope of a graph of resistive force versus squared speed. (c) Set the experimental and theoretical slopes equal to each other and proceed to calculate the drag coefficient of the filters. Model the cross-sectional area of the filters as that of a circle of radius 10.5 cm and take the density of air to be 1.20kg/m3.1.20kg/m3. (d) Arbitrarily choose the eighth data point on the graph and find its vertical separation from the line of best fit. Express this scatter as a percentage. (e) In a short paragraph, state what the graph demonstrates and compare it with the theoretical prediction. You will need to make reference to the quantities plotted on the axes, to the shape of the graph line, to the data points, and to the results of parts (c) and (d). • A seismographic station receives SS and PP waves from an earthquake, separated in time by 17.3 ss . Assume the waves have traveled over the same path at speeds of 4.50 km/skm/s and 7.80 km/skm/s . Find the distance from the seismograph to the focus of the quake. • A pion at rest (mπ=273me)(mπ=273me) decays to a muon (mμ=(mμ= 207me)me) and an antineutrino (m¯v≈0).(mv¯¯¯≈0). The reaction is writtenπ−→μ−+¯ν.tenπ−→μ−+ν¯¯¯. Find (a) the kinetic energy of the muon and (b)(b) the energy of the antineutrino in electron volts. • The relationship L=Li+αLiΔTL=Li+αLiΔT is a valid approximation when αΔTαΔT is small. If αΔTαΔT is large, one must integrate the relationship dL=αLdTdL=αLdT to determine the final length. (a) Assuming the coefficient of linear expansion of a material is constant as LL varies, determine a general expression for the final length of a rod made of the material. Given a rod of length 1.00 mm and a temperature change of 100.0∘0∘C , determine the error caused by the approximation when (b) α=2.00×10−5(∘C)−1α=2.00×10−5(∘C)−1 (a typical value for a metal) and (c)(c) when α=0.0200(∘C)−1α=0.0200(∘C)−1 (an unrealistically large value for comparison). (d) Using the equation from part (a), solve Problem 15 again to find more accurate results. • The cosmic background radiation is blackbody radiation from a source at a temperature of 2.73 KK . (a) Use Wien’s law to determine the wavelength at which this radiation has its maximum intensity. (b) In what part of the electromagnetic spectrum is the peak of the distribution? • A light spring with force constant 3.85 N/m is compressed by 8.00 cm as it is held between a 0.250−kg block on the left and a 0.500−kg block on the right, both resting on a horizontal surface. The spring exerts a force on each block, tending to push the blocks apart. The blocks are simultaneously released from rest. Find the acceleration with which each block starts to move, given that the coefficient of kinetic friction between each block and the surface is (a) 0, (b) 0.100, and (c) 0.462. • Figure P33.8P33.8 shows three lightbulbs connected to a 120−V120−V ACAC (rms) household supply voltage. Bulbs 1 and 2 have a power rating of 150 WW , and bulb 3 has a 100−W100−W rating. Find (a) the rms current in each bulb and (b) the resistance of each bulb. (c) What is the total resistance of the combination of the three lightbulbs? • It is shown in Example 25.7 that the potential at a point PP a distance aa above one end of a uniformly charged rod of length ℓℓ lying along the xx axis is V=keQℓln(ℓ+a2+ℓ2−−−−−−√a)V=keQℓln⁡(ℓ+a2+ℓ2a) Use this result to derive an expression for the yy component of the electric field at P.P. • An object disintegrates into two fragments. One fragment has mass 1.00 MeV/c2MeV/c2 and momentum 1.75 MeV/cMeV/c in the positive xx direction, and the other has mass 1.50 MeV/c2MeV/c2 and momentum 2.00 MeV/cMeV/c in the positive yy direction. Find (a) the mass and (b)(b) the speed of the original object. • The overall length of a piccolo is 32.0 cm. The resonating air column is open at both ends. (a) Find the frequency of the lowest note a piccolo can sound. (b) Opening holes in the side of a piccolo effectively shortens the length of the resonant column. Assume the highest note a piccolo can sound is 4 000 Hz. Find the distance between adjacent antinodes for this mode of vibration. • A rescue plane flies horizontally at a constant speed searching for a disabled boat. When the plane is directly above the boat, the boat’s crew blows a loud horn. By the time the plane’s sound detector receives the horn’s sound, the plane has traveled a distance equal to half its altitude above the ocean. Assuming it takes the sound 2.00 s to reach the plane, determine (a) the speed of the plane and (b) its altitude. • Why is the following situation impossible? A 1.30 -kg toaster is not plugged in. The coefficient of static friction between the toaster and a horizontal countertop is 0.350.0.350. To make the toaster start moving, you carelessly pull on its electric cord. Unfortunately, the cord has become frayed from your previous similar actions and will break if the tension in the cord exceeds 4.00 NN . By pulling on the cord at a particular angle, you successfully start the toaster moving without breaking the cord. • The current-voltage characteristic curve for a semiconductor diode as a function of temperature TT is given by I=I0(eeΔV/kBT−1).I=I0(eeΔV/kBT−1). Here the first symbol ee represents Euler’s number, the base of natural logarithms. The second ee is the magnitude of the electron charge, the kBkB stands for Boltzmann’s constant, and TT is the absolute temperature. (a) Set up a spreadsheet to calculate II and R=ΔV/IR=ΔV/I for ΔV=0.400VΔV=0.400V to 0.600V0.600V in increments of 0.005V0.005V. Assume I0=1.00nAI0=1.00nA. (b) Plot RR versus ΔVΔV for T=280K,300K,T=280K,300K, and 320K320K. • A square, single-turn wire loop ℓ=1.00cm on a side is placed inside a solenoid that has a circular cross section of radius r=3.00cm as shown in the end view of Figure P31.13. The solenoid is 20.0 cm long and wound with 100 turns of wire. (a) If the current in the solenoid is 3.00 A what is the magnetic flux through the square loop? (b) If the current in the solenoid is reduced to zero in 3.00 s, what is the magnitude of the average induced emf in the square loop? • An electron in an infinitely deep square well has a wave function that is given by ψ3(x)=√2Lsin(3πxL) for 0≤x≤L and is zero otherwise. (a) What are the most probable positions of the electron? (b) Explain how you identify them. • Why is the following situation impossible? An experimenter is accelerating electrons for use in probing a material. She finds that when she accelerates them through a potential difference of 84.0 kVkV , the electrons have half the speed she wishes. She quadruples the potential difference to 336 kVkV , and the electrons accelerated through this potential difference have her desired speed. • The electron beam emerging from a certain high-energy electron accelerator has a circular cross section of radius 1.00mm.1.00mm. (a) The beam current is 8.00μAμA . Find the current density in the beam assuming it is uniform throughout. (b) The speed of the electrons is so close to the speed of light that their speed can be taken as 300 Mm/sMm/s with negligible error. Find the electron density in the beam. (c) Over what time interval does Avogadro’s number of electrons emerge from the accelerator? • An object 10.0 cm tall is placed at the zero mark of a meterstick. A spherical mirror located at some point on the meterstick creates an image of the object that is upright. 4.00 cm tall, and located at the 42.0-cm mark of the meterstick. (a) Is the mirror convex or concave? (b) Where is the mirror? (c) What is the mirror’s focal length? • One mole of an ideal gas is contained in a cylinder with a movable piston. The initial pressure, volume, and temperature are Pi,Vi,Pi,Vi, and TiTi , respectively. Find the work done on the gas in the following processes. In operational terms, describe how to carry out each process and show each process on a PVPV diagram. (a) an isobaric compression in which the final volume is one-half the initial volume (b) an isothermal compression in which the final pressure is four times the initial pressure (c) an isovolumetric process in which the final pressure is three times the initial pressure • Consider an electron orbiting a proton and maintained in a fixed circular path of radius R=5.29×10−11mR=5.29×10−11m by the Coulomb force. Treat the orbiting particle as a current loop. Calculate the resulting torque when the electron– proton system is placed in a magnetic field of 0.400 T directed perpendicular to the magnetic moment of the loop. • Vector →A has a magnitude of 5.00 units, and vector →B has a magnitude of 9.00 units. The two vectors make an angle of 50.0∘ with each other. Find →A⋅→B . • A container in the shape of a cube 10.0 cm on each edge contains air (with equivalent molar mass 28.9 g/mol) at atmospheric pressure and temperature 300 K. Find (a) the mass of the gas, (b) the gravitational force exerted on it, and (c) the force it exerts on each face of the cube. (d) Why does such a small sample exert such a great force? • An inventive child named Nick wants to reach an apple in a tree without climbing the tree. Sitting in a chair connected to a rope that passes over a frictionless pulley (Fig. P5.57), Nick pulls on the loose end of the rope with such a force that the spring scale reads 250 N. Nick’s true weight is 320 NN , and the chair weighs 160 NN . Nick’s feet are not touching the ground. (a) Draw one pair of diagrams showing the forces for Nick and the chair considered as separate systems and another diagram for Nick and the chair considered as one system. (b) Show that the acceleration of the system is upward and find its magnitude. (c) Find the force Nick exerts on the chair. • A Chinook salmon can swim underwater at 3.58 m/s, and it can also jump vertically upward, leaving the water with a speed of 6.26 m/s. A record salmon has length 1.50 m and mass 61.0 kg. Consider the fish swimming straight upward in the water below the surface of a lake. The gravitational force exerted on it is very nearly canceled out by a buoyant force exerted by the water as we will study in Chapter 14. The fish experiences an upward force P exerted by the water on its threshing tail fin and a downward fluid friction force that we model as acting on its front end. Assume the fluid friction force disappears as soon as the fish’s head breaks the water surface and assume the force on its tail is constant. Model the gravitational force as suddenly switching full on when half the length of the fish is out of the water. Find the value of P. • The electrons in a particle beam each have a kinetic energy K. What are (a) the magnitude and (b) the direction of the electric field that will stop these electrons in a distance d? • If A sample consists of an amount nn in moles of a monatomic ideal gas. The gas expands adiabatically, with work W done on it. (Work WW is a negative number.) The initial temperature and pressure of the gas are TiTi and PiPi . Calculate (a) the final temperature and (b) the final pressure. • As a result of friction, the angular speed of a wheel changes with time according to dθdt=ω0e−σtdθdt=ω0e−σt where ω0ω0 and σσ are constants. The angular speed changes from 3.50 rad/srad/s at t=0t=0 to 2.00 rad/srad/s at t=9.30st=9.30s . (a) Use this information to determine σσ and ω0.ω0. Then determine (b) the magnitude of the angular acceleration at t=3.00st=3.00s , (c) the number of revolutions the wheel makes in the first 2.50 ss , and (d)(d) the number of revolutions it makes before coming to rest. • Model the electromagnetic wave in a microwave oven as a plane traveling wave moving to the left, with an intensity of 25.0kW/m2.25.0kW/m2. An oven contains two cubical conintensity of 25.0 kW/m2. An oven contains two cubical containers of small mass, each full of water. One has an edge length of 6.00 cm, and the other, 12.0 cm. Energy falls perpendicularly on one face of each container. The water in the smaller container absorbs 70.0% of the energy that falls on it. The water in the larger container absorbs 91.0%. That is, the fraction 0.300 of the incoming microwave energy passes through a 6.00-cm thickness of water, and the fraction (0.300)(0.300) 5 0.090 passes through a 12.0-cm thickness. Assume a negligible amount of energy leaves either container by heat. Find the temperature change of the water in each container over a time interval of 480 s. • A 60.0−Ω60.0−Ω resistor is connected in series with a 30.0−μF30.0−μF capacitor and a source whose maximum voltage is 120 VV , operating at 60.0 HzHz . Find (a) the capacitive reactance of the circuit, (b) the impedance of the circuit, and cc ) the maximum current in the circuit. (d) Does the voltage lead or lag the current? (e) How will adding an inductor in series with the existing resistor and capacitor affect the current? Explain. • Consider a container of nitrogen gas molecules at 900 KK . Calculate (a) the most probable speed, (b) the average speed, and (c)(c) the rms speed for the molecules. (d) State how your results compare with the values displayed in Active Figure 21.11.21.11. • An electric motor turns a flywheel through a drive belt that joins a pulley on the motor and a pulley that is rigidly attached to the flywheel as shown in Figure P10.37P10.37 . The flywheel is a solid disk with a mass of 80.0 kgkg and a radius R=R= 0.625m.0.625m. It turns on a frictionless axle. Its pulley has much smaller mass and a radius of r=0.230m.r=0.230m. The tension TuTu in the upper (taut) segment of the belt is 135 NN , and the flywheel has a clockwise angular acceleration of 1.67 rad/s? Find the tension in the lower (slack) segment of the belt. • The first quasar to be identified and the brightest found to date, 3C 273 in the constellation Virgo, was observed to be moving away from the Earth at such high speed that the observed blue 434-nm Hg line of hydrogen is Doppler-shifted to 510 nm, in the green portion of the spectrum (Fig. P46.43). (a) How fast is the quasar receding? (b) Edwin Hubble discovered that all objects outside the local group of galaxies are moving away from us, with speeds vv proportional to their distances RR . Hubble’s law is expressed as v=HR,v=HR, where the Hubble constant has the approximate value H≈22×10−3m/s⋅H≈22×10−3m/s⋅ Determine the distance from the Earth to this quasar. • An air-filled capacitor consists of two parallel plates, each with an area of 7.60 cm2cm2 , separated by a distance of 1.80mm.1.80mm. A 20.0−V20.0−V potential difference is applied to these plates. Calculate (a) the electric field between the plates, (b) the surface charge density, (c) the capacitance, and (d) the charge on each plate. • A 2.00 -kg particle has a velocity (2.00i−3.00ˆj)m/s(2.00i−3.00j^)m/s , and a 3.00 -kg particle has a velocity (1.00ˆi+6.00ˆj)m/s(1.00i^+6.00j^)m/s . Find (a) the velocity of the center of mass and (b)(b) the total momentum of the system. • The area of a typical eardrum is about 5.00×10−5m25.00×10−5m2. (a) Calculate the average sound power incident on an eardrum at the threshold of pain, which corresponds to an intensity of 1.00W/m2.1.00W/m2. (b) How much energy is transferred to the eardrum exposed to this sound for 1.00 minmin ? • In fair weather, the electric field in the air at a particular location immediately above the Earth’s surface is 120 N/C directed downward. (a) What is the surface charge density on the ground? Is it positive or negative? (b) Imagine the surface charge density is uniform over the planet. What then is the charge of the whole surface of the Earth? (c) What is the Earth’s electric potential due to this charge? (d) What is the difference in potential between the head and the feet of a person 1.75 m tall? (Ignore any charges in the atmosphere.) (e) Imagine the Moon, with 27.3% of the radius of the Earth, had a charge 27.3% as large, with the same sign. Find the electric force the Earth would then exert on the Moon. (f) State how the answer to part (e) compares with the gravitational force the Earth exerts on the Moon. • Two coils are close to each other. The first coil carries a current given by I(t)=5.00e−0.0250tsin120πt,I(t)=5.00e−0.0250tsin⁡120πt, where II is in amperes and tt is in seconds. At t=0.800st=0.800s , the emf measured across the second coil is −3.20V−3.20V . What is the mutual inductance of the coils? • The critical angle for total internal reflection for sapphire surrounded by air is 34.4°. Calculate the polarizing angle for sapphire. • A boy in a wheelchair (total mass 47.0 kg) has speed 1.40 m/s at the crest of a slope 2.60 m high and 12.4 m long. At the bottom of the slope his speed is 6.20 m/s. Assume air resistance and rolling resistance can be modeled as a constant friction force of 41.0 N. Find the work he did in pushing forward on his wheels during the downhill ride. • Consider Joule’s apparatus described in Figure 20.1.20.1. The mass of each of the two blocks is 1.50 kgkg , and the insulated tank is filled with 200 gg of water. What is the increase in the water’s temperature after the blocks fall through a distance of 3.00 mm ? • A 2100 -kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls 5.00 m before coming into contact with the top of the beam, and it drives the beam 12.0 cm farther into the ground before coming to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest. • An estimated force-time curve for a baseball struck by a bat is shown in Figure P9. 11. From this curve, determine (a) the magnitude of the impulse delivered to the ball and (b) the average force exerted on the ball. • Starting with Equation 43.17, show that the force exerted on an ion in an ionic solid can be written as F=−keαe2r2[1−(r0r)m−1]F=−keαe2r2[1−(r0r)m−1] where αα is the Madelung constant and r0r0 is the equilibrium separation. (b) Imagine that an ion in the solid is displaced a small distance ss from r0.r0. Show that the ion experiences a restoring force F=−Ks,F=−Ks, where K=keαe2r30(m−1)K=keαe2r03(m−1) (c) Use the result of part (b)(b) to find the frequency of vibration of a Na+Na+ ion in NaClNaCl . Take m=8m=8 and use the value α=1.7476.α=1.7476. • A system consists of a spring with force constant k=1250N/mk=1250N/m length L=1.50m,L=1.50m, and an object of mass m=5.00kgm=5.00kg attached to the end (Fig.(Fig. placed at the level of the position yi=L,yi=L, and then it is released so that it swings like a pendulum. (a) Find the yy position of the object at the lowest point. (b) Will the pendulum’s period be greater or less than the period of a simple pendulum with the same mass mm and length LL ? Explain. • Argon enters a turbine at a rate of 80.0 kg/minkg/min , a temperature of 800∘C800∘C , and a pressure of 1.50 MPaMPa . It expands adiabatically as it pushes on the turbine blades and exits at pressure 300 kPakPa (a) Calculate its temperature at exit. (b) Calculate the (maximum) power output of the turning turbine. (c) The turbine is one component of a model closed-cycle gas turbine engine. Calculate the maximum efficiency of the engine. • Find the instantaneous velocity of the particle described in Figure$\mathrm{P} 2.1$at the following times: (a)$t=1.0 \mathrm{s},$(b)$t=3.0 \mathrm{s},(\mathrm{c}) t=4.5 \mathrm{s},$and$(\mathrm{d}) t=7.5 \mathrm{s}$• Two waves are traveling in the same direction along a stretched string. The waves are 90.0° out of phase. Each wave has an amplitude of 4.00 cm. Find the amplitude of the resultant wave. • a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis as shown in Figure P6.48. So that the clothes will dry uniformly, they are made to tumble. The rate of rotation of the smooth-walled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of θ=θ= 68.0∘0∘ above the horizontal. If the radius of the tub is r=r= 0.330m,0.330m, what rate of revolution is needed? • A pendulum of length LL and mass MM has a spring of force constant kk connected to it at a distance hh below its point of suspension (Fig. Pl5.55). Find the frequency of vibration of the system for small values of the amplitude (small θ)θ) Assume the vertical suspension rod of length LL is rigid, but ignore its mass. • The instantaneous position of an object is specified by its position vector leading from a fixed origin to the location of the object, modeled as a particle. Suppose for a certain object the position vector is a function of time given by r→=4i^+3j^−2tk^,r→=4i^+3j^−2tk^, where r→r→ is in meters and tt is in seconds. (a) Evaluate dr→/dt.dr→/dt. (b) What physical quantity does dr→/dtdr→/dt represent about the object? • A roller-coaster car (Fig. P6.16) has a mass of 500 kg when fully loaded with passengers. The path of the coaster from its initial point shown in the figure to point involves only up-and-down motion (as seen by the riders), with no motion to the left or right. (a) If the vehicle has a speed of 20.0 m/s at point , what is the force exerted by the track on the car at this point? (b) What is the maximum speed the vehicle can have at point B and still remain on the track? Assume the roller-coaster tracks at points @ and (B) are parts of vertical circles of radius r1=10.0m and r2= 15.0m, respectively. • The quantity ntnt in Equations 37.17 and 37.18 is called the optical path length corresponding to the geometrical distance tt and is analogous to the quantity δδ in Equation 37.1 , the path difference. The optical path length is proportional to nn because a larger index of refraction shortens the wavelength, so more cycles of a wave fit into a particular geometrical distance. (a) Assume a mixture of corn syrup and water is prepared in a tank, with its index of refraction nn increasing uniformly from 1.33 at y=20.0cmy=20.0cm at the top to 1.90 at y=0.y=0. Write the index of refraction n(y)n(y) as a function of yy . (b) Compute the optical path length correspond- ∫200cm∫020cm (c) Suppose a narrow beam of light is directed into the mixture at a nonzero angle with respect to the normal to the surface of the mixture. Qualitatively describe its path. • A sample of a monatomic ideal gas occupies 5.00 L at atmospheric pressure and 300 K (point A in Fig. P21.63). It is warmed at constant volume to 3.00 atm (point B). Then it is allowed to expand isothermally to 1.00 atm (point C) and at last compressed isobarically to its original state. (a) Find the number of moles in the sample. Find (b) the temperature at point B, (c) the temperature at point C, and (d) the volume at point C.C. (e) Now consider the processes A→B,B→C,A→B,B→C, and C→A.C→A. Describe how to carry out each process experimentally. (f) Find Q,W,Q,W, and ΔEintΔEint for each of the processes. (g) For the whole cycle A→B→C→A,A→B→C→A, find Q,W,Q,W, and ΔEintΔEint • A Geiger–Mueller tube is a radiation detector that consists of a closed, hollow, metal cylinder (the cathode) of inner radius rara and a coaxial cylindrical wire (the anode) of radius rbrb (Fig. P25.66a). The charge per unit length on the anode is λ,λ, and the charge per unit length on the cathode is −λ.−λ. A gas fills the space between the electrodes. When the tube is in use (Fig. P25.66b) and a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the electric potential difference between the wire and the cylinder is ΔV=2keλln(rarb)ΔV=2keλln⁡(rarb) (b) Show that the magnitude of the electric field in the space between cathode and anode is E=ΔVln(ra/rb)(1r)E=ΔVln⁡(ra/rb)(1r) where rr is the distance from the axis of the anode to the point where the field is to be calculated. • A fish swimming in a horizontal plane has velocity →vi=(4.00ˆi+1.00ˆj)m/s at a point in the ocean where the position relative to a certain rock is →ri=(10.0ˆi−4.00ˆj)m After the fish swims with constant acceleration for 20.0 s its velocity is →v=(20.0ˆi−5.00ˆj)m/s (a) What are the components of the acceleration of the fish? (b) What is the direction of its acceleration with respect to unit vector ˆi ? (c) If the fish maintains constant acceleration, where is it at t=25.0s and in what direction is it moving? • A 3.50 -kN piano is lifted by three workers at constant speed to an apartment 25.0 mm above the street using a pulley system fastened to the roof of the building. Each worker is able to deliver 165 WW of power, and the pulley system is 75.0%% efficient (so that 25.0%% of the mechanical energy is transformed to other forms due to friction in the pulley). Neglecting the mass of the pulley, find the time required to lift the piano from the street to the apartment. • An infinitely long, straight wire carrying a current I1I1 is partially surrounded by a loop as shown in Figure P30.66P30.66 . The loop has a length LL and radius R,R, and it carries a current I2.I2. The axis of the loop coincides with the wire. Calculate the magnetic force exerted on the loop. • An all-electric car (not a hybrid) is designed to run from a bank of 12.0−V12.0−V batteries with total energy storage of 2.00×107J2.00×107J . If the electric motor draws 8.00 kWkW as the car moves at a steady speed of 20.0m/s,20.0m/s, (a) what is the current delivered to the motor? (b) How far can the car travel before it is “out of juice”? • Why is the following situation impossible? Two samples of water are mixed at constant pressure inside an insulated container: 1.00 kgkg of water at 10.0∘0∘C and 1.00 kgkg of water at 30.0∘C30.0∘C . Because the container is insulated, there is no exchange of energy by heat between the water and the environment. Furthermore, the amount of energy that leaves the warm water by heat is equal to the amount that enters the cool water by heat. Therefore, the entropy change of the Universe is zero for this process. • The half-life of 131 II is 8.04 days. (a) Calculate the decay constant for this nuclide. (b) Find the number of 13 II nuclei necessary to produce a sample with an activity of 6.40 mCimCi . (c) A sample of 131I131I with this initial activity decays for 40.2 dd . What is the activity at the end of that period? • A 60.0-kg woman stands at the western rim of a horizontal turntable having a moment of inertia of 500 kg⋅m2kg⋅m2 and a radius of 2.00m.2.00m. The turntable is initially at rest and is free to rotate about a friction less, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. Consider the woman–turntable system as motion begins. (a) Is the mechanical energy of the system constant? (b) Is the momentum of the system constant? (c) Is the angular momentum of the system constant? (d) In what direction and with what angular speed does the turntable rotate? (e) How much chemical energy does the woman’s body convert into mechanical energy of the woman–turntable system as the woman sets herself and the turntable into motion? • Kathy tests her new sports car by racing with Stan, an experienced racer. Both start from rest, but Kathy leaves the starting line 1.00 s after Stan does. Stan moves with a constant acceleration of$3.50 \mathrm{m} / \mathrm{s}^{2},$while Kathy maintains an acceleration of 4.90$\mathrm{m} / \mathrm{s}^{2}$. Find$(\mathrm{a})$the time at which Kathy overtakes Stan, (b) the distance she travels before she catches him, and (c) the speeds of both cars at the instant Kathy overtakes Stan. • The combination of an applied force and a friction force produces a constant total torque of 36.0 N⋅mN⋅m on a wheel rotating about a fixed axis. The applied force acts for 00 s. During this time, the angular speed of the wheel increases from 0 to 10.0 rad/srad/s . The applied force is then removed, and the wheel comes to rest in 60.0 s. Find (a) the moment of inertia of the wheel, (b) the magnitude of the torque due to friction, and (c) the total number of revolutions of the wheel during the entire interval of 66.0 ss . • Calculate the magnitude of the orbital angular momentum for a hydrogen atom in (a) the 4 dd state and (b) the 6 ff state. • Assume an x-ray technician takes an average of eight x-rays per workday and receives a dose of 5.0 rem/yr as a result. (a) Estimate the dose in rem per x-ray taken. (b) Explain how the technician’s exposure compares with low-level background radiation. • Find the energy released in the fusion reaction 11H+21H→32He+γ11H+21H→32He+γ • A standing wave is set up in a string of variable length and tension by a vibrator of variable frequency. Both ends of the string are fixed. When the vibrator has a frequency f,f, in a string of length LL and under tension T,nT,n antinodes are set up in the string. (a) If the length of the string is doubled, by what factor should the frequency be changed so that the same number of antinodes is produced? (b) If the frequency and length are held constant, what tension will produce n+1n+1 antinodes? (c) If the frequency is tripled and the length of the string is halved, by what factor should the tension be changed so that twice as many antinodes are produced? • Consider the circuit shown in Figure P32.17P32.17 . (a) When the switch is in position a,a, for what value of RR will the circuit have a time constant of 15.0μsμs ? (b) What is the current in the inductor at the instant the switch is thrown to position bb ? • Show that a magnetic dipole in a uniform magnetic field, displaced from its equilibrium orientation and released, can oscillate as a torsional pendulum (Section 15.5) in simple harmonic motion. (b) Is this statement true for all angular displacements, for all displacements less than 180∘180∘ , or only for small angular displacements? Explain. (c) Assume the dipole is a compass needle-a light bar magnet-with a magnetic moment of magnitude μ.μ. It has moment of inertia II about its center, where it is mounted on a friction less, vertical axle, and it is placed in a horizontal magnetic field of magnitude B. Determine its frequency of oscillation. (d) Explain how the compass needle can be conveniently used as an indicator of the magnitude of the external magnetic field. (e) If its frequency is 0.680 Hz in the Earth’s local field, with a horizontal component of 39.2μTμT , what is the magnitude of a field parallel to the needle in which its frequency of oscillation is 4.90 Hz2Hz2 • A solid sphere of mass mm and radius rr rolls without slipping along the track shown in Figure P10.77P10.77 . It starts from rest with the lowest point of the sphere at height hh above the bottom of the loop of radius R,R, much larger than rr . (a) What is the minimum value of hh (in terms of R)R) such that the sphere completes the loop? (b) What are the force components on the sphere at the point PP if h=3Rh=3R ? • Suppose a luminous sphere of radius R1R1 (such as the Sun is surrounded by a uniform atmosphere of radius R2>R2> R1R1 and index of refraction nn . When the sphere is viewed from a location far away in vacuum, what is its apparent radius (a) when R2>nR1R2>nR1 and (b) when R2<nR1R2<nR1 . • A velocity selector consists of electric and magnetic fields described by the expressions →E=EˆkE→=Ek^ and →B=Bˆj,B→=Bj^, with B=15.0B=15.0 mTmT . Find the value of EE such that a 750 -eV electron moving in the negative xx direction is undeflected. • A thief hides a precious jewel by placing it on the bottom of a public swimming pool. He places a circular raft on the surface of the water directly above and centeredover the jewel as shown in figure P35.55. The surface of the water is calm. The raft, of diameter d=4.54md=4.54m , prevents the jewel from being seen by any observer above the water, either on the raft or on the side of the pool. What is the maximum depth hh of the pool for the jewel to remain unseen? • The wave function for a quantum particle is given by ψ(x)=Axψ(x)=Ax between x=0x=0 and x=1.00x=1.00 , and ψ(x)=0ψ(x)=0 elsewhere. Find (a)(a) the value of the normalization constant AA (b) the probability that the particle will be found between x=0.300x=0.300 and x=0.400,x=0.400, and (c)(c) the expectation value of the particle’s position. • The LCLC circuit of a radar transmitter oscillates at 9.00 GHzGHz . (a) What inductance is required for the circuit to resonate at this frequency if its capacitance is 2.00 pFpF ? (b) What is the inductive reactance of the circuit at this frequency? • In a 30.0 -s interval, 500 hailstones strike a glass window of area 0.600 m2m2 at an angle of 45.0∘0∘ to the window surface. Each hailstone has a mass of 5.00 g and a speed of 8.00 m/s. Assuming the collisions are elastic, find (a) the average force and (b) the average pressure on the window during this interval. • A steel wire of length 30.0 m and a copper wire of length 20.0m,20.0m, both with 1.00 -mm diameters, are connected end to end and stretched to a tension of 150 NN . During what time interval will a transverse wave travel the entire length of the two wires? • Two points in a plane have polar coordinates (2.50m,30.0∘)(2.50m,30.0∘) and (3.80m,120.0∘).(3.80m,120.0∘). Determine (a) the Cartesian coordinates of these points and (b) the distance between them. • Sinusoidal waves 5.00 cmcm in amplitude are to be transmitted along a string that has a linear mass density of 4.00×10−2kg/m.4.00×10−2kg/m. The source can deliver a maximum power of 300 WW , and the string is under a tension of 100 NN . What is the highest frequency ff at which the source can operate? • In a purely inductive AC circuit as shown in Figure P33.10,ΔVmax=100VP33.10,ΔVmax=100V . (a) The maximum current is 7.50 AA at 50.0 HzHz . Calculate the inductance L.L. (b) What If? At what angular frequency ωω is the maximum current 2.50 AA ? • You attach an object to the bottom end of a hanging vertical spring. It hangs at rest after extending the spring 18.3 cm. You then set the object vibrating. (a) Do you have enough information to find its period? (b) Explain your answer and state whatever you can about its period. • A 6000-kg freight car rolls along rails with negligible friction. The car is brought to rest by a combination of two coiled springs as illustrated in Figure P7.27. Both springs are described by Hooke’s law and have spring constants k1=1600N/m and k2=3400N/m. After the first spring compresses a distance of 30.0cm, the second spring acts with the first to increase the force as additional compression occurs as shown in the graph. The car comes to rest 50.0 cm after first contacting the two-spring system. Find the car’s initial speed. • Calculate the minimum energy required to remove a neutron from the 43 CaCa nucleus. • Astronomers observe a series of spectral lines in the light from a distant galaxy. On the hypothesis that the lines form the Lyman series for a (new?) one-electron atom, they start to construct the energy-level diagram shown in Figure P42.70, which gives the wavelengths of the first four lines and the short-wavelength limit of this series. Based on this information, calculate (a) the energies of the ground state and first four excited states for this one-electron atom and (b) the wavelengths of the first three lines and the short-wavelength limit in the Balmer series for this atom. (c) Show that the wavelengths of the first four lines and the short-wavelength limit of the Lyman series for the hydrogen atom are all 60.0% of the wavelengths for the Lyman series in the one-electron atom in the distant galaxy. (d) Based on this observation, explain why this atom could be hydrogen. • Determine the acceleration of the center of mass of a uniform solid disk rolling down an incline making angle θθ with the horizontal. (b) Compare the acceleration found in part (a) with that of a uniform hoop. (c) What is the minimum coefficient of friction required to maintain pure rolling motion for the disk? • Tritium has a half-life of 12.33 years. What fraction of the nuclei in a tritium sample will remain (a) after 5.00 yr? (b) After 10.0 yr? (c) After 123.3 yr? (d) According to Equation 44.6, an infinite amount of time is required for the entire sample to decay. Discuss whether that is realistic. • Suppose a quantum particle is in its ground state in a box that has infinitely high walls (see Active Fig. 41.4a). Now suppose the left-hand wall is suddenly lowered to a finite height and width. (a) Qualitatively sketch the wave function for the particle a short time later. (b) If the box has a length L, what is the wavelength of the wave that penetrates the left-hand wall? • A large block P attached to a light spring executes horizontal, simple harmonic motion as it slides across a frictionless surface with a frequency f. Block B rests on it as shown in Figure P15.53, and the coefficient of static friction between the two is μsμs. What maximum amplitude of oscillation can the system have if block B is not to slip? • Calculate the electric potential 0.250 cm from an electron. (b) What is the electric potential difference between two points that are 0.250 cm and 0.750 cm from an electron? (c) How would the answers change if the electron were replaced with a proton? • The student engineer of a campus radio station wishes to verify the effectiveness of the lightning rod on the antenna mast (Fig. P28.73). The unknown resistance RxRx is between points CC and E.E. Point EE is a true ground, but it is inaccessible for direct measurement because this stratum is several meters below the Earth’s surface. Two identical rods are driven into the ground at AA and BB , introducing an unknown resistance RyRy . The procedure is as follows. Measure resistance R1R1 between points AA and B,B, then connect AA and BB with a heavy conducting wire and measure resistance R2R2 between points AA and C.C. (a) Derive an equation for RxRx in terms of the observable resistances, R1R1 and R2R2 . (b) AA satisfactory ground resistance would be Rx<2.00ΩRx<2.00Ω . Is the grounding of the station adequate if measurements give R1=13.0ΩR1=13.0Ω and R2=6.00ΩR2=6.00Ω ? Explain. • Estimate the mass of the air in your bedroom. State the quantities you take as data and the value you measure or estimate for each. • Magnetic field values are often determined by using a device known as a search coil. This technique depends onthe measurement of the total charge passing through a coil in a time interval during which the magnetic flux linking the windings changes either because of the coil’s motion or because of a change in the value of B. (a) Show that as the flux through the coil changes from Φ1 to Φ2, the charge transferred through the coil is given by Q=N(Φ2−Φ1)/R, where R is the resistance of the coil and N is the number of turns. (b) As a specific example, calculate B when a total charge of 5.00×10−4C passes through a 100 -turn coil of resistance 200Ω and cross-sectional area 40.0 cm2 as it is rotated in a uniform field from a position where the plane of the coil is perpendicular to the field to a position where it is parallel to the field. • In the circuit of Figure P28.39P28.39 , the switch SS has been open for a long time. It is then suddenly closed. Take ε=10.0Vε=10.0V , R1=50.0kΩ,R2=100kΩ,R1=50.0kΩ,R2=100kΩ, and C=10.0μFC=10.0μF . Determine the time constant (a) before the switch is closed and (b) after the switch is closed. (c) Let the switch be closed at t=0t=0 . Determine the current in the switch as a function of time. • A series RLCRLC circuit has components with the following values: L=20.0mH,C=100nF,R=20.0Ω,L=20.0mH,C=100nF,R=20.0Ω, and ΔVmax=ΔVmax= 100V,100V, with Δv=ΔVmaxsinωtΔv=ΔVmaxsinωt . Find (a)(a) the resonant frequency of the circuit, (b) the amplitude of the current at the resonant frequency, (c) the QQ of the circuit, and (d) the amplitude of the voltage across the inductor at resonance. • A Hall-effect probe operates with a 120 -mA current. When the probe is placed in a uniform magnetic field of magnitude 0.0800 TT , it produces a Hall voltage of 0.700μVμV . (a) When it is used to measure an unknown magnetic field, the Hall voltage is 0.330μVμV . What is the magnitude of the unknown field? (b) The thickness of the probe in the direction of →BB→ is 2.00 mm. Find the density of the charge carriers, each of which has charge of magnitude ee . • Two identical loudspeakers are placed on a wall 2.00 m apart. A listener stands 3.00 m from the wall directly in front of one of the speakers. A single oscillator is driving the speakers at a frequency of 300 Hz. (a) What is the phase difference in radians between the waves from the speakers when they reach the observer? (b) What If? What is the frequency closest to 300 Hz to which the oscillator may be adjusted such that the observer hears minimal sound? • A light, rigid rod is 77.0 cmcm long. Its top end is pivoted on a frictionless, horizontal axle. The rod hangs straight down at rest with a small, massive ball attached to its bottom end. You strike the ball, suddenly giving it a horizontal velocity so that it swings around in a full circle. What minimum speed at the bottom is required to make the ball go over the top of the circle? • In Example 4.6,4.6, we found the centripetal acceleration of the Earth as it revolves around the Sun. From information on the endpapers of this book, compute the centripetal acceleration of a point on the surface of the Earth at the equator caused by the rotation of the Earth about its axis. • A certain grade of crude oil has an index of refraction of 1.25.1.25. A ship accidentally spills 1.00 m3m3 of this oil into the ocean, and the oil spreads into a thin, uniform slick. If the film produces a first-order maximum of light of wavelength 500 nmnm normally incident on it, how much surface area of the ocean does the oil slick cover? Assume the index of refraction of the ocean water is 1.34 . • Vector →AA→ has a negative xx component 3.00 units in length and a positive yy component 2.00 units in length. (a) Determine an expression for →AA→ in unit-vector notation. (b) Determine the magnitude and direction of →AA→ . (c) What vector →BB→ when added to →AA→ gives a resultant vector with no xx component and a negative yy component 4.00 units in length? • Why is the following situation impossible? The centers of two homogeneous spheres are 1.00 mm apart. The spheres are each made of the same element from the periodic table. The gravitational force between the spheres is 1.00 N. • The visible section of the Universe is a sphere centered on the bridge of your nose, with radius 13.7 billion light-years. (a) Explain why the visible Universe is getting larger, with its radius increasing by one light-year in every year. (b) Find the rate at which the volume of the visible section of the Universe is increasing. • Spherical waves of wavelength 45.0 cm propagate outward from a point source. (a) Explain how the intensity at a distance of 240 cm compares with the intensity at a distance of 60.0 cm. (b) Explain how the amplitude at a distance of 240 cm compares with the amplitude at a distance of 60.0 cm. (c) Explain how the phase of the wave at a distance of 240 cm compares with the phase at 60.0 cm at the same moment. • A light ray enters a rectangular block of plastic at an angle θ1=45.0∘θ1=45.0∘ and emerges at an angle θ2=θ2= 76.0∘0∘ as shown in Figure P35.69. (a) Determine the index of refraction of the plastic. (b) If the light ray enters the plastic at a point L=50.0cmL=50.0cm from the bottom edge, what time interval is required for the light ray to travel through the plastic? • A camera is being used with a correct exposure at$f / 4$and a shutter speed of$\frac{1}{15}$s. In addition to the$f$-numbers listed in Section$36.6,$this camera has$f$-numbers$f / 1, f / 1.4,$and$f / 2 .$To photograph a rapidly moving subject, the shutter speed is changed to$\frac{1}{125}$. Find the new$f$-number setting needed on this camera to maintain satisfactory exposure. • A uniform beam resting on two pivots has a length L=6.00mL=6.00m and mass M=90.0kgM=90.0kg . The pivot under the left end exerts a normal force n1n1 on the beam, and the second pivot located a distance ℓ=4.00mℓ=4.00m from the left end exerts a normal force n2.n2. A woman of mass m=55.0kgm=55.0kg steps onto the left end of the beam and begins walking to the right as in Figure P12.38P12.38 . The goal is to find the woman’s position when the beam begins to tip. (a) What is the appropriate analysis model for the beam before it begins to tip? (b) Sketch a force diagram for the beam, labeling the gravitational and normal forces acting on the beam and placing the woman a distance xx to the right of the first pivot, which is the origin. (c) Where is the woman when the normal force n1n1 is the greatest? (d) What is n1n1 when the beam is about to tip? (e) Use Equation 12.1 to find the value of n2n2 when the beam is about to tip. ( ff ) Using the result of part (d) and Equation 12.2 , with torques computed around the second pivot, find the woman’s position xx when the beam is about to tip. (g) Check the answer to part (e) by computing torques around the first pivot point. • Why is the following situation impossible? An inquisitive physics student takes a 100−W100−W lightbulb out of its socket and measures its resistance with an ohmmeter. He measures a value of 10.5ΩΩ . He is able to connect an ammeter to the lightbulb socket to correctly measure the current drawn by the bulb while operating. Inserting the bulb back into the socket and operating the bulb from a 120−V120−V source, he measures the current to be 11.4 AA . • Suppose the target in a laser fusion reactor is a sphere of solid hydrogen that has a diameter of 1.50×10−4m and a density of 0.200g/cm3. Assume half of the nuclei are 2 H and half are 3H . (a) If 1.00% of a 200−kJ laser pulse is delivered to this sphere, what temperature does the sphere reach? (b) If all the hydrogen fuses according to the D−T reaction, how many joules of energy are released? • A steel wire and a copper wire, each of diameter 2.000mm,2.000mm, are joined end to end. At 40.0∘C,40.0∘C, each has an unstretched length of 2.000 mm . The wires are connected between two fixed supports 4.000 mm apart on a tabletop. The steel wire extends from x=−2.000mx=−2.000m to x=0,x=0, the copper wire extends from x=0x=0 to x=2.000m,x=2.000m, and the tension is negligible. The temperature is then lowered to 20.0∘C20.0∘C . Assume the average coefficient of linear expansion of steel is 11.0×10−6(∘C)−111.0×10−6(∘C)−1 and that of copper is 17.0×17.0× 10−6C∘C−110−6C∘C−1 . Take Young’s modulus for steel to be 20.0×20.0× 1010N/m21010N/m2 and that for copper to be 11.0×1010N/m2.11.0×1010N/m2. At this lower temperature, find (a) the tension in the wire and (b) the xx coordinate of the junction between the wires. • Heedless of danger, a child leaps onto a pile of old mattresses to use them as a trampoline. His motion between two particular points is described by the energy conservation equation 12(46.0kg)(2.40m/s)2+(46.0kg)(9.80m/s2)(2.80m+x)=12(1.94×104N/m)x212(46.0kg)(2.40m/s)2+(46.0kg)(9.80m/s2)(2.80m+x)=12(1.94×104N/m)x2 (a) Solve the equation for x.x. (b) Compose the statement of a problem, including data, for which this equation gives the solution. (c) Add the two values of xx obtained in part (a) and divide by 2.2. (d) What is the significance of the resulting value in part (c)? • Three identical 60.0-W, 120-V lightbulbs are connected across a 120-V power source as shown in Figure P28.66. Assuming the resistance of each lightbulb is constant (even though in reality the resistance might increase markedly with current), find (a) the total power supplied by the power source and (b) the potential difference across each lightbulb. • The radioactive isotope 137 Ba has a relatively short half- life and can be easily extracted from a solution containing its parent 137Cs . This barium isotope is commonly used in an undergraduate laboratory exercise for demonstrating the radioactive decay law. Undergraduate students using modest experimental equipment took the data presented in Figure P44.72. Determine the half-life for the decay of 137 Ba using their data. • Show that the wave function ψ=Aei(kx−ωt) is a solution to the Schrödinger equation (Eq. 41.15), where k=2π/λ and U=0. • In Figure P37.16,P37.16, let L=120cmL=120cm and d=0.250cm.d=0.250cm. The slits are illuminated with coherent 600 -nm light. Calculate the distance yy from the central maximum for which the average intensity on the screen is 75.0%% of the maximum. • Helium gas is sold in steel tanks that will rupture if subjected to tensile stress greater than its yield strength of 5×108N/m25×108N/m2 . If the helium is used to inflate a balloon, could the balloon lift the spherical tank the helium came in? Justify your answer. Suggestion: You may consider a spherical steel shell of radius rr and thickness tt having the density of iron and on the verge of breaking apart into two hemispheres because it contains helium at high pressure. • Consider a ring of radius RR with the total charge QQ spread uniformly over its perimeter. What is the potential difference between the point at the center of the ring and a point on its axis a distance 2RR from the center? • A truck covers 40.0 m in 8.50 s while smoothly slowing down to a final speed of 2.80 m/s. (a) Find its original speed. (b) Find its acceleration. • A piece of insulated wire is shaped into a figure eight as shown in Figure P31.18 . For simplicity, model the two halves of the figure eight as circles. The radius of the upper circle is 5.00 cm and that of the lower circle is 9.00cm. The wire has a uniform resistance per unit length of 3.00Ω/m . A uniform magnetic field is applied perpendicular to the plane of the two circles, in the direction shown. The magnetic field is increasing at a constant rate of 2.00 T/s . Find (a) the magnitude and (b) the direction of the induced current in the wire. • An aluminum wire with a diameter of 0.100 mm has a uniform electric field of 0.200 V/m imposed along its entire length. The temperature of the wire is 50.0∘0∘C . Assume one free electron per atom. (a) Use the information in Table 27.2 to determine the resistivity of aluminum at this temperature. (b) What is the current density in the wire? (c) What is the total current in the wire? (d) What is the drift speed of the conduction electrons? (e) What potential difference must exist between the ends of a 2.00−m2.00−m length of the wire to produce the stated electric field? • A house has well-insulated walls. It contains a volume of 100 m3m3 of air at 300 KK . (a) Calculate the energy required to increase the temperature of this diatomic ideal gas by 1.00∘1.00∘C. (b) What If? If all this energy could be used to lift an object of mass mm through a height of 2.00m,2.00m, what is the value of m?m? • An electron moves in a three-dimensional box of edge length LL and volume L3L3 . The wave function of the particle is ψ=Asin(kxx)sin(kyy)sin(kzz).ψ=Asin⁡(kxx)sin⁡(kyy)sin⁡(kzz). Show that its energy is given by Equation 43.20 , E=ℏ2π22meL2(n2x+n2y+n2z)E=ℏ2π22meL2(nx2+ny2+nz2) where the quantum numbers (nx,ny,nz)(nx,ny,nz) are integers ≥1.≥1. Suggestion: The Schrodinger equation in three dimensions may be written ℏ22m(∂2ψ∂x2+∂2ψ∂y2+∂2ψ∂z2)=(U−E)ψℏ22m(∂2ψ∂x2+∂2ψ∂y2+∂2ψ∂z2)=(U−E)ψ • A person wears a hearing aid that uniformly increases the sound level of all audible frequencies of sound by 30.0 dBdB . The hearing aid picks up sound having a frequency of 250 HzHz at an intensity of 3.0×10−11W/m2.3.0×10−11W/m2. What is the intensity delivered to the eardrum? • A particle with charge qq is located at x=−R,x=−R, and a particle with charge −2q−2q is located at the origin. Prove that the equipotential surface that has zero potential is a sphere centered at (−4R/3,0,0)(−4R/3,0,0) and having a radius r=23R.r=23R. • A certain toaster has a heating element made of Nichrome wire. When the toaster is first connected to a 120-V source (and the wire is at a temperature of 20.0°C), the initial current is 1.80 A. The current decreases as the heating element warms up. When the toaster reaches its final operating temperature, the current is 1.53 A. (a) Find the power delivered to the toaster when it is at its operating temperature. (b) What is the final temperature of the heating element? • A particle of mass 0.400 kg is attached to the 100-cm mark of a meter stick of mass 0.100 kg. The meter stick rotates on the surface of a friction less, horizontal table with an angular speed of 4.00 rad/s. Calculate the angular momentum of the system when the stick is pivoted about an axis (a) perpendicular to the table through the 50.0-cm mark and (b) perpendicular to the table through the 0-cm mark. • A sample of radioactive material contains 1.00×10151.00×1015 atoms and has an activity of 6.00×1011Bq.6.00×1011Bq. What is its half-life? • A thin, square, conducting plate 50.0 cm on a side lies in the xyxy plane. A total charge of 4.00×10−8C4.00×10−8C is placed on the plate. Find (a) the charge density on each face of the plate, (b) the electric field just above the plate, and (c) the electric field just below the plate. You may assume the charge density is uniform. • A cube of wood having an edge dimension of 20.0$\mathrm{cm}$and a density of 650$\mathrm{kg} / \mathrm{m}^{3}$floats on water. (a) What is the distance from the horizontal top surface of the cube to the water level? (b) What mass of lead should be placed on the cube so that the top of the cube will be just level with the water surface? • Can a particle moving with instantaneous speed 3.00 m/sm/s on a path with radius of curvature 2.00 mm have an acceleration of magnitude 6.00 m/s2?m/s2? (b) Can it have an acceleration of magnitude 4.00 m/s2?m/s2? In each case, if the answer is yes, explain how it can happen; if the answer is no, explain why not. • A particle with a mass of 2.00×10−16kg and a charge of 30.0 nC starts from rest, is accelerated through a potential difference ΔV and is fired from a small source in a region containing a uniform, constant magnetic field of magnitude 0.600 T . The particle’s velocity is perpendicular to the magnetic field lines. The circular orbit of the particle as it returns to the location of the source encloses a magnetic flux of 15.0μWb . (a) Calculate the particle’s speed. (b) Calculate the potential difference through which the particle was accelerated inside the source. • The graph in Figure P7.30 specifies a functional relationship between the two variables u and v . (a) Find ∫baudv⋅(b) Find∫abudv (c) Find ∫bavdu. • Two spheres have radii a and b, and their centers are a distance d apart. Show that the capacitance of this system is C=4πϵ01a+1b−2dC=4πϵ01a+1b−2d provided dd is large compared with aa and b.b. Suggestion: Because the spheres are far apart, assume the potential of each equals the sum of the potentials due to each sphere. (b) Show that as dd approaches infinity, the above result reduces to that of two spherical capacitors in series. • A sinusoidal wave in a string is described by the wave function y=0.150sin(0.800x−50.0t)y=0.150sin⁡(0.800x−50.0t) where xx and yy are in meters and tt is in seconds. The mass per length of the string is 12.0g/m.12.0g/m. (a) Find the maximum transverse acceleration of an element of this string. (b) Determine the maximum transverse force on a 1.00-cmcm segment of the string. (c) State how the force found in part (b) compares with the tension in the string. • Ocean waves with a crest-to-crest distance of 10.0 m can be described by the wave function y(x,t)=0.800sin[0.628(x−vt)]y(x,t)=0.800sin⁡[0.628(x−vt)] where xx and yy are in meters, tt is in seconds, and v=v= 1.20 m/sm/s . (a) Sketch y(x,t)y(x,t) at t=0.t=0. (b) Sketch y(x,t)y(x,t) at t=t= 2.00 ss . (c) Compare the graph in part (b) with that for part (a) and explain similarities and differences. (d) How has he wave moved between graph (a) and graph (b)? • In an HCl molecule, take the ClCl atom to be the isotope 35Cl35Cl The equilibrium separation of the H and Cl atoms is 0.127 46 nm. The atomic mass of the H atom is 1.007 825 u and that of the 35Cl35Cl atom is 34.968 853 u. Calculate the longest wavelength in the rotational spectrum of this molecule. (b) What If? Repeat the calculation in part (a), but take the Cl atom to be the isotope 37Cl37Cl which has atomic mass 36.965 903 u. The equilibrium separation distance is the same as in part (a). (c) Naturally occurring chlorine contains approximately three parts of 55Cl55Cl to one part of 37Cl37Cl . Because of the two different Cl masses, each line in the microwave rotational spectrum of HCl is split into a doublet as shown in Figure P 43.19. Calculate the separation in wavelength between the doublet lines for the longest wavelength. • Figure P28.70 shows a circuit model for the transmission of an electrical signal such as cable TV to a large number of subscribers. Each subscriber connects a load resistance RL between the transmission line and the ground. The ground is assumed to be at zero potential and able to carry any current between any ground connections with negligible resistance. The resistance of the transmission line between the connection points of different subscribers is modeled as the constant resistance RT . Show that the equivalent resistance across the signal source is Req=12[(4RTRL+R2T)1/2+RT]Req=12[(4RTRL+RT2)1/2+RT] • Consider a series RCRC circuit as in Figure P28.34P28.34 for which R=1.00MΩ,C=5.00μF,R=1.00MΩ,C=5.00μF, and ε=30.0Vε=30.0V . Find (a)(a) the time constant of the circuit and (b)(b) the maximum charge on the capacitor after the switch is thrown closed. (c) Find the current in the resistor 10.0 s after the switch is closed. • The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle. The speed of the airplane is 300 mi/h at the top of the loop and 450 mi/h at the bottom, and the radius of the circle is 1 200 ft. (a) What is the pilot’s apparent weight at the lowest point if his true weight is 160 lb? (b) What is his apparent weight at the highest point? (c) What If? Describe how the pilot could experience weightlessness if both the radius and the speed can be varied. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body. • Assume the intensity of sunlight is 1.00$\mathrm{kW} / \mathrm{m}^{2}$at a particular location. A highly reflecting concave mirror is to be pointed toward the Sun to produce a power of at least 350$\mathrm{W}$at the image point. (a) Assuming the disk of the Sun subtends an angle of$0.533^{\circ}$at the Earth, find the required radius$R_{a}$of the circular face area of the mirror. (b) Now suppose the light intensity is to be at least 120$\mathrm{kW} / \mathrm{m}^{2}$at the image. Find the required relationship between$R_{a}$and the radius of curvature$R$of the mirror. • Evaluate AA in the scalar equality 4(7+3)=4(7+3)=A. (b) Evaluate A,B,A,B, and CC in the vector equality 700i^+3.00k^=Ai^+Bj^+Ck^.700i^+3.00k^=Ai^+Bj^+Ck^. (c) Explain how you arrive at the answers to convince a student who thinks that you cannot solve a single equation for three different unknowns. (d) What If? The functional equality or identity A+Bcos(Cx+Dt+E)=7.00cos(3x+4t+2)A+Bcos⁡(Cx+Dt+E)=7.00cos⁡(3x+4t+2) is true for all values of the variables xx and t,t, measured in meters and in seconds, respectively. Evaluate the constants A,B,C,D,A,B,C,D, and E.E. (e) Explain how you arrive at your answers to part (d). • A negatively charged rod of finite length carries charge with a uniform charge per unit length. Sketch the electric field lines in a plane containing the rod. • Why is the following situation impossible? A normally proportioned adult walks briskly along a straight line in the +x+x direction, standing straight up and holding his right arm vertical and next to his body so that the arm does not swing. His right hand holds a ball at his side a distance hh above the floor. When the ball passes above a point marked as x=0x=0 on the horizontal floor, he opens his fingers to release the ball from rest relative to his hand. The ball strikes the ground for the first time at position x=7.00h.x=7.00h. • A biology laboratory is maintained at a constant temperature of 7.00∘00∘C by an air conditioner, which is vented to the air outside. On a typical hot summer day, the outside temperature is 27,0∘C27,0∘C and the air-conditioning unit emits energy to the outside at a rate of 10.0 kWkW . Model the unit as having a coefficient of performance (COP) equal to 40.0%% of the COP of an ideal Carnot device. (a) At what rate does the air conditioner remove energy from the laboratory? (b) Calculate the power required for the work input. (c) Find the change in entropy of the Universe produced by the air conditioner in 1.00 hh . (d) What If? The outside temperature increases to 32.0∘C32.0∘C . Find the fractional change in the COP of the air conditioner. • A step-down transformer is used for recharging the batteries of portable electronic devices. The turns ratio N2/N1N2/N1 for a particular transformer used in a DVD player is 1:131:13 . When used with 120−V120−V (rms) household service, the transformer draws an rms current of 20.0 mAmA from the house outlet. Find (a) the rms output voltage of the transformer and (b) the power delivered to the DVD player. • A small piece of Styrofoam packing material is dropped from a height of 2.00 mm above the ground. Until it reaches terminal speed, the magnitude of its acceleration is given by a=g−Bva=g−Bv . After falling 0.500 mm , the Styrofoam effectively reaches terminal speed and then takes 5.00 s more to reach the ground. (a) What is the value of the constant BB ? (b) What is the acceleration at t=0?t=0? (c) What is the acceleration when the speed is 0.150 m/sm/s ? • The tensile stress in a thick copper bar is 99.5%% of its elastic breaking point of 13.0×1010N/m2.13.0×1010N/m2. If a 500−Hz500−Hz sound wave is transmitted through the material, (a) what displacement amplitude will cause the bar to break? (b) What is the maximum speed of the clements of copper at this moment? (c) What is the sound intensity in the bar? • Considering an undamped, forced oscillator (b=0)(b=0) show that Equation 15.35 is a solution of Equation 15.34,15.34, with an amplitude given by Equation 15.36.15.36. • The RCRC high-pass filter shown in Figure P33.53P33.53 has a resistance R=0.500ΩR=0.500Ω and a capacitance C=613μFC=613μF . What is the ratio of the amplitude of the output voltage to that of the input voltage for this filter for a source frequency of 600 HzHz ? • An object has a kinetic energy of 275 JJ and a momentum of magnitude 25.0 kg⋅m/skg⋅m/s . Find the speed and mass of the object. • What If? The block of mass m=200gm=200g described in Problem 41 (Fig. P8.41) is released from rest at point QQ , and the surface of the bowl is rough. The block’s speed at point is 1.50 m/s. (a) What is its kinetic energy at point ? (b) How much mechanical energy is transformed into internal energy as the block moves from point to point ? (c) Is it possible to determine the coefficient of friction from these results in any simple manner? (d) Explain your answer to part (c). • The mean free path , of a molecule is the average distance that a molecule travels before colliding with another molecule. It is given by ℓ=1√2πd2NVℓ=12–√πd2NV where dd is the diameter of the molecule and NVNV is the number of molecules per unit volume. The number of collisions that a molecule makes with other molecules per unit time, or collision frequency f,f, is given by f=vargℓf=vargℓ (a) If the diameter of an oxygen molecule is 2.00×10−10m,2.00×10−10m, find the mean free path of the molecules in a scuba tank that has a volume of 12.0 LL and is filled with oxygen at a gauge pressure of 100 atmatm at a temperature of 25.0∘0∘C . (b) What is the average time interval between molecular collisions for a molecule of this gas? • In the circuit diagrammed in Figure P32.21, assume the switch has been closed for a long time interval and is opened at t=0.t=0. Also assume R=4.00Ω,L=1.00H,R=4.00Ω,L=1.00H, and E=10.0VE=10.0V . (a) Before the switch is opened, does the inductor behave as an open circuit, a short circuit, a resistor of some particular resistance, or none of those choices? (b) What current does the inductor carry? (c) How much energy is stored in the inductor for t<0t<0 ? (d) After the switch is opened, what happens to the energy previously stored in the inductor? (e) Sketch a graph of the current in the inductor for t≥0t≥0 . Label the initial and final values and the time constant. • How much charge can be placed on a capacitor with air between the plates before it breaks down if the area of each plate is 5.00 cm2?cm2? (b) What If? Find the maximum charge if polystyrene is used between the plates instead of air. • If it has enough kinetic energy, a molecule at the surface of the Earth can “escape the Earth’s gravita-tion” in the sense that it can continue to move away from the Earth forever as discussed in Section 13.6. Using the principle of conservation of energy, show that the minimum kinetic energy needed for “escape” is m0gRE,m0gRE, where m0m0 is the mass of the molecule, gg is the free-fall acceleration at the surface, and RERE is the radius of the Earth. (b) Calculate the temperature for which the minimum escape kinetic energy is ten times the average kinetic energy of an oxygen molecule. • A simple pendulum makes 120 complete oscillations in 3.00 minmin at a location where g=9.80m/s2.g=9.80m/s2. Find (a) the period of the pendulum and (b)(b) its length. • The total energy of a proton is twice its rest energy. Find the momentum of the proton in MeV/c units. • Two identical, flat, circular coils of wire each have 100 turns and radius R=0.500mR=0.500m . The coils are arranged as a set of Helmholtz coils so that the separation distance between the coils is equal to the radius of the coils (see Fig. P30.58P30.58 ). Each coil carries current I=10.0I=10.0 A. Determine the magnitude of the magnetic field at a point on the common axis of the coils and halfway between them. • Find the net electric flux through the cube shown in Figure P 24.14. (b) Can you use Gauss’s law to find the electric field on the surface of this cube? Explain. • Police radar detects the speed of a car (Fig. P39.23 on page 1180 ) as follows. Microwaves of a precisely known frequency are broadcast toward the car. The moving car reflects the microwaves with a Doppler shift. The reflected waves are received and combined with an attenuated version of the transmitted wave. Beats occur between the two microwave signals. The beat frequency is measured. (a) For an electro- magnetic wave reflected back to its source from a mirror approaching at speed vv , show that the reflected wave has frequency f′=c+vc−vff′=c+vc−vf where ff is the source frequency. (b) Noting that vv is much less than c,c, show that the beat frequency can be written as ff beat =2v/λ=2v/λ . (c) What beat frequency is measured for a car speed of 30.0 m/sm/s if the microwaves have frequency 10.0 GHzGHz (d) If the beat frequency measurement in part (c) is accurate to ±5.0Hz±5.0Hz , how accurate is the speed measurement? • If the average density of the Universe is small compared with the critical density, the expansion of the Universe described by Hubble’s law proceeds with speeds that are nearly constant over time. (a) Prove that in this case the age of the Universe is given by the inverse of the Hubble constant. (b) Calculate 1/H/H and express it in years. • A 1.50 -kg object is held 1.20 mm above a relaxed massless, vertical spring with a force constant of 320N/m.320N/m. The object is dropped onto the spring. (a) How far does the object compress the spring? (b) What If? Repeat part (a), but this time assume a constant air-resistance force of 0.700 NN acts on the object during its motion. (c) What If? How far does the object compress the spring if the same experiment is performed on the Moon, where g=1.63m/s2g=1.63m/s2 and air resistance is neglected? • The latent heat of vaporization for water at room temperature is 2430 J/gJ/g . Consider one particular molecule at the surface of a glass of liquid water, moving upward with sufficiently high speed that it will be the next molecule to join the vapor. (a) Find its translational kinetic energy. (b) Find its speed. Now consider a thin gas made only. of molecules like that one. (c) What is its temperature? (d) Why are you not burned by water evaporating from a vessel at room temperature? • Lisa in her Lamborghini accelerates at the rate of (3.00ˆi−2.00ˆj)m/s2,(3.00i^−2.00j^)m/s2, while Jill in her Jaguar accelerates at (1.00ˆi+3.00ˆj)m/s2(1.00i^+3.00j^)m/s2 . They both start from rest at the origin of an xyxy coordinate system. After 5.00s,5.00s, (a) what is Lisa’s speed with respect to Jill, (b) how far apart are they, and (c)(c) what is Lisa’s acceleration relative to Jill? • The large quadriceps muscle in the upper leg terminates at its lower end in a tendon attached to the upper end of the tibia (Fig. Pl2.52 a). The forces on the lower leg when the leg is extended are modeled as in Figure P12.52bP12.52b , where T¯¯¯¯T¯ is the force in the tendon, F→glegF→gleg is the gravitational force acting on the lower leg, and F→g.footF→g.foot is the gravitational force acting on the foot. Find TT when the tendon is at an angle of ϕ=25.0∘ϕ=25.0∘ with the tibia, assuming Fgleg=30.0N,Fgleg=30.0N, Fg.foot=12.5N,Fg.foot=12.5N, and the leg is extended at an angle θ=θ= 40.0∘0∘ with respect to the vertical. Also assume the center of gravity of the tibia is at its geometric center and the tendon attaches to the lower leg at a position one-fifth of the way down the leg. • A fission reactor is hit by a missile, and 5.00×106Ci of 90Sr, with half-life 29.1yr, evaporates into the air. The strontium falls out over an area of 104km2 . After what time interval will the activity of the “So Sr reach the agriculturally “safe” level of 2.00μCi/m27 • A long solenoid that has 1000 turns uniformly distributed over a length of 0.400 mm produces a magnetic field of magnitude 1.00×10−4T1.00×10−4T at its center. What current is required in the windings for that to occur? • A block of mass m=5.00kgm=5.00kg is released from point QQ and slides on the frictionless track shown in Figure P8.6. Determine (a) the block’s speed at points BB and CC and (b)(b) the net work done by the gravitational force on the block as it moves from point @@ to point CC . • For each of the following forbidden decays, determine what conservation laws are violated. (a)μ−→e−+γ(c)Λ0→p+π0(b)n→p+e−+νe(d)p→e++π0(a)μ−→e−+γ(b)n→p+e−+νe(c)Λ0→p+π0(d)p→e++π0 (e) Ξ0→n+π0Ξ0→n+π0 • An antelope is at a distance of 20.0$\mathrm{m}$from a converging lens of focal length$30.0 \mathrm{cm} .$The lens forms an image of the animal. (a) If the antelope runs away from the lens at a speed of 5.00$\mathrm{m} / \mathrm{s}, how fast does the image move? (b) Does the image move toward or away from the lens? • Consider the Bohr model of the hydrogen atom, with the electron in the ground state. The magnetic field at the nucleus produced by the orbiting electron has a value of 12.5 T. (See Problem 4 in Chapter 30. ) The proton can have its magnetic moment aligned in either of two directions perpendicular to the plane of the electron’s orbit. The interaction of the proton’s magnetic moment with the electron’s magnetic field causes a difference in energy between the states with the two different orientations of the proton’s magnetic moment. Find that energy difference in electron volts. • The pressure gauge on a tank registers the gauge pressure, which is the difference between the interior pressure and exterior pressure. When the tank is full of oxygen (O2),(O2), it contains 12.0 kgkg of the gas at a gauge pressure of 40.0 atmatm . Determine the mass of oxygen that has been with drawn from the tank when the pressure reading is 25.0 atmatm . Assume the temperature of the tank remains constant. • We have all complained that there aren’t enough hours in a day. In an attempt to fix that, suppose all the people in the world line up at the equator and all start running east at 2.50 m/s relative to the surface of the Earth. By how much does the length of a day increase? Assume the world population to be 7.00×1097.00×109 people with an average mass of 55.0 kg each and the Earth to be a solid homogeneous sphere. In addition, depending on the details of your solution, you may need to use the approximation 1/(1−x)≈1+x/(1−x)≈1+x for small x.x. • A boy starts at rest and slides down a frictionless slide as in Figure P8.43. The bottom of the track is a height h above the ground. The boy then leaves the track horizontally, striking the ground at a distance d as shown. Using energy methods, determine the initial height H of the boy above the ground in terms of h and d. • Consider the system pictured in Figure P 29.40. A 15.0-cm horizontal wire of mass 15.0 g is placed between two thin, vertical conductors, and a uniform magnetic field acts perpendicular to the page. The wire is free to move vertically without friction on the two vertical conductors. When a 5.00-A current is directed as shown in the figure, the horizontal wire moves upward at constant velocity in the presence of gravity. (a) What forces act on the horizontal wire, and (b) under what condition is the wire able to move upward at constant velocity? (c) Find the magnitude and direction of the minimum magnetic field required to move the wire at constant speed. (d) What happens if the magnetic field exceeds this minimum value? • The electric potential everywhere on the xyxy plane is V=36(x+1)2+y2−−−−−−−−−−−√−45×2+(y−2)2−−−−−−−−−−−√V=36(x+1)2+y2−45×2+(y−2)2 where VV is in volts and xx and yy are in meters. Determine the position and charge on each of the particles that create this potential. • A simple pendulum with a length of 2.23 m and a mass of 6.74 kg is given an initial speed of 2.06 m/s at its equilibrium position. Assume it undergoes simple harmonic motion. Determine (a) its period, (b) its total energy, and (c) its maximum angular displacement. • A smaller disk of radius rr and mass mm is attached rigidly to the face of a second larger disk of radius RR and mass MM as shown in Figure P15.72. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle θθ from its equilibrium position and released. (a) Show that the speed of the center of the small disk as it passes through the equilibrium position is v=2[Rg(1−cosθ)(M/m)+(r/R)2+2]1/2v=2[Rg(1−cos⁡θ)(M/m)+(r/R)2+2]1/2 (b) Show that the period of the motion is T=2π[(M+2m)R2+mr22mgR]1/2T=2π[(M+2m)R2+mr22mgR]1/2 • The coordinates of an object moving in the xy plane vary with time according to the equations x=−5.00sinωt and y=4.00−5.00cosωt, where ω is a constant, x and y are in meters, and t is in seconds. (a) Determine the components of velocity of the object at t=0. (b) Determine the components of acceleration of the object at t=0. (c) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time l>0 . (d) Describe the path of the object in an xy plot. • Energy transfers by heat through the exterior walls and roof of a house at a rate of 5.00×103J/s=5.00kW5.00×103J/s=5.00kW when the interior temperature is 22.0∘0∘C and the outside temperature is −5.00∘C−5.00∘C . (a) Calculate the electric power required to maintain the interior temperature at 22.0∘C22.0∘C if the power is used in electric resistance heaters that convert all the energy transferred in by electrical transmission into internal energy. (b) What If? Calculate the electric power required to maintain the interior temperature at 22.0∘C22.0∘C if the power is used to drive an electric motor that operates the compressor of a heat pump that has a coefficient of performance equal to 60.0%% of the Carnot-cycle value. • In Figure P23.21, determine the point (other than infinity) at which the electric field is zero. • Spacecraft I, containing students taking a physics exam, approaches the Earth with a speed of 0.600cc (relative to the Earth), while spacecraft II, containing professors proctoring the exam, moves at 0.280cc (relative to the Earth) directly toward the students. If the professors stop the exam after 50.0 min have passed on their clock, for what time interval does the exam last as measured by (a) the students and (b) an observer on the Earth? • Why is the following situation impossible? The effective force constant of a vibrating HCl molecule is k=480N/m.k=480N/m. A beam of infrared radiation of wavelength 6.20×103nm6.20×103nm is directed through a gas of HCl molecules. As a result, the molecules are excited from the ground vibrational state to the first excited vibrational state. • A series RLCRLC circuit consists of an 8.00−Ω8.00−Ω resistor, a 5.00−μF5.00−μF capacitor, and a 50.0−mH50.0−mH inductor. A variable frequency source applies an emf of 400 VV (rms) across the combination. Assuming the frequency is equal to one-half the resonance frequency, determine the power delivered to the circuit. • The nonrelativistic expression for the momentum of a particle, p=mu,p=mu, agrees with experiment if u<<c.u<<c. For what speed does the use of this equation give an error in the measured momentum of (a) 1.00%% and (b) 10.0%% ? • An electric motor rotating a workshop grinding wheel at 1.00×1021.00×102 rev/min is switched off. Assume the wheel has a constant negative angular acceleration of magnitude 2.00 rad/s2rad/s2 . (a) How long does it take the grinding wheel to stop? (b) Through how many radians has the wheel turned to during the time interval found in part (a)? • An isolated atom of a certain element emits light of wave- length 520 nmnm when the atom falls from its fifth excited state into its second excited state. The atom emits a photon of wavelength 410 nmnm when it drops from its sixth excited state into its second excited state. Find the wavelength of the light radiated when the atom makes a transition from its sixth to its fifth excited state. • A charged capacitor is connected to a resistor and switch as in Figure P28.41.P28.41. The circuit has a time constant of 1.50 ss .Soon after the switch is closed, the charge on the capaci- tor is 75.0%% of its initial charge. (a) Find the time interval required for the capacitor to reach this charge. (b) If R=R= 250kΩ,250kΩ, what is the value of C?C? • A 7.80 -g bullet moving at 575 m/s strikes the hand of a superhero, causing the hand to move 5.50 cm in the direction of the bullet’s velocity before stopping. (a) Use work and energy considerations to find the average force that stops the bullet. (b) Assuming the force is constant, determine how much time elapses between the moment the bullet strikes the hand and the moment it stops moving. • The gravitational force exerted on a baseball is −F8ˆj . A pitcher throws the ball with velocity vi by uniformly accelerating it along a straight horizontal line for a time interval of Δt=t−0=t (a) Starting from rest, through what distance does the ball move before its release? (b) What force does the pitcher exert on the ball? • An experiment is conducted to measure the electrical resistivity of Nichrome in the form of wires with different lengths and cross-sectional areas. For one set of measurements, a student uses 30-gauge wire, which has a cross-sectional area of 7.30×10−8m2.7.30×10−8m2. The student measures the potential difference across the wire and the current in the wire with a voltmeter and an ammeter, respectively. (a) For each set of measurements given in the table taken on wires of three different lengths, calculate the resistance of the wires and the corresponding values of the resistivity. (b) What is the average value of the resistivity? (c) Explain how this value compares with the value given in Table 27.2.27.2. L(m)ΔV(V)I(A)R(Ω)ρ(Ω⋅m)0.5405.220.721.0285.820.4141.5435.940.281L(m)0.5401.0281.543ΔV(V)5.225.825.94I(A)0.720.4140.281R(Ω)ρ(Ω⋅m) • A 10.0−g10.0−g piece of Styrofoam carries a net charge of −0.700μC−0.700μC and is suspended in equilibrium above the center of a large, horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet? • A child lying on her back experiences 55.0 N tension in the muscles on both sides of her neck when she raises her head to look past her toes. Later, sliding feet first down a water slide at terminal speed 5.70 m/s and riding high on the outside wall of a horizontal curve of radius 2.40 m, she raises her head again to look forward past her toes. Find the tension in the muscles on both sides of her neck while she is sliding. • Using the graph in Figure 44.5 , estimate how much energy is released when a nucleus of mass number 200 fissions into two nuclei each of mass number 100 . • The flexible loop in Figure P31.3P31.3 has a radius of 12.0 cmcm and is in a magnetic field of magnitude 0.150 TT . The loop is grasped at points AA and BB and stretched until its area is nearly zero. If it takes 0.200 ss to close the loop, what is the magnitude of the average induced emf in it during this time interval? • In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is 5.29×10−11m.5.29×10−11m. (a) Find the magnitude of the electric force exerted on each particle. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron? • An AC voltage of the form Δv=100sin1000t,Δv=100sin1000t, where ΔvΔv is in volts and tt is in seconds, is applied to a series RLC circuit. Assume the resistance is 400Ω, the capacitance is 5.00μF, and the inductance is 0.500 H. Find the average power delivered to the circuit. • A 25.0 -mW laser beam of diameter 2.00 mmmm is reflected at normal incidence by a perfectly reflecting mirror. Calculate the radiation pressure on the mirror. • A watch balance wheel (Fig. P15.35) has a period of oscillation of 0.250 s. The wheel is constructed so that its mass of 20.0 g is concentrated around a rim of radius 0.500 cm. What are (a) the wheel’s moment of inertia and (b) the torsion constant of the attached spring? • Assume a photomultiplier tube has seven dynodes with potentials of 100,200,300,…,700V as shown in Figure P45.66 . The average energy required to free an electron from the dynode surface is 10.0 eV. Assume only one electron is incident and the tube functions with 100% efficiency. (a) How many electrons are freed at the first dynode at 100 V ? (b) How many electrons are collected at the last dynode? (c) What is the energy available to the counter for all the electrons arriving at the last dynode? • The normalized wave functions for the ground state, ψ0(x), and the first excited state, ψ1(x), of a quantum harmonic oscillator are ψ0(x)=(aπ)1/4e−ax2/2ψ1(x)=(4a3π)1/4xe−ax2/2 where a=mω/ℏ . A mixed state, ψ01(x), is constructed from these states: ψ01(x)=1√2[ψ0(x)+ψ1(x)] ψ0(x), and the first excited state, ψ1(x), of a quantum harmonic oscillator are ψ0(x)=(aπ)1/4e−ax2/2ψ1(x)=(4a3π)1/4xe−ax2/2 where a=mω/ℏ . A mixed state, ψ01(x), is constructed from these states: ψ01(x)=1√2[ψ0(x)+ψ1(x)] The symbol ⟨q⟩s denotes the expectation value of the quantity q for the state ψs(x). Calculate the expectation values (a) ⟨x⟩0, (b) ⟨x⟩1, and (c)⟨x⟩ • Both sides of a uniform film that has index of refraction nn and thickness dd are in contact with air. For normal incidence of light, an intensity minimum is observed in the reflected light at λ2λ2 and an intensity maximum is observed at λ1,λ1, where λ1>λ2λ1>λ2 (a) Assuming no intensity minima are observed between λ1λ1 and λ2,λ2, find an expression for the integer mm in Equations 37.17 and 37.18 in terms of the wave- lengths λ1λ1 and λ2.λ2. (b) Assuming n=1.40,λ1=500nm,n=1.40,λ1=500nm, and λ2=370nm,λ2=370nm, determine the best estimate for the thickness of the film. • An electron is accelerated through 2.40×103V2.40×103V from rest and then enters a uniform 1.70 – T magnetic field. What are (a) the maximum and (b) the minimum values of the magnetic force this particle experiences? • Nuclei having the same mass numbers are called isobars. The isotope 13957La13957La is stable. A radioactive isobar, 13959Pr,13959Pr, is located below the line of stable nuclei as shown in Figure P44.19P44.19 and decays by e+e+ emission. Another radioactive isobar of 13957La,139BJCs,13957La,139BJCs, decays by e−e− emission and is located above the line of stable nuclei in Figure P44.19.P44.19. (a) Which of these three isobars has the highest neutron-to-proton ratio? (b) Which has the greatest binding energy per nucleon? (c) Which do you expect to be heavier, 13959Pr13959Pr or 139 Cs2Cs2 • Why is the following situation impossible? An air rifle is used to shoot 1.00 – gg particles at a speed of vx=100m/svx=100m/s . The rifle’s barrel has a diameter of 2.00mm.2.00mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00cm.1.00cm. • Three solid plastic cylinders all have radius 2.50 cmcm and length 6.00cm.6.00cm. Find the charge of each cylinder given the following additional information about each one. Cylinder (a) carries charge with uniform density 15.0 nC/m2nC/m2 everywhere on its surface. Cylinder (b) carries charge with uniform density 15.0 nC/m2nC/m2 on its curved lateral surface only. Cylinder (c) carries charge with uniform density 500 nC/m3nC/m3 throughout the plastic. • A certain superconducting magnet in the form of a solenoid of length 0.500 mm can generate a magnetic field of 9.00 TT in its core when its coils carry a current of 75.0 AA . Find the number of turns in the solenoid. • A simple pendulum has a mass of 0.250 kg and a length of 1.00 m. It is displaced through an angle of 15.08 and then released. Using the analysis model of a particle in simple harmonic motion, what are (a) the maximum speed of the bob, (b) its maximum angular acceleration, and (c) the maximum restoring force on the bob? (d) What If? Solve parts (a) through (c) again by using analysis models introduced in earlier chapters. (e) Compare the answers. • An unstable atomic nucleus of mass 17.0×10−27kg17.0×10−27kg initially at rest disintegrates into three particles. One of the particles, of mass 5.00×10−27kg5.00×10−27kg , moves in the yy direction with a speed of 6.00×106m/s6.00×106m/s . Another particle, of mass 40×10−27kg8.40×10−27kg , moves in the xx direction with a speed of 4.00×106m/s4.00×106m/s . Find (a)(a) the velocity of the third particle and (b)(b) the total kinetic energy increase in the process. • An air column in a glass tube is open at one end and closed at the other by a movable piston. The air in the tube is warmed above room temperature, and a 384-Hz tuning fork is held at the open end. Resonance is heard when the piston is at a distance d1=22.8cmd1=22.8cm from the open end and again when it is at a distance d2=68.3cmd2=68.3cm from the open end. (a) What speed of sound is implied by these data? (b) How far from the open end will the piston be when the next resonance is heard? • With a particular fingering, a flute produces a note with frequency 880 Hz at 20.0°C. The flute is open at both ends. (a) Find the air column length. (b) At the beginning of the halftime performance at a late-season football game, the ambient temperature is −5.00∘C−5.00∘C and the flutist has not had a chance to warm up her instrument. Find the frequency the flute produces under these conditions. • A toroid has a major radius R and a minor radius r and is tightly wound with N turns of wire on a hollow cardboard torus. Figure P32.12 shows half of this toroid, allowing us to see its cross section. If R>>r,R>>r, the magnetic field in the region enclosed by the wire is essentially the same as the magnetic field of a solenoid that has been bent into a large circle of radius RR . Modeling the field as the uniform field of a long solenoid, show that the inductance of such a toroid is approximately L≈12μ0N2r2RL≈12μ0N2r2R • A 4.00-m length of light nylon cord is wound around a uniform cylindrical spool of radius 0.500 m and mass 1.00 kg. The spool is mounted on a frictionless axle and is initially at rest. The cord is pulled from the spool with a constant acceleration of magnitude 2.50 m/s2m/s2 . (a) How much work has been done on the spool when it reaches an angular speed of 8.00 rad/srad/s ? (b) How long does it take the spool to reach this angular speed? (c) How much cord is left on the spool when it reaches this angular speed? • Why is the following situation impossible? A series circuit consists of an ideal AC source (no inductance or capacitance in the source itself) with an rms voltage of ΔV at a frequency f and a magnetic buzzer with a resistance R and an inductance L. By carefully adjusting the inductance L of the circuit, a power factor of exactly 1.00 is attained. • An auditorium has dimensions 10.0m×20.0m×10.0m×20.0m× 30.0m.30.0m. How many molecules of air fill the auditorium at 20.0∘0∘C and a pressure of 101 kPa(1.00atm)?kPa(1.00atm)? • Big Ben (Fig. P 10.45, page 312), the Parliament tower clock in London, has hour and minute hands with lengths of 2.70 m and 4.50 m and masses of 60.0 kg and 100 kg, respectively. Calculate the total angular momentum of these hands about the center point. (You may model the hands as long, thin rods rotating about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively.) • Consider a “crystal” consisting of two fixed ions of charge +e and two electrons as shown in Figure P41.60. (a) Taking into account all the pairs of interactions, find the potential energy of the system as a function of d. (b) Assuming the electrons to be restricted to a one-dimensional box of length 3d, find the minimum kinetic energy of the two electrons. (c) Find the value of d for which the total energy is a minimum. (d) State how this value of d compares with the spacing of atomsin lithium, which has a density of 0.530 g/cm3 and a molar mass of 6.94g/mol. • One side of a plant shelf is supported by a bracket mounted on a vertical wall by a single screw as shown in Figure P12.55. Ignore the weight of the bracket. (a) Find the horizontal component of the force that the screw exerts on the bracket when an 80.0 NN vertical force is applied as shown. (b) As your grandfather waters his geraniums, the 80.0 – NN load force is increasing at the rate 0.150 N/sN/s . At what rate is the force exerted by the screw changing? Suggestion: Imagine that the bracket is slightly loose. • Identify the particles corresponding to the quark states (a) suu, (b) u¯¯¯d,(c)sd¯¯¯¯¯,u¯d,(c)sd¯, and (d)(d) ssd. • Calculate the energy of a conduction electron in silver at 800 K, assuming the probability of finding an electron in that state is 0.950. The Fermi energy of silver is 5.48 eV at this temperature. • In the Davisson-Germer experiment, 54.0 -eV electrons were diffracted from a nickel lattice. If the first maximum in the diffraction pattern was observed at ϕ=50.0∘ϕ=50.0∘ (Fig. P40.41P40.41 , what was the lattice spacing aa between the vertical columns of atoms in the figure? • Two electrons in the same atom both have n=3n=3 and ℓ=1.ℓ=1. Assume the electrons are distinguishable, so that interchanging them defines a new state. (a) How many states of the atom are possible considering the quantum numbers these two electrons can have? (b) What If? How many states would be possible if the exclusion principle were inoperative? • Assume the region to the right of a certain plane contains a uniform magnetic field of magnitude 1.00 mT and the field is zero in the region to the left of the plane as shown in Figure P 29.22. An electron, originally traveling perpendicular to the boundary plane, passes into the region of the field. (a) Determine the time interval required for the electron to leave the “field-filled” region, noting that the electron’s path is a semicircle. (b) Assuming the maximum depth of penetration into the field is 2.00 cm, find the kinetic energy of the electron. • A block with a speaker bolted to it is connected to a spring having spring constant k=20.0N/mk=20.0N/m and oscillates as shown in Figure P17.39.P17.39. The total mass of the block and speaker is 5.00 kgkg , and the amplitude of this unit’s motion is 0.500 mm . The speaker emits sound waves of frequency 440 HzHz . Determine (a) the highest and (b) the lowest frequencies heard by the person to the right of the speaker. (c) If the maximum sound level heard by the person is 60.0 dBdB when the speaker is at its closest distance d=d= 1.00 mm from him, what is the minimum sound level heard by the observer? • An undersea earthquake or a landslide can produce an ocean wave of short duration carrying great energy, called a tsunami. When its wavelength is large compared to the ocean depth dd, the speed of a water wave is given approximately by v=gd−−√v=gd . Assume an earthquake occurs all along a tectonic plate boundary running north to south and produces a straight tsunami wave crest moving everywhere to the west. (a) What physical quantity can you consider to be constant in the motion of any one wave crest?(b) Explain why the amplitude of the wave increases as the wave approaches shore. (c) If the wave has amplitude 1.80 mm when its speed is 200 m/sm/s , what will be its amplitude where the water is 9.00 mm deep? (d) Explain why the amplitude at the shore should be expected to be still greater, but cannot be meaningfully predicted by your model. • As shown in Figure P9.26P9.26 a bullet of mass mm and speed vv passes completely through a pendulum bob of mass MM . The bullet emerges with a speed of v/2.v/2. The pendulum bob is suspended by a stiff rod (nota string) of length(nota string) of length ℓℓ and negligible mass. What is the minimum value of vv such that the pendulum bob will barely swing through a complete vertical circle? • A motorboat cuts its engine when its speed is 10.0 m/s and then coasts to rest. The equation describing the motion of the motorboat during this period is v=vie−ct,v=vie−ct, where vv is the speed at time t,vit,vi is the initial speed at t=0,t=0, and cc is a constant. At t=20.0st=20.0s , the speed is 5.00 m/sm/s . (a) Find the constant c.c. (b) What is the speed at t=40.0st=40.0s ? (c) Differentiate the expression for v(t)v(t) and thus show that the acceleration of the boat is proportional to the speed at any time. • Determine the maximum magnetic flux through an inductor connected to a North American electrical outlet (ΔVrms=120V,f=60.0Hz)(ΔVrms=120V,f=60.0Hz) • When a particular wire is vibrating with a frequency of 4.00 HzHz , a transverse wave of wavelength 60.0 cmcm is produced. Determine the speed of waves along the wire. • Three long, parallel conductors each carry a current of I=I= 2.00 A. Figure P30.15P30.15 is an end view of the conductors, with each current coming out of the page. Taking a=1.00cm,a=1.00cm, determine the magnitude and direction of the magnetic field at (a) point A,(b)A,(b) point B,B, and (c) point C.C. • A quantum particle in an infinitely deep square well has a wave function that is given by ψ1(x)=√2Lsin(πxL) for 0≤x≤L and is zero otherwise. (a) Determine the probability of finding the particle between x=0 and x=13L (b) Use the result of this calculation and a symmetry argument to find the probability of finding the particle between x=13L and x=23L. Do not re-evaluate the integral. • For a technology project, a student has built a vehicle, of total mass 6.00 kg, that moves itself. As shown in Figure P9.45, it runs on four light wheels. A reel is attached to one of the axles, and a cord originally wound on the reel goes up over a pulley attached to the vehicle to support an elevated load. After the vehicle is released from rest, the load descends very slowly, unwinding the cord to turn the axle and make the vehicle move forward (to the left in Fig. P9.45). Friction is negligible in the pulley and axle bearings. The wheels do not slip on the floor. The reel has been constructed with a conical shape so that the load descends at a constant low speed while the vehicle moves horizontally across the floor with constant acceleration, reaching a final velocity of 3.00i^m/s. (a) Does the floor impart impulse to the vehicle? If so, how much? (b) Does the floor do work on the vehicle? If so, how much? (c) Does it make sense to say that the final momentum of the vehicle came from the floor? If not, where did it come from? (d) Does it make sense to say that the final kinetic energy of the vehicle came from the floor? If not, where did it come from? (e) Can we say that one particular force causes the forward acceleration of the vehicle? What does cause it? • A 500 -W heating coil designed to operate from 110 VV is made of Nichrome wire 0.500 mmmm in diameter. (a) Assuming the resistivity of the Nichrome remains constant at its 20.0∘0∘C value, find the length of wire used. (b) What If? Now consider the variation of resistivity with temperature. What power is delivered to the coil of part (a) when it is warmed to 1200∘C?1200∘C? • A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from four identical spiral springs under tension, one near each corner of the shelf. Each tray is rectangular, 45.3 cm by 35.6cm,0.450cm thick, and with mass 580 g. (a) Demonstrate that the top tray in the stack can always be at the same height above the floor, however many trays are in the dispenser. (b) Find the spring constant each spring should have for the dispenser to function in this convenient way. (c) Is any piece of data unnecessary for this determination? • A particle moves according to the equationx=10 t^{2}$, where$x$is in meters and$t$is in seconds. (a) Find the average velocity for the time interval from 2.00$\mathrm{s}$to 3.00$\mathrm{s}$. (b) Find the average velocity for the time interval from 2.00 to 2.10$\mathrm{s}$. • For bacteriological testing of water supplies and in medical clinics, samples must routinely be incubated for 24 hh at 37∘C37∘C . Peace Corps volunteer and MIT engineer Amy Smith invented a low-cost, low-maintenance incubator. The incubator consists of a foam-insulated box containing a waxy material that melts at 37.0°C interspersed among tubes, dishes, or bottles containing the test samples and growth medium (bacteria food). Outside the box, the waxy material is first melted by a stove or solar energy collector. Then the waxy material is put into the box to keep the test samples warm as the material solidifies. The heat of fusion of the phase-change material is 205 kJ/kgkJ/kg . Model the insulation as a panel with surface area 0.490 m2m2 , thickness 4.50 cm,cm, and conductivity 0.0120 W/m⋅∘CW/m⋅∘C . Assume the exterior temperature is 23.0∘0∘C for 12.0 hh and 16.0∘C16.0∘C for 12.0 hh . (a) What mass of the waxy material is required to conduct the bacteriological test? (b) Explain why your calculation can be done without knowing the mass of the test samples or of the insulation. • An unstable particle, initially at rest, decays into a positively charged particle of charge + eand rest energy E+E+ and a negatively charged particle of charge – ee and rest energy E−.E−. A uniform magnetic field of magnitude BB exists perpendicular to the velocities of the created particles. The radius of curvature of each track is r.r. What is the mass of the original unstable particle? • A flat cushion of mass mm is released from rest at the corner of the roof of a building, at height h.h. A wind blowing along the side of the building exerts a constant horizontal force of magnitude FF on the cushion as it drops as shown in Figure P5.65. force. (a) Show that the path of the cushion is a straight line. (b) Does the cushion fall with constant velocity? Explain. (c) If m=1.20kg,h=8.00m,m=1.20kg,h=8.00m, and F=2.40N,F=2.40N, how far from the building will the cushion hit the level ground? What If? (d) If the cushion is thrown downward with a nonzero speed at the top of the building, what will be the shape of its trajectory? Explain. • A typical nuclear fission power plant produces approximately 1.00 GWGW of electrical power. Assume the plant has an overall efficiency of 40.0%% and each fission reaction produces 200 MeVMeV of energy. Calculate the mass of 255U consumed each day. • For a Maxwellian gas, use a computer or programmable calculator to find the numerical value of the ratio Nv(v)/Nv(vmp)Nv(v)/Nv(vmp) for the following values of v:(a)v=(vmp/50.0)v:(a)v=(vmp/50.0) (b) (vmp/10.0),(c)(vmp/2.00),(d)vmp,(e)2.00vmp,(f)10.0vmp(vmp/10.0),(c)(vmp/2.00),(d)vmp,(e)2.00vmp,(f)10.0vmp and (g) 50.0vmpvmp . Give your results to three significant figures. • Two blocks of masses m1m1 and m2m2 are placed on a table in contact with each other as discussed in Example 5.7 and shown in Active Figure 5.12 a. The coefficient of kinetic friction between the block of mass m1m1 and the table is μ1μ1 , and that between the block of mass m2m2 and the table is μ2.μ2. A horizontal force of magnitude FF is applied to the block of mass m1⋅m1⋅ We wish to find P,P, the magnitude of the contact force between the blocks. (a) Draw diagrams showing the forces for each block. (b) What is the net force on the system of two blocks? (c) What is the net force acting on m1m1 ? (d) What is the net force acting on m2?m2? (e) Write Newton’s second law in the xx direction for each block. (f) Solve the two equations in two unknowns for the acceleration aa of the blocks in terms of the masses, the applied force F,F, the coefficients of friction, and gg . (g) Find the magnitude PP of the contact force between the blocks in terms of the same quantities. • A rectangular coil of 60 turns, dimensions 0.100 m by 0.200m, and total resistance 10.0Ω rotates with angular speed 30.0 rad/s about the y axis in a region where a 1.00−T magnetic field is directed along the x axis. The time t=0 is chosen to be at an instant when the plane of the coil is perpendicular to the direction of →B . Calculate (a) the maximum induced emf in the coil, (b) the maximum rate of change of magnetic flux through the coil, (c) the induced emf at t=0.0500s , and (d) the torque exerted by the magnetic field on the coil at the instant when the emf is a maximum. • Why is the following situation impossible? An electron enters a region of uniform electric field between two parallel plates. The plates are used in a cathode-ray tube to adjust the position of an electron beam on a distant fluorescent screen. The magnitude of the electric field between the plates is 200 N/CN/C . The plates are 0.200 mm in length and are separated by 1.50cm.1.50cm. The electron enters the region at a speed of 3.00×106m/s3.00×106m/s , traveling parallel to the plane of the plates in the direction of their length. It leaves the plates heading toward its correct location on the fluorescent screen. • A step-up transformer is designed to have an output voltage of 2 200 V (rms) when the primary is connected across a 110-V (rms) source. (a) If the primary winding has exactly 80 turns, how many turns are required on the secondary? (b) If a load resistor across the secondary draws a current of 1.50 A, what is the current in the primary, assuming ideal conditions? (c) What If? If the transformer actually has an efficiency of 95.0%, what is the current in the primary when the secondary current is 1.20 A? • The string shown in Figure P16.11P16.11 is driven at a frequency of 5.00 HzHz . The amplitude of the motion is A=12.0cm,A=12.0cm, and the wave speed is v=20.0m/sv=20.0m/s . Furthermore, the wave is such that y=0y=0 at x=0x=0 and t=0.t=0. Determine (a) the angular frequency and (b) the wave number for this wave. (c) Write an expression for the wave function. Calculate (d) the maximum transverse speed and (e) the maximum transverse acceleration of an element of the string. • The magnetic moment of the Earth is approximately 8.00×1022A⋅8.00×1022A⋅m2. Imagine that the planetary magnetic field were caused by the complete magnetization of a huge iron deposit with density 7900 kg/m3kg/m3 and approximately 8.50 ×1028×1028 iron atoms/m^{3} . (a) How many unpaired electrons, each with a magnetic moment of 9.27 ×10−24A⋅m2×10−24A⋅m2 , would participate? (b) At two unpaired electrons per iron atom, how many kilograms of iron would be present in the deposit? • A lightbulb marked ” 75 WW [at] 120 VnVn is screwed into a socket at one end of a long extension cord, in which each of the two conductors has resistance 0.800ΩΩ . The other end of the extension cord is plugged into a 120−V120−V outlet. (a) Explain why the actual power delivered to the lightbulb cannot be 75 WW in this situation. (b) Draw a circuit diagram. (c) Find the actual power delivered to the light bulb in this circuit. • A piece of putty is initially located at point A on the rim of a grinding wheel rotating at constant angular speed about a horizontal axis. The putty is dislodged from point A when the diameter through A is horizontal. It then rises vertically and returns to A at the instant the wheel completes one revolution. From this information, we wish to find the speed vv of the putty when it leaves the wheel and the force holding it to the wheel. (a) What analysis model is appropriate for the motion of the putty as it rises and falls? (b) Use this model to find a symbolic expression for the time interval between when the putty leaves point AA and when it arrives back at A,A, in terms of vv and g. (c) What is the appropriate analysis model to describe point AA on the wheel? (d) Find the period of the motion of point AA in terms of the tangential speed vv and the radius RR of the wheel. (e) Set the time interval from part (b) equal to the period from part (d) and solve for the speed vv of the putty as it leaves the wheel. (f) If the mass of the putty is mm, what is the magnitude of the force that held it to the wheel before it was released? • Two identical loudspeakers 10.0 m apart are driven by the same oscillator with a frequency of f=21.5Hzf=21.5Hz (Fig. P 18.11) in an area where the speed of sound is 344 m/s. (a) Show that a receiver at point A records a minimum in sound intensity from the two speakers. (b) If the receiver is moved in the plane of the speakers, show that the path it should take so that the intensity remains at a minimum is along the hyperbola 9×2−16y2=1449×2−16y2=144 (shown in red-brown in Fig. P 18.11). (c) Can the receiver remain at a minimum and move very far away from the two sources? If so, determine the limiting form of the path it must take. If not, explain how far it can go. • A uniform rod of weight FgFg and length LL is supported at its ends by a frictionless trough as shown in Figure P12.64 (a) Show that the center of gravity of the rod must be vertically over point OO when the rod is in equilibrium. (b) Determine the equilibrium value of the angle θ.θ. (c) Is the equilibrium of the rod stable or unstable? • Four identical charged particles (q=+10.0μC)(q=+10.0μC) are located on the corners of a rectangle as shown in Figure P23.62P23.62 . The dimensions of the rectangle are L=60.0cmL=60.0cm and W=15.0cm.W=15.0cm. Calculate (a)(a) the magnitude and (b)(b) the dirce exerted on the charge at the lower left corner by the other three charges. • A small airplane with a wingspan of 14.0 m is flying due north at a speed of 70.0 m/s over a region where the vertical component of the Earth’s magnetic field is 1.20μT downward. (a) What potential difference is developed between the airplane’s wingtips? (b) Which wingtip is at higher potential? (c) What If? How would the answers to parts (a) and (b) change if the plane turned to fly due east? (d) Can this emf be used to power a lightbulb in the passenger compartment? Explain your answer. • A gasoline engine has a compression ratio of 6.00.6.00. (a) What is the efficiency of the engine if it operates in an idealized Otto cycle? (b) What If? If the actual efficiency is 15.0%% , what fraction of the fuel is wasted as a result of friction and energy transfers by heat that could be avoided in a reversible engine? Assume complete combustion of the air–fuel mixture. • A 45.0 -kg girl is standing on a 150−kg150−kg plank. Both are originally at rest on a frozen lake that constitutes a frictionless, flat surface. The girl begins to walk along the plank at a constant velocity of 1.50 i m/sm/s relative to the plank. (a) What is the velocity of the plank relative to the ice surface? (b) What is the girl’s velocity relative to the ice surface? • Show that Equation 15.32 is a solution of Equation 15.31 provided that b2<4mkb2<4mk • A particle with charge +q+q is at the origin. A particle with charge −2q−2q is at x=2.00mx=2.00m on the xx axis. (a) For what finite value(s) of xx is the electric field zero? (b) For what finite value(s) of xx is the electric potential zero? • Two lenses made of kinds of glass having different indices of refraction$n_{1}$and$n_{2}$are cemented together to form an optical doublet. Optical doublets are often used to correct chromatic aberrations in optical devices. The first lens of a certain doublet has index of refraction$n_{1}$, one flat side, and one concave side with a radius of curvature of magnitude$R$. The second lens has index of refraction$n_{2}$and two convex sides with radii of curvature also of magnitude$R$. Show that the doublet can be modeled as a single thin lens with a focal length described by 1f=2n2−n1−1R1f=2n2−n1−1R • Figure P35.8P35.8 shows a refracted light beam in linseed oil making an angle of α=20.0∘α=20.0∘ with the normal line NN′.NN′. The index of refraction of linseed oil is 48. Determine the angles (a) θ and (b) θ′ . • A hollow aluminum cylinder 20.0 cmcm deep has an internal capacity of 2.000 LL at 20.0∘0∘C . It is completely filled with turpentine at 20.0∘C20.0∘C . The turpentine and the aluminum cylinder are then slowly warmed together to 80.0∘C.80.0∘C. (a) How much turpentine overflows? (b) What is the volume of turpentine remaining in the cylinder at 80.0∘C80.0∘C ? (c) If the combination with this amount of turpentine is then cooled back to 20.0∘C20.0∘C , how far below the cylinder’s rim does the turpentine’s surface recede? • At t=0,t=0, a transverse pulse in a wire is described by the function y=6.00×2+3.00y=6.00×2+3.00 where xx and yy are in meters. If the pulse is traveling in the positive xx direction with a speed of 4.50 m/sm/s , write the function y(x,t)y(x,t) that describes this pulse. • A conducting rod of length ℓ moves with velocity →v parallel to a long wire carrying a steady current I . The axis of the rod is maintained perpendicular to the wire with the near end a distance r away (Fig. P31.65). Show that the magnitude of the emf induced in the rod is |ε|=μ0Iv2πln(1+ℓr) • How much work is done (by a battery, generator, or some other source of potential difference) in moving Avogadro’s number of electrons from an initial point where the electric potential is 9.00 V to a point where the electric potential is 25.00 V? (The potential in each case is measured relative to a common reference point.) • Two particles with masses m and 3m are moving toward each other along the x axis with the same initial speeds vi . Particle m is traveling to the left, and particle 3m is traveling to the right. They undergo an elastic glancing collision such that particle m is moving in the negative y direction after the collision at a right angle from its initial direction. (a) Find the final speeds of the two particles in terms of vi . (b) What is the angle θ at which the particle 3m is scattered? • The maximum distance from the Earth to the Sun (at aphelion) is 1.521×1011m,1.521×1011m, and the distance of closest approach (at perihelion) is 1.471×1011m.1.471×1011m. The Earth’s orbital speed at perihelion is 3.027×104m/s3.027×104m/s . Determine (a) the Earth’s orbital speed at aphelion and the kinetic and potential energies of the Earth-Sun system (b) at perihelion, and (c)(c) at aphelion. (d) Is the total energy of the system constant? Explain. Ignore the effect of the Moon and other planets. • Carbon detonations are powerful nuclear reactions that temporarily tear apart the cores inside massive stars late in their lives. These blasts are produced by carbon fusion, which requires a temperature of approximately 6×108K to overcome the strong Coulomb repulsion between carbon nuclei. (a) Estimate the repulsive energy barrier to fusion, using the temperature required for carbon fusion. (In other words, what is the average kinetic energy of a carbon nucleus at 6×108K2 (b) Calculate the energy (in MeV) released in each of these “carbon-burning” reactions: 12C+12C→20Ne+4He12C+12C→24Mg+γ (c) Calculate the energy in kilowatt-hours given off when 2.00 kg of carbon completely fuse according to the first reaction. • A police car is traveling east at 40.0 m/s along a straight road, overtaking a car ahead of it moving east at 30.0 m/s. The police car has a malfunctioning siren that is stuck at 1 000 Hz. (a) What would be the wavelength in air of the siren sound if the police car were at rest? (b) What is the wavelength in front of the police car? (c) What is it behind the police car? (d) What is the frequency heard by the driver being chased? • Why is the following situation impossible? You are in the high-speed package delivery business. Your competitor in the next building gains the right-of-way to build an evacuated tunnel just above the ground all the way around the Earth. By firing packages into this tunnel at just the right speed, your competitor is able to send the packages into orbit around the Earth in this tunnel so that they arrive on the exact opposite side of the Earth in a very short time interval. You come up with a competing idea. Figuring that the distance through the Earth is shorter than the distance around the Earth, you obtain permits to build an evacuated tunnel through the center of the Earth (Fig. P15.74). By simply dropping packages into this tunnel, they fall down-ward and arrive at the other end of your tunnel, which is in a building right next to the other end of your competitor’s tunnel. Because your packages arrive on the other side of the Earth in a shorter time interval, you win the competition and your business flourishes. Note: An object at a distance r from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius r (the reddish region in Fig. P15.74). Assume the Earth has uniform density. • Over a certain region of space, the electric potential is V=V= 5x−3x2y+2yz2.5x−3x2y+2yz2. (a) Find the expressions for the x,y,x,y, and zz components of the electric field over this region. (b) What is the magnitude of the field at the point PP that has coordinates (1.00,0,−2.00)m?(1.00,0,−2.00)m? • An electric utility company supplies a customer’s house from the main power lines (120V)(120V) with two copper wires, each of which is 50.0 mm long and has a resistance of 0.108ΩΩ per 300 mm . (a) Find the potential difference at the customer’s house for a load current of 110 AA . For this load current, find (b) the power delivered to the customer and (c) the rate at which internal energy is produced in the copper wires. • In the What If? section of Example 12.2,12.2, let dd represent the distance in meters between the person and the hinge at the left end of the beam. (a) Show that the cable tension is given by T=93.9d+125,T=93.9d+125, with TT in newtons. (b) Show that the direction angle θθ of the hinge force is described by tanθ=(323d+4−1)tan53.0∘tan⁡θ=(323d+4−1)tan⁡0∘ (c) Show that the magnitude of the hinge force is given by R=8.82×103d2−9.65×104d+4.96×105−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√R=8.82×103d2−9.65×104d+4.96×105 (d) Describe how the changes in T,θ,T,θ, and RR as dd increases differ from one another. • Carbon- 14 and carbon- 12 ions (each with charge of magnitude ee ) are accelerated in a cyclotron. If the cyclotron has a magnetic field of magnitude 2.40 T, what is the difference in cyclotron frequencies for the two ions? • How many electrons should be removed from an initially uncharged spherical conductor of radius 0.300 mm to produce a potential of 7.50 kVkV at the surface? • A pinhole camera has a small circular aperture of diameter D. Light from distant objects passes through the aperture into an otherwise dark box, falling on a screen located a distance LL away. If DD is too large, the display on the screen will be fuzzy because a bright point in the field of view will send light onto a circle of diameter slightly larger than D. On the other hand, if D is too small, diffraction will blur the display on the screen. The screen shows a reasonably sharp image if the diameter of the central disk of the diffraction pattern, specified by Equation 38.6, is equal to D at the screen. (a) Show that for monochromatic light with plane wave fronts and L>>D,L>>D, the condition for a sharp view is fulfilled if D2=2.44λL.D2=2.44λL. (b) Find the optimum pinhole diameter for 500-nm light projected onto a screen 15.0 cm away. • Why is the following situation impossible? In an effort to study positronium, a scientist places 57 Co and 14 C in proximity. The 57 Co nuclei decay by e+ emission, and the 14 C nuclei decay by e− emission. Some of the positrons and electrons from these decays combine to form sufficient amounts of positronium for the scientist to gather data. • A merry-go-round is stationary. A dog is running around the merry-go-round on the ground just outside its circumference, moving with a constant angular speed of 0.750 rad/srad/s . The dog does not change his pace when he sees what he has been looking for: a bone resting on the edge of the merry-go-round one-third of a revolution in front of him. At the instant the dog sees the bone (t=0)(t=0) the merry-go-round begins to move in the direction the dog is running, with a constant angular acceleration of 0.0150 rad/s2rad/s2 . ( a) At what time will the dog first reach the bone? (b) The confused dog keeps running and passes the bone. How long after the merry-go-round starts to turn do the dog and the bone draw even with each other for the second time? • As it plows a parking lot, a snowplow pushes an ever-growing pile of snow in front of it. Suppose a car moving through the air is similarly modeled as a cylinder of area A pushing a growing disk of air in front of it. The originally stationary air is set into motion at the constant speed v of the cylinder as shown in Figure 54. In a time interval ΔtΔt a new disk of air of mass ΔmΔm must be moved a distance vΔtvΔt and hence must be given a kinetic energy 12(Δm)v2.12(Δm)v2. Using this model, show that the car’s power loss owing to air resistance is 12ρAv312ρAv3 and that the resistive force acting on the car is 12ρAv2,12ρAv2, where ρρ is the density of air. Compare this result with the empirical expression 12DρAv212DρAv2 for the resistive force. • As you enter a fine restaurant, you realize that you have accidentally brought a small electronic timer from home instead of your cell phone. In frustration, you drop the timer into a side pocket of your suit coat, not realizing that the timer is operating. The arm of your chair presses the light cloth of your coat against your body at one spot. Fabric with a length L hangs freely below that spot, with the timer at the bottom. At one point during your dinner, the timer goes off and a buzzer and a vibrator turn on and off with a frequency of 1.50 Hz. It makes the hanging part of your coat swing back and forth with remarkably large amplitude, drawing everyone’s attention. Find the value of L. • A proton moving with velocity →v=viˆiv→=vii^ experiences a magnetic force →F=FˆjF→=Fj^ . Explain what you can and cannot infer about →BB→ from this information. (b) What If? In terms of Fi,Fi, what would be the force on a proton in the same field moving with velocity →v=−viˆiv→=−vii^ (c) What would be the force on an electron in the same field moving with velocity →v=−viˆiv→=−vii^ ? • Show that the ratio of the Compton wavelength λCλC to the de Broglie wavelength λ=h/pλ=h/p for a relativistic electron is λCλ=[(Emec2)2−1]1/2λCλ=[(Emec2)2−1]1/2 where EE is the total energy of the electron and meme is its mass. • An enemy spacecraft moves away from the Earth at a speed of v=0.800c(Fig.P39.29).v=0.800c(Fig.P39.29). A galactic patrol spacecraft pursues at a speed of u=0.900cu=0.900c relative to the Earth. Observers on the Earth measure the patrol craft to be over- taking the enemy craft at a relative speed of 0.100cc . With what speed is the patrol craft overtaking the enemy craft as measured by the patrol craft’s crew? • Figure$\mathrm{P} 14.68$shows the essential parts of a hydraulic brake system. The area of the piston in the master cylinder is 1.8$\mathrm{cm}^{2}$and that of the piston in the brake cylinder is$6.4 \mathrm{cm}^{2} .$The coefficient of friction between shoe and wheel drum is$0.50 .$If the wheel has a radius of$34 \mathrm{cm},$determine the frictional torque about the axle when a force of 44$\mathrm{N}$is exerted on the brake pedal. • The circuit in Figure P26.39 consists of two identical, parallel metal plates connected to identical metal springs, a switch, and a 100-V battery. With the switch open, the plates are uncharged, are separated by a distance d 5 8.00 mm, and have a capacitance C=2.00μFC=2.00μF . When the switch is closed, the distance between the plates decreases by a factor of 0.500.0.500. (a) How much charge collects on each plate? (b) What is the spring constant for each spring? • Electrons are ejected from a metallic surface with speeds of up to 4.60×105m/s4.60×105m/s when light with a wavelength of 625 nmnm is used. (a) What is the work function of the surface? (b) What is the cutoff frequency for this surface? • At point AA in a Carnot cycle, 2.34 molmol of a monatomic ideal gas has a pressure of 1400 kPakPa , a volume of 10.0 LL , and a temperature of 720 KK . The gas expands isothermally to point BB and then expands adiabatically to point C,C, where its volume is 24.0 LL . An isothermal compression brings it to point D,D, where its volume is 15.0 LL . An adiabatic process returns the gas to point A.A. (a) Determine all the unknown pressures, volumes, and temperatures as you fill in the following table: 0L720KBC24.0LD15.0LABCDP1400kPaV10.0L24.0L15.0LT720K (b) Find the energy added by heat, the work done by the engine, and the change in internal energy for each of the steps A→B,B→C,C→D,A→B,B→C,C→D, and D→A.D→A. (c) Calculate the efficiency Wnet/|Qh|⋅(d) Show that the efficiency isWnet/|Qh|⋅(d) Show that the efficiency is equal to 1−TC/Td,1−TC/Td, the Carnot efficiency. • As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0∘0∘ north of west with a speed of 41.0 km/h. (a) What is the unit-vector expression for the velocity of the hurricane? It maintains this velocity for 3.00 h, at which time the course of the hurricane suddenly shifts due north, and its speed slows to a constant 25.0 km/h. This new velocity is maintained for 1.50 h. (b) What is the unit-vector expression for the new velocity of the hurricane? (c) What is the unit-vector expression for the displacement of the hurricane during the first 3.00 h? (d) What is the unit-vector expression for the displacement of the hurricane during the latter 1.50 h? (e) How far from Grand Bahama is the eye 4.50 h after it passes over the island? • The height of a helicopter above the ground is given by$h=3.00 t^{3},$where$h$is in meters and$t$is in seconds. At$t=2.00 \mathrm{s}$, the helicopter releases a small mailbag. How long after its release does the mailbag reach the ground? • A capacitor of unknown capacitance has been charged to a potential difference of 100 VV and then disconnected from the battery. When the charged capacitor is then connected in parallel to an uncharged 10.0−μF10.0−μF capacitor, the potential difference across the combination is 30.0 VV . Calculate the unknown capacitance. • Classical general relativity views the structure of space-time as deterministic and well defined down to arbitrarily small distances. On the other hand, quantum general relativity forbids distances smaller than the Planck length given by L=(ℏG/c3)1/2L=(ℏG/c3)1/2 (a) Calculate the value of the Planck length. The quantum limitation suggests that after the big bang, when all the presently observable section of the Universe was contained within a point-like singularity, nothing could be observed until that singularity grew larger than the Planck length. Because the size of the singularity grew at the speed of light, we can infer that no observations were possible during the time interval required for light to travel the Planck length. (b) Calculate this time interval, known as the Planck time T,T, and state how it compares with the ultrahot epoch mentioned in the text. • An interstate highway has been built through a neighborhood in a city. In the afternoon, the sound level in an apartment in the neighborhood is 80.0 dB as 100 cars pass outside the window every minute. Late at night, the traffic flow is only five cars per minute. What is the average late-night sound level? • What maximum current is delivered by an AC source with ΔVmax=48.0VΔVmax=48.0V and f=90.0Hzf=90.0Hz when connected across a 3.70−μF3.70−μF capacitor? • A transparent cylinder of radius R=2.00m has a mirrored surface on its right half as shown in Figure P35.66. A light ray traveling in air is incident on the left side of the cylinder. The incident light ray and exiting light ray are parallel, and d=2.00m. Determine the index of refraction of the material. • A small light fixture on the bottom of a swimming pool is 1.00 m below the surface. The light emerging from the still water forms a circle on the water surface. What is the diameter of this circle? • Why is the following situation impossible? A freight train is lumbering along at a constant speed of 16.0$\mathrm{m} / \mathrm{s}$. Behind the freight train on the same track is a passenger train traveling in the same direction at 40.0$\mathrm{m} / \mathrm{s}$. When the front of the passenger train is 58.5$\mathrm{m}$from the back of the freight train, the engineer on the passenger train recognizes the danger and hits the brakes of his train, causing the train to move with acceleration$-3.00 \mathrm{m} / \mathrm{s}^{2}$. Because of the engineer’s action, the trains do not collide. • Free neutrons have a characteristic half-life of 10.4 min. What fraction of a group of free neutrons with kinetic energy 0.0400 eV decays before traveling a distance of 10.0 km ? • As shown in Figure P36.34, Ben and Jacob check out an aquarium that has a curved front made of plastic with uniform thickness and a radius of curvature of magnitude R 5 2.25 m. (a) Locate the images of fish that are located (i) 5.00 cm and (ii) 25.0 cm from the front wall of the aquarium. (b) Find the magnification of images (i) and (ii) from the previous part. (See Problem 32 to find an expression for the magnification of an image formed by a refracting surface.) (c) Explain why you don’t need to know the refractive index of the plastic to solve this problem. (d) If this aquarium were very long from front to back, could the image of a fish ever be farther from the front surface than the fish itself is? (e) If not, explain why not. If so, give an example and find the magnification. • Miranda, a satellite of Uranus, is shown in Figure P13.11a. It can be modeled as a sphere of radius 242 km and mass 6.68×1019kg.6.68×1019kg. (a) Find the free-fall acceleration on its surface. (b) A cliff on Miranda is 5.00 km high. It appears on the limb at the 11 o’clock position in Figure P13.11a and is magnified in Figure P13.11b. If a devotee of extreme sports runs horizontally off the top of the cliff at 8.50 m/s, for what time interval is he in flight? (c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? (d) What will be his vector impact velocity? • Example 23.8 derives the exact expression for the electric field at a point on the axis of a uniformly charged disk. Consider a disk of radius R 5 3.00 cm having a uniformly distributed charge of +5.20μC+5.20μC . (a) Using the result of Example 23.8 , compute the electric field at a point on the axis and 3.00 mmmm from the center. (b) What If? Explain how the answer to part (a) compares with the field computed from the near-field approximation E=σ/2ϵ0E=σ/2ϵ0 . (We will derive this expression in Chapter 24.)24.) (c) Using the result of Example 23.8, compute the electric field at a point on the axis and 30.0 cm from the center of the disk. (d) What If? Explain how the answer to part (c) compares with the electric field obtained by treating the disk as a +5.20−μC+5.20−μC charged particle at a distance of 30.0cm.30.0cm. • A 15.0-mW helium–neon laser emits a beam of circular cross section with a diameter of 2.00 mm. (a) Find the maximum electric field in the beam. (b) What total energy is contained in a 1.00-m length of the beam? (c) Find the momentum carried by a 1.00-m length of the beam. • An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from zero to 230 N . (a) What is the equivalent spring constant of the bow? (b) How much work does the archer do on the string in drawing the bow? • The mass of the blue puck in Figure P9.34 is 20.0% greater than the mass of the green puck. Before colliding, the pucks approach each other with momenta of equal magnitudes and opposite directions, and the green puck has an initial speed of 10.0 m/s. Find the speeds the pucks have after the collision if half the kinetic energy of the system becomes internal energy during the collision. • A diode is at room temperature so that kBT=0.0250eVkBT=0.0250eV. Taking the applied voltages across the diode to be +1.00V+1.00V (under forward bias) and −1.00V−1.00V (under reverse bias), calculate the ratio of the forward current to the reverse current if the diode is described by Equation 43.27 . • In an insulated vessel, 250 g of ice at 0°C is added to 600 g of water at 18.0°C. (a) What is the final temperature of the system? (b) How much ice remains when the system reaches equilibrium? • The four particles in Figure P10.25P10.25 are connected by rigid rods of negligible mass. The origin is at the center of the rectangle. The system rotates in the xyxy plane about the zz axis with an angular speed of 6.00 rad/srad/s . Calculate (a) the moment of inertia of the system about the zz axis and (b) the rotational kinetic energy of the system. • A vector is given by R→=2i^+j^+3k^.R→=2i^+j^+3k^. Find (a) the magnitudes of the x,y,x,y, and zz components; (b) the magnitude of R→;R→; and (c) the angles between R→R→ and the x,y,x,y, and zz axes. • A transverse traveling wave on a taut wire has an amplitude of 0.200 mmmm and a frequency of 500 HzHz . It travels with a speed of 196 m/sm/s . (a) Write an equation in SI units of the form y=Asin(kx−ωt)y=Asin⁡(kx−ωt) for this wave. (b) The mass per unit length of this wire is 4.10 g/mg/m . Find the tension in the wire. • An unstable particle, initially at rest, decays into a proton (rest energy 938.3 MeV) and a negative pion (rest energy 139.6 MeVMeV ). A uniform magnetic field of 0.250 T exists perpendicular to the velocities of the created particles. The radius of curvature of each track is found to be 1.33 mm . What is the mass of the original unstable particle? • For an electron with magnetic moment →μsμ⃗s in a magnetic field →BB→, Section 29.5 showed the following. The electron–field system can be in a higher energy state with the z component of the electron’s magnetic moment opposite the field or a lower energy state with the z component of the magnetic moment in the direction of the field. The difference in energy between the two states is2μBBμBB. Under high resolution, many spectral lines are observed to be doublets. The most famous doublet is the pair of two yellow lines in the spectrum of sodium (the D lines), with wavelengths of 588.995 nm and 589.592 nm. Their existence was explained in 1925 by Goudsmit and Uhlenbeck, who postulated that an electron has intrinsic spin angular momentum. When the sodium atom is excited with its outermost electron in a 3p state, the orbital motion of the outermost electron creates a magnetic field. The atom’s energy is somewhat different depending on whether the electron is itself spin-up or spin-down in this field. Then the photon energy the atom radiates as it falls back into its ground state depends on the energy of the excited state. Calculate the magnitude of the internal magnetic field, mediating this so-called spin-orbit coupling. • Liquid nitrogen has a boiling point of −195.81∘C−195.81∘C at atmospheric pressure. Express this temperature (a) in degrees Fahrenheit and (b) in kelvins. • A ball of mass m=300gm=300g is connected by a strong string of length L=80.0cmL=80.0cm to a pivot and held in place with the string vertical. A wind exerts constant force FF to the right on the ball as shown in Figure P8.78P8.78 . The ball is released from rest. The wind makes it swing up to attain maximum height HH above its starting point before it swings down again. (a) Find HH as a function of FF . Evaluate HH for (b)F=(b)F= 1.00 NN and (c)F=10.0N(c)F=10.0N . How does HH behave (d) as FF approaches zero and (e) as FF approaches infinity? (f) Now consider the equilibrium height of the ball with the wind blowing. Determine it as a function of FF . Evaluate the equilibrium height for (g)F=10N(g)F=10N and (h)(h) Fgoing to infinity. • Derive the equation for the Compton shift (Eq. 40.11) from Equations 40.12 through 40.14. • A uniform sign of weight FεFε and width 2LL hangs from a light, horizontal beam hinged at the wall and supported by a cable (Fig. Pl2.45). Determine (a) the tension in the cable and (b) the components of the reaction force exerted by the wall on the beam in terms of Fg,d,L,Fg,d,L, and θ.θ. • Turn on your desk lamp. Pick up the cord, with your thumb and index finger spanning the width of the cord. (a) Compute an order-of-magnitude estimate for the current in your hand. Assume the conductor inside the lamp cord next to your thumb is at potential ~ 102 V at a typical instant and the conductor next to your index finger is at ground potential (0 V). The resistance of your hand depends strongly on the thickness and the moisture content of the outer layers of your skin. Assume the resistance of your hand between fingertip and thumb tip is ~ 104 V. You may model the cord as having rubber insulation. State the other quantities you measure or estimate and their values. Explain your reasoning. (b) Suppose your body is isolated from any other charges or currents. In order-of-magnitude terms, estimate the potential difference between your thumb where it contacts the cord and your finger where it touches the cord. • Two identical steel balls, each of diameter 25.4 mm and moving in opposite directions at 5 m/s , run into each other head-on and bounce apart. Prior to the collision, one of the balls is squeezed in a vise while precise measurements are made of the resulting amount of compression. The results show that Hooke’s law is a fair model of the ball’s elastic behavior. For one datum, a force of 16 kN exerted by each jaw of the vise results in a 0.2−mm reduction in the diameter. The diameter returns to its original value when the force is removed. (a) Modeling the ball as a spring, find its spring constant. (b) Does the interaction of the balls during the collision last only for an instant or for a nonzero time interval? State your evidence. (c) Compute an estimate for the kinetic energy of each of the balls before they collide. (d) Compute an estimate for the maximum amount of compression each ball undergoes when the balls collide. (e) Compute an order-of-magnitude estimate for the time interval for which the balls are in contact. (In Chapter 15, you will learn to calculate the contact time interval precisely.) • In a cyclotron (one type of particle accelerator), a deuteron (of mass 2.00 u) reaches a final speed of 10.0% of the speed of light while moving in a circular path of radius 0.480 m. What magnitude of magnetic force is required to maintain the deuteron in a circular path? • Estimate the force with which a karate master strikes a board, assuming the hand’s speed at the moment of impact is 10.0 m/sm/s and decreases to 1.00 m/sm/s during a 0.00200−s0.00200−s time interval of contact between the hand and the board. The mass of his hand and arm is 1.00kg.1.00kg. (b) Estimate the shear stress, assuming this force is exerted on a 1.00−cm−1.00−cm− thick pine board that is 10.0 cmcm wide. (c) If the maximum shear stress a pine board can support before breaking is 3.60×106N/m2,3.60×106N/m2, will the board break? • An amateur skater of mass M is trapped in the middle of an ice rink and is unable to return to the side where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return to safety by throwing her gloves of mass mm in the direction opposite the safe side. (a) She throws the gloves as hard as she can, and they leave her hand with a horizontal velocity →vgloves.v→gloves. Explain whether or not she moves. If she does move, calculate her velocity →vgitv→git relative to the Earth after she throws the gloves. (b) Discuss her motion from the point of view of the forces acting on her. • X-rays with a wavelength of 120.0 pm undergo Compton scattering. (a) Find the wavelengths of the photons scattered at angles of 30.0∘,60.0∘,90.0∘,120∘,150∘,30.0∘,60.0∘,90.0∘,120∘,150∘, and 180∘.180∘. (b) Find the energy of the scattered electron in each case. (c) Which of the scattering angles provides the electron with the greatest energy? Explain whether you could answer this question without doing any calculations. • A satellite of mass 200 kg is placed into Earth orbit at a height of 200 km above the surface. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite’s speed? (c) Starting from the satellite on the Earth’s surface, what is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet’s daily rotation. • Why is the following situation impossible? A technician is sending laser light of wavelength 632.8 nm through a pair of slits separated by 30.0μm.30.0μm. Each slit is of width 2.00μm.2.00μm. The screen on which he projects the pattern is not wide enough, so light from the m=15m=15 interference maximum misses the edge of the screen and passes into the next lab station, startling a coworker. • In Figure P33.71,P33.71, find the rms current delivered by the 45.0−V45.0−V (rms) power supply when (a) the frequency is very large and (b) the frequency is very small. • A wire of density ρρ is tapered so that its cross-sectional area varies with xx according to A=1.00×10−5x+1.00×10−6A=1.00×10−5x+1.00×10−6 where AA is in meters squared and xx is in meters. The tension in the wire is TT (a) Derive a relationship for the speed of a wave as a function of position. (b) What If? Assumed the wire is aluminum and is under a tension T=24.0NT=24.0N . Determine the wave speed at the origin and at x=10.0mx=10.0m . • When two unknown resistors are connected in series with a battery, the battery delivers 225 WW and carries a total current of 5.00 A. For the same total current, 50.0 WW is delivered when the resistors are connected in parallel. Deter- mine the value of each resistor. • The voltage phasor diagram for a certain series RLCRLC circuit is shown in Figure P33.59. The resistance of the circuit is 75.0Ω,75.0Ω, and the frequency is 60.0 HzHz . Find (a) the maximum voltage ΔVmax,ΔVmax, (b) the phase angle (c) the maximum current, (d) the impedance, (e) the capacitance and (f) the inductance of the circuit, and (g) the average power delivered to the circuit. • In Figure P32.76,P32.76, the battery has emf E=18.0VE=18.0V and the other circuit elements have values L=0.400H,R1=L=0.400H,R1= 2.00kΩ,2.00kΩ, and R2=6.00kΩ.R2=6.00kΩ. The switch is closed for t<0,t<0, and steady-state conditions are established. The switch is then opened at t=0.t=0. (a) Find the emf across LL immediately after t=0.t=0. (b) Which end of the coil, aa or b,b, is at the higher potential? (c) Make graphs of the currents in R1R1 and in R2R2 as a function of time, treating the steady-state directions as positive. Show values before and after t=0t=0 . (d) At what moment after t=0t=0 does the current in R2R2 have the value 2.00 mAmA ? • A modified oscilloscope is used to perform an electron interfence experiment. Electrons are incident on a pair of narrow slits 0.0600μmμm apart. The bright bands in the interference pattern are separated by 0.400 mmmm on a screen 20.0 cmcm from the slits. Determine the potential difference through which the electrons were accelerated to give this pattern. • A student claims that he has found a vector →AA→ such that (2ˆi−3ˆj+4ˆk)×→A=(4ˆi+3ˆj−ˆk)⋅(2i^−3j^+4k^)×A→=(4i^+3j^−k^)⋅ (a) Do you believe this claim? (b) Explain why or why not. • Why is the following situation impossible? A solid copper sphere of radius 15.0 cm is in electrostatic equilibrium and carries a charge of 40.0 nC. Figure P 24.38 shows the magnitude of the electric field as a function of radial position rr measured from the center of the sphere. • Consider the sinusoidal wave of Example 16.2 with the wave function y=0.150cos(15.7x−50.3t)y=0.150cos⁡(15.7x−50.3t) where xx and yy are in meters and tt is in seconds. At a certain instant, let point AA be at the origin and point BB be the closest point to AA along the xx axis where the wave is 60.0∘0∘ out of phase with A.A. What is the coordinate of B?B? • The athlete shown in Figure P4.27 rotates a 1.00-kg discus along a circular path of radius 1.06 m. The maximum speed of the discus is 20.0 m/sm/s . Determine the magnitude of the maximum radial acceleration of the discus. • Equation 17.13 states that at distance r away from a point source with power (Power) avg, the wave intensity is I=(Power)arg4πr2I=(Power)arg4πr2 Study Active Figure 17.10 and prove that at distance rr straight in front of a point source with power (Power) arg moving with constant speed vSvS the wave intensity is I=(Power)avg4πr2(v−vSv)I=(Power)avg4πr2(v−vSv) • Consider a solenoid of length ℓℓ and radius aa containing NN closely spaced turns and carrying a steady current I.I. (a) In terms of these parameters, find the magnetic field at a point along the axis as a function of position xx from the end of the solenoid. (b) Show that as ℓℓ becomes very long, BB approaches μ0NI/2ℓμ0NI/2ℓ at each end of the solenoid. • The switch in Figure P32.49P32.49 is connected to position aa for a long time interval. At t=0,t=0, the switch is thrown to position b. After this time, what are (a) the frequency of oscillation of the LCLC circuit, (b)(b) the maximum charge that appears on the capacitor, (c) the maximum current in the inductor, and (d) the total energy the circuit possesses at t=3.00st=3.00s ? • A superconducting ring of niobium metal 2.00 cm in diameter is immersed in a uniform 0.020 0-T magnetic field directed perpendicular to the ring and carries no current. Determine the current generated in the ring when the magnetic field is suddenly decreased to zero. The inductance of the ring is 3.10×10−8H.3.10×10−8H. • A physicist drives through a stop light. When he is pulled over, he tells the police officer that the Doppler shift made the red light of wavelength 650 nmnm appear green to him, with a wavelength of 520 nmnm . The police officer writes out a traffic citation for speeding. How fast was the physicist traveling, according to his own testimony? • Two vectors are given by →A=ˆi+2ˆjA→=i^+2j^ and →B=−2ˆi+3ˆjB→=−2i^+3j^ Find (a) →A×→BA→×B→ and (b) the angle between →AA→ and →B.B→. • A proton moving at 4.00×106m/s4.00×106m/s through a magnetic field of magnitude 1.70 TT experiences a magnetic force of magnitude 8.20×10−13N8.20×10−13N. What is the angle between the proton’s velocity and the field? • A particle with charge −3.00nC−3.00nC is at the origin, and a particle with negative charge of magnitude QQ is at x=50.0cm.x=50.0cm. A third particle with a positive charge is in equilibrium at x=20.9cm.x=20.9cm. What is QQ ? • An electron moves in a circular path perpendicular to a uniform magnetic field with a magnitude of 2.00 mTmT . If the speed of the electron is 1.50×107m/s1.50×107m/s , determine (a) the radius of the circular path and (b) the time interval required to complete one revolution. • You can think of the work–kinetic energy theorem as a second theory of motion, parallel to Newton’s laws in describing how outside influences affect the motion of an object. In this problem, solve parts (a), (b), and (c) separately from parts (d) and (e) so you can compare the predictions of the two theories. A 15.0 -g bullet is accelerated from rest to a speed of 780 m/s in a rifle barrel of length 72.0cm. (a) Find the kinetic energy of the bullet as it leaves the barrel. (b) Use the work-kinetic energy theorem to find the net work that is done on the bullet. (c) Use your result to part (b) to find the magnitude of the average net force that acted on the bullet while it was in the barrel. (d) Now model the bullet as a particle under constant acceleration. Find the constant acceleration of a bullet that starts from rest and gains a speed of 780 m/s over a distance of 72.0cm. (e) Modeling the bullet as a particle under a net force, find the net force that acted on it during its acceleration. (f) What conclusion can you draw from comparing your results of parts (c) and (e)? • A long, uniform rod of length LL and mass MM is pivoted about a frictionless, horizontal pin through one end. The rod is nudged from rest in a vertical position as shown in Figure P10.67. At the instant the rod is horizontal, find (a) its angular speed, (b) the magnitude of its angular acceleration, (c) the xx and yy components of the acceleration of its center of mass, and (d) the components of the reaction force at the pivot. • If |→A×→B|=→A⋅→B,|A→×B→|=A→⋅B→, what is the angle between →AA→ and →BB→? • Two insulating spheres have radii r1r1 and r2,r2, masses m1m1 and m2,m2, and uniformly distributed charges −q1−q1 and q2.q2. They are released from rest when their centers are separated by a distance d.d. (a) How fast is each moving when they collide? (b) What If? If the spheres were conductors, would their speeds be greater or less than those calculated in part (a)? Explain. • Use the uncertainty principle to show that if an electron were confined inside an atomic nucleus of diameter on the order of 10−14m10−14m , it would have to be moving relativistically, whereas a proton confined to the same nucleus can be moving nonrelativistically. • Jane waits on a railroad platform while two trains approach from the same direction at equal speeds of 8.00 m/s. Both trains are blowing their whistles (which have the same frequency), and one train is some distance behind the other. After the first train passes Jane but before the second train passes her, she hears beats of frequency 4.00 Hz. What is the frequency of the train whistles? • The energy absorbed by an engine is three times greater than the work it performs. (a) What is its thermal efficiency?(b) What fraction of the energy absorbed is expelled to the cold reservoir? • Air in a thundercloud expands as it rises. If its initial temperature is 300 KK and no energy is lost by thermal conduction on expansion, what is its temperature when the initial volume has doubled? • Figure$\mathrm{P} 2.15$shows a graph of$v_{x}$versus$t$for the motion of a motorcyclist as he starts from rest and moves along the road in a straight line. (a) Find the average acceleration for the time interval$t=0$to$t=6.00$s. (b) Estimate the time at which the acceleration has its greatest positive value and the value of the acceleration at that instant. (c) When is the acceleration zero? (d) Estimate the maximum negative value of the acceleration and the time at which it occurs. • The tank in Figure$\mathrm{P} 14.13$is filled with water of depth$d .$At the bottom of one sidewall is a rectangular hatch of height$h$and width$w$that is hinged at the top of the hatch. (a) Determine the magnitude of the force the water exerts on the hatch. (b) Find the magnitude of the torque exerted by the water about the hinges. • Two adjacent natural frequencies of an organ pipe are determined to be 550 Hz and 650 Hz. Calculate (a) the fundamental frequency and (b) the length of this pipe. • A proton moves at 4.50×105m/s4.50×105m/s in the horizontal direction. It enters a uniform vertical electric field with a magnitude of 9.60×103N/C9.60×103N/C . Ignoring any gravitational effects, find (a)(a) the time interval required for the proton to travel 5.00 cmcm horizontally, (b) its vertical displacement during the time interval in which it travels 5.00 cmcm horizontally, and (c)(c) the horizontal and vertical components of its velocity after it has traveled 5.00 cmcm horizontally. • A 9.00-kg object starting from rest falls through a viscous medium and experiences a resistive force given by Equation 6.2. The object reaches one half its terminal speed in 5.54 s. (a) Determine the terminal speed. (b) At what time is the speed of the object three-fourths the terminal speed? (c) How far has the object traveled in the first 5.54 s of motion? • A nuclear power plant operates by using the energy released in nuclear fission to convert liquid water at Tc into steam at Th. How much water could theoretically be converted to steam by the complete fissioning of a mass m of 235U if the energy released per fission event is E? • A cylinder is closed by a piston connected to a spring of constant 2.00×2.00× 103N/m103N/m (see Fig. Pl9.60) With the spring relaxed, the cylinder is filled with 5.00 L of gas at a pressure of 1.00 atm and a temperature of 20.0∘0∘C . (a) If the piston has a cross-sectional area of 0.0100 m2m2 and negligible mass, how high will it rise when the temperature is raised to 250∘C250∘C ? (b) What is the pressure of the gas at 250∘C250∘C ? • Does your bathroom mirror show you older or younger than you actually are? (b) Compute an order-of-magnitude estimate for the age difference based on data you specify. • Figure P10.16 shows the drive train of a bicycle that has wheels 67.3 cm in diameter and pedal cranks 17.5 cm long. The cyclist pedals at a steady cadence of 76.0 rev/min. The chain engages with a front sprocket 15.2 cmcm in diameter and a rear sprocket 7.00 cmcm in diameter. Calculate (a) the speed of a link of the chain relative to the bicycle frame, (b) the angular speed of the bicycle wheels, and (c) the speed of the bicycle relative to the road. (d) What pieces of data, if any, are not necessary for the calculations? • In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression x=5.00cos(2t+π6)x=5.00cos(2t+π6) where xx is in centimeters and tt is in seconds. At t=0t=0 , find (a) the position of the particle, (b)(b) its velocity, and (c)(c) its acceleration. Find (d)(d) the period and (e)(e) the amplitude of the motion. • The waves from a radio station can reach a home receiver by two paths. One is a straight-line path from transmitter to home, a distance of 30.0km.30.0km. The second is by reflection from the ionosphere (a layer of ionized air molecules high in the atmosphere). Assume this reflection takes place at a point midway between receiver and transmitter, the wavelength broadcast by the radio station is 350m,350m, and no phase change occurs on reflection. Find the minimum height of the ionospheric layer that could produce destructive interference between the direct and reflected beams. • A sled of mass mm is given a kick on a frozen pond. The kick imparts to the sled an initial speed of vv . The coefficient of kinetic friction between sled and ice is μk.μk. Use energy considerations to find the distance the sled moves before it stops. • As shown in Figure P25.70P25.70 two large, parallel, vertical conducting plates separated by distance dd are charged so that their potentials are +V0+V0 and −V0.−V0. A small conducting ball of mass mm and radius RR (where R<<R<< plates. The thread of length LL supporting the ball is a conducting wire connected to ground, so the potential of the ball is fixed at V=0.V=0. The ball hangs straight down in stable equilibrium when V0V0 is sufficiently small. Show that the equilibrium of the ball is unstable if V0V0 exceeds the critical value kdd2mg/(4RL).kdd2mg/(4RL). Suggestion: Consider the forces on the ball when it is displaced a distance x<<L.x<<L. • An object of height 2.00$\mathrm{cm}$is placed 30.0$\mathrm{cm}$from a convex spherical mirror of focal length of magnitude$10.0 \mathrm{cm} .$(a) Find the location of the image. (b) Indicate whether the image is upright or inverted. (c) Determine the height of the image. • A cord is wrapped around a pulley that is shaped like a disk of mass mm and radius rr . The cord’s free end is connected to a block of mass MM . The block starts from rest and then slides down an incline that makes an angle θθ with the horizontal as shown in Figure P10.86P10.86 . The coefficient of kinetic friction between block and incline is μ.μ. (a) Use energy methods to show that the block’s speed as a function of position dd down the incline is v=4Mgd(sinθ−μcosθ)m+2M−−−−−−−−−−−−−−−−−−√v=4Mgd(sin⁡θ−μcos⁡θ)m+2M (b) Find the magnitude of the acceleration of the block in terms of μ,m,M,g,μ,m,M,g, and θ.θ. • Starting with Equation 19.10 , show that the total pressure PP in a container filled with a mixture of several ideal gases is P=P1+P2+P3+…,P=P1+P2+P3+…, where P1,P2,…P1,P2,… are the pressures that each gas would exert if it alone filled the container. (These individual pressures are called the partial pressures of the respective gases). This result is known as Dalton’s law of partial pressures. • Why is the following situation impossible? A mischievous child goes to an amusement park with his family. On one ride, after a severe scolding from his mother, he slips out of his seat and climbs to the top of the ride’s structure, which is shaped like a cone with its axis vertical and its sloped sides making an angle of θ=20.0∘θ=20.0∘ with the horizontal as shown in Figure P6.46.P6.46. This part of the structure rotates about the vertical central axis when the ride operates. The child sits on the sloped surface at a point d=5.32md=5.32m down the sloped side from the center of the cone and pouts. The coefficient of static friction between the boy and the cone is 0.700 . The ride operator does not notice that the child has slipped away from his seat and so continues to operate the ride. As a result, the sitting, pouting boy rotates in a circular path at a speed of 3.75 m/sm/s. • A system consists of three particles, each of mass 5.00 g, located at the corners of an equilateral triangle with sides of 30.0 cm. (a) Calculate the potential energy of the system. (b) Assume the particles are released simultaneously. Describe the subsequent motion of each. Will any collisions take place? Explain. • The fret closest to the bridge on a guitar is 21.4 cm from the bridge as shown in Figure P 18.57. When the thinnest string is pressed down at this first fret, the string produces the highest frequency that can be played on that guitar, 2 349 Hz. The next lower note that is produced on the string has frequency 2 217 Hz. How far away from the first fret should the next fret be? • As the Earth moves around the Sun, its orbits are quantized. (a) Follow the steps of Bohr’s analysis of the hydrogen atom to show that the allowed radii of the Earth’s orbit are given by r=n2ℏ2GMSM2Er=n2ℏ2GMSM2E where nn is an integer quantum number, MsMs is the mass of the Sun, and MEME is the mass of the Earth. (b) Calculate the numerical value of nn for the Sun-Earth system. (c) Find the distance between the orbit for quantum number nn and the next orbit out from the Sun corresponding to the quantum number n+1.n+1. (d) Discuss the significance of your results from parts (b) and (c). • In Figure P30.23P30.23 , the current in the long, straight wire is I1I1 and the wire lies in the plane of a rectangular loop, which carries a current I2I2 . The loop is of length ℓℓ and width a. Its left end is a distance cc from the wire. Find the magnitude and direction of the net force exerted on the loop by the magnetic field created by the wire. • A baseball outfielder throws a 0.150 -kg baseball at a speed of 40.0 m/s and an initial angle of 30.0∘ to the horizontal. What is the kinetic energy of the baseball at the highest point of its trajectory? • A cosmic-ray proton in interstellar space has an energy of 10.0 MeV and executes a circular orbit having a radius equal to that of Mercury’s orbit around the Sun (5.80×1010m).(5.80×1010m). What is the magnetic field in that region of space? • An object is placed a distance$p$to the left of a diverging lens of focal length$f_{1}$. A converging lens of focal length$f_{2}$is placed a distance$d$to the right of the diverging lens. Find the distance$d$so that the final image is infinitely far away to the right. • Use Equation 43.18 to calculate the ionic cohesive energy for NaClNaCl . Take α=1.7476,r0=0.281nm,α=1.7476,r0=0.281nm, and m=8m=8. • You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the xx axis and at a fixed height of 7.60×103m.7.60×103m. At time t=0,t=0, the airplane is directly above you so that the vector leading from you to it is →P0=7.60×103ˆjm.P→0=7.60×103j^m. At t=30.0st=30.0s , the position vector leading from you to the airplane is →P30=(8.04×103ˆi+7.60×103j)mP→30=(8.04×103i^+7.60×103j)m as suggested in Figure P 3.43. Determine the magnitude and orientation of the airplane’s position vector at t=45.0st=45.0s . • A cylinder of mass 10.0 kg rolls without slipping on a horizontal surface. At a certain instant, its center of mass has a speed of 10.0 m/s. Determine (a) the translational kinetic energy of its center of mass, (b) the rotational kinetic energy about its center of mass, and (c) its total energy. • A 10.0-kg monkey climbs a uniform ladder with weight 1.20×1.20× 102N102N and length L=3.00mL=3.00m as shown in Figure P12.24P12.24 . The ladder rests against the wall and makes an angle of θ=60.0∘θ=60.0∘ with the ground. The upper and lower ends of the ladder rest on frictionless surfaces. The lower end is connected to the wall by a horizontal rope that is frayed and can support a maximum tension of only 80.0 NN . (a) Draw a force diagram for the ladder. (b) Find the normal force exerted on the bottom of the ladder. (c) Find the tension in the rope when the monkey is two-thirds of the way up the ladder. (d) Find the maximum distance dd that the monkey can climb up the ladder before the rope breaks. (e) If the horizontal surface were rough and the rope were removed, how would your analysis of the problem change? What other information would you need to answer parts (c) and (d)? • An aluminum rod has a resistance of 1.23ΩΩ at 20.0∘C20.0∘C . Calculate the resistance of the rod at 120∘C120∘C by accounting for the changes in both the resistivity and the dimensions of the rod. The coefficient of linear expansion for aluminum is 2.40×10−6(∘C)−12.40×10−6(∘C)−1 . • A pendulum of length LL and mass MM has a spring of force constant kk connected to it at a distance hh below its point of suspension (Fig. P15.55P15.55 ). Find the frequency of vibration of the system for small values of the amplitude (small θθ ). Assume the vertical suspension rod of length LL is rigid, but ignore its mass. • Yellow light of wavelength 589 nm is used to view an object under a microscope. The objective lens diameter is 9.00 mm. (a) What is the limiting angle of resolution? (b) Suppose it is possible to use visible light of any wave-length. What color should you choose to give the smallest possible angle of resolution, and what is this angle? (c) Suppose water fills the space between the object and the objective. What effect does this change have on the resolving power when 589-nm light is used? • The position vector of a particle of mass 2.00 kg as a function of time is given by →r=(6.00ˆi+5.00tˆj),r→=(6.00i^+5.00tj^), where →rr→ is in meters and tt is in seconds. Determine the angular momentum of the particle about the origin as a function of time. • When high-energy charged particles move through a transparent medium with a speed greater than the speed of light in that medium, a shock wave, or bow wave, of light is produced. This phenomenon is called the Cerenkov effect. When a nuclear reactor is shielded by a large pool of water, Cerenkov radiation can be seen as a blue glow in the vicinity of the reactor core due to high-speed electrons moving through the water (Fig. 17.38). In a particular case, the Cerenkov radiation produces a wave front with an apex half-angle of 53.0∘.53.0∘. Calculate the speed of the electrons in the water. The speed of light in water is 2.25×2.25× 108m/s108m/s. • Consider the filter circuit shown in Figure P33.56. (a) Show that the ratio of the amplitude of the output voltage to that of the input voltage is ΔVoutΔVin=1/ωC√R2+(1ωC2)ΔVoutΔVin=1/ωCR2+(1ωC2)−−−−−−−−−−√ (b) What value does this ratio approach as the frequency decreases toward zero? (c) What value does this ratio approach as the frequency increases without limit? (d) At what frequency is the ratio equal to one-half? • A child’s toy consists of a small wedge that has an acute angle θθ (Fig. P6.42). The sloping side of the wedge is frictionless, and an object of mass mm on it remains at constant height if the wedge is spun at a certain constant speed. as an axis, a vertical rod that is as an axis, a vertical rod that is firmly attached to the wedge at the bottom end. Show that, when the boject sits at rest at a point at distance LL up along the wedge, the speed of the object must be v=(gLsinθ)1/2v=(gLsin⁡θ)1/2 • In 1983 , the United States began coining the one-cent piece out of copper-clad zinc rather than pure copper. The mass of the old copper penny is 3.083$\mathrm{g}$and that of the new cent is 2.517$\mathrm{g}$. The density of copper is 8.920$\mathrm{g} / \mathrm{cm}^{3}$and that of$\mathrm{zinc}$is$7.133 \mathrm{g} / \mathrm{cm}^{3} .$The new and old coins have the same volume. Calculate the percent of zinc (by volume) in the new cent. • You unconsciously estimate the distance to an object from the angle it subtends in your field of view. This angle$\theta$in radians is related to the linear height of the object$h$and to the distance$d$by$\theta=h / d$. Assume you are driving a car and another car, 1.50$\mathrm{m}$high, is 24.0$\mathrm{m}$behind you. ( a) Suppose your car has a flat passenger-side rearview mirror, 1.55$\mathrm{m}$from your eyes. How far from your eyes is the image of the car following you: (b) What angle does the image subtend in your field of view? (c) What If? Now suppose your car has a convex rearview mirror with a radius of curvature of magnitude 2.00$\mathrm{m}$(as suggested in Fig.$36.15 ) .$How far from your eyes is the image of the car behind you? (d) What angle does the image subtend at your eyes? (e) Based on its angular size, how far away does the following car appear to be? • As a sound wave passes through a gas, the compressions are either so rapid or so far apart that thermal conduction is prevented by a negligible time interval or by effective thickness of insulation. The compressions and rarefactions are adiabatic. (a) Show that the speed of sound in an ideal gas is v=√γRTMv=γRTM−−−−√ where MM is the molar mass. The speed of sound in a liquid is given by Equation 17.8;17.8; use that equation and the definition of the bulk modulus from Section 12.4.12.4. (b) Compute the theoretical speed of sound in air at 20.0∘0∘C and state how it compares with the value in Table 17.1.17.1. Take M=M= 28.9 g/molg/mol (c) Show that the speed of sound in an ideal gas is v=√γkBTm0v=γkBTm0−−−−−√ where m0m0 is the mass of one molecule. (d) State how the result in part (c) compares with the most probable, average, and rms molecular speeds. • A 0.500 -kg block rests on the frictionless, icy surface of a frozen pond. If the location of the block is measured to a precision of 0.150 cmcm and its mass is known exactly, what is the minimum uncertainty in the block’s speed? • A uniform disk of mass 10.0$\mathrm{kg}$and radius 0.250$\mathrm{m}$spins at 300$\mathrm{rev} / \mathrm{min}$on a low-friction axle. It must be brought to a stop in 1.00$\mathrm{min}$by a brake pad that makes contact with the disk at an average distance 0.220$\mathrm{m}$from the axis. The coefficient of friction between pad and disk is$0.500 .$A piston in a cylinder of diameter 5.00$\mathrm{cm}$presses the brake pad against the disk. Find the pressure required for the brake fluid in the cylinder. • The nuclear potential energy that binds protons and neutrons in a nucleus is often approximated by a square well. Imagine a proton confined in an infinitely high square well of length 10.0 fm, a typical nuclear diameter. Assuming the proton makes a transition from the n 5 2 state to the ground state, calculate (a) the energy and (b) the wavelength of the emitted photon. (c) Identify the region of the electromagnetic spectrum to which this wavelength belongs. • A 2.00 -n FF parallel-plate capacitor is charged to an initial potential difference ΔVi=100VΔVi=100V and is then isolated. The dielectric material between the plates is mica, with a dielectric constant of 5.00 . (a) How much work is required to withdraw the mica sheet? (b) What is the potential difference across the capacitor after the mica is withdrawn? • Consider the cyclic process depicted in Figure P20.26P20.26 . If QQ is negative for the process BCBC and ΔEintΔEint is negative for the process CA,CA, what are the signs of Q,W,Q,W, and ΔEintΔEint that are associated with each of the three processes? • A flow calorimeter is an apparatus used to measure the specific heat of a liquid. The technique of flow calorimetry involves measuring the temperature difference between the input and output points of a flowing stream of the liquid while energy is added by heat at a known rate. A liquid of density r flows through the calorimeter with volume flow rate R. At steady state, a temperature difference DT is established between the input and output points when energy is supplied at the rate P. What is the specific heat of the liquid? • In a manufacturing process, a large, cylindrical roller is used to flatten material fed beneath it. The diameter of the roller is 1.00 m, and, while being driven into rotation around a fixed axis, its angular position is expressed as θ=2.50t2−0.600t3θ=2.50t2−0.600t3 where θθ is in radians and tt is in seconds. (a) Find the maximum angular speed of the roller. (b) What is the maximum tangential speed of a point on the rim of the roller? (c) At what time tt should the driving force be removed from the roller so that the roller does not reverse its direction of rotation? (d) Through how many rotations has the roller turned between t=0t=0 and the time found in part (c)? • A rock rests on a concrete sidewalk. An earthquake strikes, making the ground move vertically in simple harmonic motion with a constant frequency of 2.40 Hz and with gradually increasing amplitude. (a) With what amplitude does the ground vibrate when the rock begins to lose contact with the sidewalk? Another rock is sitting on the concrete bottom of a swimming pool full of water. The earthquake produces only vertical motion, so the water does not slosh from side to side. (b) Present a convincing argument that when the ground vibrates with the amplitude found in part (a), the submerged rock also barely loses contact with the floor of the swimming pool. • Two slits are separated by 0.180 mmmm . An interference pattern is formed on a screen 80.0 cmcm away by 656.3 -nm light. Calculate the fraction of the maximum intensity a distance y=0.600cmy=0.600cm away from the central maximum. • A woman is draining her fish tank by siphoning the water into an outdoor drain as shown in Figure$\mathrm{P} 14.74$. The rectangular tank has footprint area$A$and depth$h$. The drain is located a distance$d$below the surface of the water in the tank, where$d>>h$. The cross-sectional area of the siphon tube is$A^{\prime} .$Model the water as flowing without friction. Show that the time interval required to empty the tank is given by Δt=AhA′2gd−−−√Δt=AhA′2gd • Problems 69 through 72 apply ideas from this and earlier chapters to some properties of superconductors, which were introduced in Section 27.5. In an experiment carried out by S. C. Collins between 1955 and 1958 , a current was maintained in a superconducting lead ring for 2.50 yr with no observed loss, even though there was no energy input. If the inductance of the ring were 3.14×10−8H3.14×10−8H and the sensitivity of the experiment were 1 part in 109,109, what was the maximum resistance of the ring? Suggestion: Treat the ring as an RLRL circuit carrying decaying current and recall that the approximation e−x≈1−xe−x≈1−x is valid for small xx . • A 1.00 -km steel railroad rail is fastened securely at both ends when the temperature is 20.0∘0∘C . As the temperature increases, the rail buckles, taking the shape of an arc of a vertical circle. Find the height hh of the center of the rail when the temperature is 25.0∘C25.0∘C . (You will need to solve a transcendental equation.) • A refrigerator has a coefficient of performance equal to 5.00 . The refrigerator takes in 120 JJ of energy from a cold reservoir in each cycle. Find (a) the work required in each cycle and (b) the energy expelled to the hot reservoir. • A particle with charge qq and kinetic energy KK travels in a uniform magnetic field of magnitude BB . If the particle moves in a circular path of radius R,R, find expressions for (a) its speed and (b) its mass. • A horizontal plank of mass 5.00 kgkg and length 2.00 mm is pivoted at one The plank’s other end. is supported by a spring of force constant 100 N/mN/m is displaced by a small angle θθ from its horizontal equilibrium position and released. Find the angular frequency with which the plank moves with simple harmonic motion. • When a falling meteoroid is at a distance above the Earth’s surface of 3.00 times the Earth’s radius, what is its acceleration due to the Earth’s gravitation? • A screen is placed 50.0 cm from a single slit, which is illuminated with light of wavelength 690 nm. If the distance between the first and third minima in the diffraction pattern is 3.00 mm, what is the width of the slit? • A sphere of radius R=1.00mR=1.00m surrounds a particle with charge Q=50.0μCQ=50.0μC located at its center as shown in Figure P 24.47. Find the electric flux through a circular cap of half-angle θ=45.0∘.θ=45.0∘. • Protons are projected with an initial speed vi=vi= 9.55 km/skm/s from a field-free region through a plane and into a region where a uniform electric field E=−720j^N/CE=−720j^N/C is present above the plane as shown in Figure P23.48P23.48 . The initial velocity vector of the protons makes an angle θθ with the plane. The protons are to hit a target that lies at a horizontal distance of R=1.27mmR=1.27mm from the point where the protons cross the plane and enter the electric field. We wish to find the angle θθ at which the protons must pass through the plane to strike the target. (a) What analysis model describes the horizontal motion of the protons above the plane? (b) What analysis model describes the vertical motion of the protons above the plane? (c) Argue that Equation 4.13 would be applicable to the protons in this situation. (d) Use Equation 4.13 to write an expression for RR in terms of vi,E,vi,E, the charge and mass of the proton, and the angle θ.θ. (e) Find the two possible values of the angle θ.θ. (f) Find the time interval during which the proton is above the plane in Figure P23.48P23.48 for each of the two possible values of θ.θ. • A neutral pion at rest decays into two photons according to π0→γ+γπ0→γ+γ . Find the (a) energy, (b) momentum, and (c) frequency of each photon. • How many cubic meters of helium are required to lift a balloon with a 400-kg payload to a height of 8 000 m? Take$\rho_{\mathrm{He}}=0.179 \mathrm{kg} / \mathrm{m}^{3} .$Assume the balloon maintains a constant volume and the density of air decreases with the altitude$z$according to the expression$\rho_{\text { air }}=\rho_{0} e^{-z / 8000}$, where$z$is in meters and$\rho_{0}=1.20 \mathrm{kg} / \mathrm{m}^{3}$is the density of air at sea level. • A ray of light passes from air into water. For its deviation angle δ=|θ1−θ2|δ=|θ1−θ2| to be 10.0∘,10.0∘, what must its angle of incidence be? • A violin string has a length of 0.350 m and is tuned to concert G,G, with fG=392HzfG=392Hz . (a) How far from the end of the string must the violinist place her finger to play concert A, with fA=440Hz2fA=440Hz2 (b) If this position is to remain correct to one-half the width of a finger (that is, to within 0.600 cm), what is the maximum allowable percentage change in the string tension? • A small object of mass 3.80 gg and charge −18.0μC−18.0μC is suspended motionless above the ground when immersed in a uniform electric field perpendicular to the ground. What are the magnitude and direction of the electric field? • A proton having an initial velvocity of 20.0ˆiMm/si^Mm/s enters a uniform magnetic field of magnitude 0.300 TT with a direction perpendicular to the proton’s velocity. It leaves the field-filled region with velocity −20.0→jMm/s−20.0j→Mm/s . Determine (a) the direction of the magnetic field, (b) the radius of curvature of the proton’s path while in the field, (c) the distance the proton traveled in the field, and (d) the time interval during which the proton is in the field. • Accelerating charges radiate electromagnetic waves. Calculate the wavelength of radiation produced by a proton of mass mpmp moving in a circular path perpendicular to a magnetic field of magnitude BB . • The active element of a certain laser is made of a glass rod 30.0 cmcm long and 50 cmcm in diameter. Assume the average coefficient of linear expansion of the glass is equal to 9.00×10−6(∘C)−19.00×10−6(∘C)−1 . If the temperature of the rod increases by 65.0∘C,65.0∘C, what is the increase in (a) its length, (b) its diameter, and (c) its volume? • A surveyor measures the distance across a straight river by the following method (Fig. P 3.7). Starting directly across from a tree on the opposite bank, she walks d=100md=100m along the riverbank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is θ=35.0∘.θ=35.0∘. How wide is the river? • A ball is tossed from an upper-story window of a building. The ball is given an initial velocity of 8.00 m/sm/s at an angle of 20.0∘0∘ below the horizontal. It strikes the ground 3.00 s later. (a) How far horizontally from the base of the building does the ball strike the ground? (b) Find the height from which the ball was thrown. (c) How long does it take the ball to reach a point 10.0 mm below the level of launching? • A plane electromagnetic wave varies sinusoidally at 90.0 MHz as it travels through vacuum along the positive x direction. The peak value of the electric field is 2.00 mV/m, and it is directed along the positive y direction. Find (a) the wavelength, (b) the period, and (c) the maximum value of the magnetic field. (d) Write expressions in SI units for the space and time variations of the electric field and of the magnetic field. Include both numerical values and unit vectors to indicate directions. (e) Find the average power per unit area this wave carries through space. (f) Find the average energy density in the radiation (in joules per cubic meter). (g) What radiation pressure would this wave exert upon a perfectly reflecting surface at normal incidence? • A pail of water is rotated in a vertical circle of radius 1.00 m. (a) What two external forces act on the water in the pail? (b) Which of the two forces is most important in causing the water to move in a circle? (c) What is the pail’s minimum speed at the top of the circle if no water is to spill out? (d) Assume the pail with the speed found in part (c) were to suddenly disappear at the top of the circle. Describe the subsequent motion of the water. Would it differ from the motion of a projectile? • An important news announcement is transmitted by radio waves to people sitting next to their radios 100 km from the station and by sound waves to people sitting across the newsroom 3.00 m from the newscaster. Taking the speed of sound in air to be 343 m/s, who receives the news first? Explain. • A common demonstration, illustrated in Figure P10.74P10.74 , consists of a ball resting at one end of a uniform board of length ℓℓ that is hinged at the other end and elevated at an angle θ.θ. A light cup is attached to the board at rcrc so that it will catch the ball when the support stick is removed suddenly. (a) Show that the ball will lag behind the falling board when θθ is less than 35.3∘.35.3∘. (b) Assuming the board is 1.00 mm long and is supported at this limiting angle, show that the cup must be 18.4 cmcm from the moving end. • A light ray traveling in air is incident on one face of a right-angle prism with index of refraction n=1.50n=1.50 as shown in Figure P35.61P35.61 , and the ray follows the path shown in the figure. Assuming θ=60.0∘θ=60.0∘ and the base of the prism is mirrored, determine the angle ϕϕ made by the outgoing ray with the normal to the right face of the prism. • An electric dipole in a uniform horizontal electric field is displaced slightly from its equilibrium position as shown in Figure P23.79,P23.79, where θθ is small. The sepa- ration of the charges is 2a,2a, and each of the two particles has mass mm (a) Assuming the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency f=12πqEma−−−−√f=12πqEma What If? (b) Suppose the masses of the two charged particles in the dipole are not the same even though each particle continues to have charge qq . Let the masses of the particles be m1m1 and m2m2 . Show that the frequency of the oscillation in this case is f=12πqE(m1+m2)2am1m2−−−−−−−−−−−−√f=12πqE(m1+m2)2am1m2 • A brick of mass MM has been placed on a rubber cushion of mass mm . Together they are sliding to the right at constant velocity on an ice-covered parking lot. (a) Draw a free-body diagram of the brick and identify each force acting on it. (b) Draw a free-body diagram of the cushion and identify each force acting on it. (c) Identify all of the action-reaction pairs of forces in the brick-cushion-planet system. • All atoms have the same size, to an order of magnitude. (a) To demonstrate this fact, estimate the atomic diameters for aluminum (with molar mass 27.0 g/molg/mol and density 2.70 g/cm3g/cm3 ) and uranium (molar mass 238 g/molg/mol and density 18.9g/cm3).18.9g/cm3). (b) What do the results of part (a) imply about the wave functions for inner-shell electrons as we progress to higher and higher atomic mass atoms? • For the arrangement shown in Figure P 18.60, the inclined plane and the small pulley are friction less; the string supports the object of mass MM at the bottom of the plane; and the string has mass mm . The system is in equilibrium, and the vertical part of the string has a length hh . We wish to study standing waves set up in the vertical section of the string. (a) What analysis model describes the object of mass M?M? (b) What analysis model describes the waves on the vertical part of the string? (c) Find the tension in the string. (d) Model the shape of the string as one leg and the hypotenuse of a right triangle. Find the whole length of the string. (e) Find the mass per unit length of the string. (f) Find the speed of waves on the string. (g) Find the lowest frequency for a standing wave on the vertical section of the string. (h) Evaluate this result for M=1.50kg,m=0.750g,h=0.500m,M=1.50kg,m=0.750g,h=0.500m, and θ=30.0∘.θ=30.0∘. (i) Find the numerical value for the lowest frequency for a standing wave on the sloped section of the string. • An older-model car accelerates from 0 to speed vv in a time interval of Δt.Δt. A newer, more powerful sports car accelerates from 0 to 2vv in the same time period. Assuming the energy coming from the engine appears only as kinetic energy of the cars, compare the power of the two cars. • For the potential energy curve shown in Figure P7.52 , (a) determine whether the force Fx is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for Fx versus x from x=0 to x=9.5m. • An AC power supply produces a maximum voltage ΔVmax=ΔVmax= 100V.100V. This power supply is connected to a resistor R=R= 24.0 ΩΩ , and the current and resistor voltage are measured with an ideal AC ammeter and voltmeter as shown in Figure P33. 3 . An ideal ammeter has. Aero resistance, and an ideal voltmeter has infinite resistance. What is the reading on (a) the ammeter and (b) the voltmeter? • An isolated, charged conducting sphere of radius 12.0 cmcm creates an electric field of 4.90×104N/C4.90×104N/C at a distance 21.0 cmcm from its center. (a) What is its surface charge density? (b) What is its capacitance? • A 9.00 -kg hanging object is connected by a light, inextensible cord over a light, frictionless pulley to a 5.00 -kg block that is sliding on a flat table (Fig. P5.28). Taking the coefficient of kinetic friction as 0.200 , find the tension in the string. • A ball swings in a vertical circle at the end of a rope 1.50 mm long. When the ball is 36.9∘9∘ past the lowest point on its way up, its total acceleration is (−22.5ˆi+20.2ˆj)m/s2.(−22.5i^+20.2j^)m/s2. For that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball. • A long solenoid has n=400 turns per meter and carries a current given by I=30.0(1−e−1.60t), where I is in amperes and t is in seconds. Inside the solenoid and coaxial with it is a coil that has a radius of R=6.00cm and consists of a total of N=250 turns of fine wire (Fig. P31.14). What emf is induced in the coil by the changing current? • Two ideal inductors, L1L1 and L2L2, have zero internal resistance and are far apart, so their magnetic fields do not influence each other. (a) Assuming these inductors are connected in series, show that they are equivalent to a single ideal inductor having Leq=L1+L2Leq=L1+L2 (b) Assuming these same two inductors are connected in parallel, show that they are equivalent to a single ideal inductor having 1/Leq=1/L1+1/L2.(c)1/Leq=1/L1+1/L2.(c) What If? Now consider two inductors L1L1 and L2L2 that have nonzero internal resistances R1R1 and R2,R2, respectively. Assume they are still far apart, so their mutual inductance is zero, and assume they are connected in series. Show that they are equivalent to a single inductor having Leq=L1+L2Leq=L1+L2 and Req=R1+R2.Req=R1+R2. (d) If these same inductors are now connected in parallel, is it necessarily true that they are equivalent to a single ideal inductor having1/Leq=1/L1+1/L2ing⁡1/Leq=1/L1+1/L2 and 1/Req=1/R1+1/R2?1/Req=1/R1+1/R2? Explain your answer. • A series RLCRLC circuit has a resistance of 22.0ΩΩ and an impedance of 80.0ΩΩ . If the rms voltage applied to the circuit is 160V,160V, what average power is delivered to the circuit? • At major league baseball games, it is commonplace to flash on the scoreboard a speed for each pitch. This speed is determined with a radar gun aimed by an operator positioned behind home plate. The gun uses the Doppler shift of microwaves reflected from the baseball, an effect we will study in Chapter 39. The gun determines the speed at some particular point on the baseball’s path, depending on when the operator pulls the trigger. Because the ball is subject to a drag force due to air proportional to the square of its speed given by R=kmv2,R=kmv2, it slows as it travels 18.3 mm toward the plate according to the formula v=vie−kxv=vie−kx . Suppose the ball leaves the pitcher’s hand at 90.0mi/h=90.0mi/h= 40.2 m/sm/s . Ignore its vertical motion. Use the calculation of RR for baseballs from Example 6.11 to determine the speed of the pitch when the ball crosses the plate. • Vector →AA→ has a magnitude of 35.0 units and points in the direction 325∘325∘ counterclockwise from the positive xx axis. Calculate the xx and yy components of this vector. • The mass of a roller-coaster car, including its passengers, is 500 kgkg . Its speed at the bottom of the track in Figure P6.16P6.16 is 19 m/sm/s . The radius of this section of the track is r1=25m.r1=25m. Find the force that a seat in the roller-coaster car exerts on a 50−kg50−kg passenger at the lowest point. • A particle with electric charge q moves along a straight line in a uniform electric field →EE→ with speed u.u. The electric force exerted on the charge is q→EqE→ . The velocity of the particle and the electric field are both in the xx direc- (a) Show that the acceleration of the particle in the xx direction is given by a=dudt=qEm(1−u2c2)3/2a=dudt=qEm(1−u2c2)3/2 (b) Discuss the significance of the dependence of the acceleration on the speed. (c) What If? If the particle starts from rest at x=0x=0 at t=0t=0 , how would you proceed to find the speed of the particle and its position at time tt ? • In Example 11.9, we investigated an elastic collision between a disk and a stick lying on a friction less surface. Suppose everything is the same as in the example except that the collision is perfectly inelastic so that the disk adheres to the stick at the endpoint at which it strikes. Find (a) the speed of the center of mass of the system and (b) the angular speed of the system after the collision. • Four resistors are connected in parallel across a 9.20-V battery. They carry currents of 150 mA, 45.0 mA, 14.0 mA, and 4.00 mA. If the resistor with the largest resistance is replaced with one having twice the resistance, (a) what is the ratio of the new current in the battery to the original current? (b) What If? If instead the resistor with the smallest resistance is replaced with one having twice the resistance, what is the ratio of the new total current to the original current? (c) On a February night, energy leaves a house by several energy leaks, including 1.50×103W1.50×103W by conduction through the ceiling, 450 WW by infiltration (air- flow) around the windows, 140 WW by conduction through the basement wall above the foundation sill, and 40.0 WW by conduction through the plywood door to the attic. To produce the biggest saving in heating bills, which one of these energy transfers should be reduced first? Explain how you decide. Clifford Swartz suggested the idea for this problem. • In the What If? section of Example 4.5 , it was claimed that the maximum range of a ski jumper occurs for a launch angle θθ given by θ=45∘−ϕ2θ=45∘−ϕ2 where ϕϕ is the angle the hill makes with the horizontal in Figure 4.14 . Prove this claim by deriving the equation above. • A 1200−N1200−N uniform boom at ϕ=65∘ϕ=65∘ to the vertical is supported by a cable at an angle θ=25.0∘θ=25.0∘ to the horizontal as shown in Figure P12.46P12.46 . The boom is pivoted at the bottom, and an object of weight m=2000Nm=2000N hangs from its top. Find (a) the tension in the support cable and (b) the components of the reaction force exerted by the floor on the boom. • Electric charge can accumulate on an airplane in flight. You may have observed needle-shaped metal extensions on the wing tips and tail of an airplane. Their purpose is to allow charge to leak off before much of it accumulates. The electric field around the needle is much larger than the field around the body of the airplane and can become large enough to produce dielectric breakdown of the air, discharging the airplane. To model this process, assume two charged spherical conductors are connected by a long conducting wire and a 1.20−μC1.20−μC charge is placed on the combination. One sphere, representing the body of the airplane, has a radius of 6.00 cmcm ; the other, representing the tip of the needle, has a radius of 2.00cm.2.00cm. (a) What is the electric potential of each sphere? (b) What is the electric field at the surface of each sphere? • Two parallel wires are separated by 6.00 cm, each carrying 3.00 A of current in the same direction. (a) What is the magnitude of the force per unit length between the wires? (b) Is the force attractive or repulsive? • Two vertical radio-transmitting antennas are separated by half the broadcast wavelength and are driven in phase with each other. In what horizontal directions are (a) the strongest and (b) the weakest signals radiated? • Given two particles with 2.00−μC2.00−μC charges as shown in Figure P25.15P25.15 and a particle with charge q=1.28×10−18Cq=1.28×10−18C at the origin, ( a) what is the net force exerted by the two 2.00−μC2.00−μC charges on the test charge q2q2 (b) What is the electric field at the origin due to the two 2.00−μC2.00−μC particles? (c) What is the electric potential at the origin due to the two 2.00−μC2.00−μC particles? • A uniform pole is propped between the floor and the ceiling of a room. The height of the room is 7.80 ftft , and the coefficient of static friction between the pole and the ceiling is 0.576.0.576. The coefficient of static friction between the pole and the floor is greater than that between the pole and the ceiling. What is the length of the longest pole that can be propped between the floor and the ceiling? • Two coplanar and concentric circular loops of wire carry currents of I1=5.00AI1=5.00A and I2=3.00AI2=3.00A in opposite directions as in Figure P30.64P30.64 . If r1=12.0cmr1=12.0cm and r2=9.00cm,r2=9.00cm, what are (a) the magnitude and (b) the direction of the net magnetic field at the center of the two loops? (c) Let r1r1 remain fixed at 12.0 cmcm and let r2r2 be a variable. Determine the value of r2r2 such that the net field at the center of the loops is zero. • A fly lands on one wall of a room. The lower-left corner of the wall is selected as the origin of a two-dimensional Cartesian coordinate system. If the fly is located at the point having coordinates (2.00, 1.00) m, (a) how far is it from the origin? (b) What is its location in polar coordinates? • A pulsed laser emits light of wavelength λ.λ. For a pulse of duration ΔtΔt having energy TER,TER, find (a)(a) the physical length of the pulse as it travels through space and (b) the number of photons in it. (c) The beam has a circular cross section having diameter d.d. Find the number of photons per unit volume. • The wave function for a particle is given by ψ(x)=Ae−|x|/aψ(x)=Ae−|x|/a , where A and a are constants. (a) Sketch this function for values of x in the interval −3a<x<3a . (b) Determine the value of A . (c) Find the probability that the particle will be found in the interval −a<x<a . • Determine the possible values of the quantum numbers ℓℓ and mℓmℓ for the He ++ ion in the state corresponding to n=n= 3. (b) What is the energy of this state? • During periods of high activity, the Sun has more sunspots than usual. Sunspots are cooler than the rest of the luminous layer of the Sun’s atmosphere (the photosphere). Paradoxically, the total power output of the active Sun is not lower than average but is the same or slightly higher than average. Work out the details of the following crude model of this phenomenon. Consider a patch of the photosphere with an area of 5.10×1014m2.5.10×1014m2. Its emissivity is 0.965 . (a) Find the power it radiates if its temperature is uniformly 5800 KK , corresponding to the quiet Sun. (b) To represent a sunspot, assume 10.0%% of the patch area is at 4 800 K and the other 90.0% is at 5 890 K. Find the power output of the patch. (c) State how the answer to part (b) compares with the answer to part (a). (d) Find the average temperature of the patch. Note that this cooler temperature results in a higher power output. (The next sunspot maximum is expected around the year 2012.) • The speed of a nerve impulse in the human body is about 100$\mathrm{m} / \mathrm{s}$. If you accidentally stub your toe in the dark, estimate the time it takes the nerve impulse to travel to your • Ganymede is the largest of Jupiter’s moons. Consider a rocket on the surface of Ganymede, at the point farthest from the planet (Fig. P13.41). Model the rocket as a particle. (a) Does the presence of Ganymede make Jupiter exert a larger, smaller, or same size force on the rocket compared with the force it would exert if Ganymede were not interposed? (b) Determine the escape speed for the rocket from the planet–satellite system. The radius of Ganymede is 2.64×106m,2.64×106m, and its mass is 1.495×1023kg.1.495×1023kg. The distance between Jupiter and Ganymede is 1.071×109m,1.071×109m, and the mass of Jupiter is 1.90×1027kg1.90×1027kg . Ignore the motion of Jupiter and Ganymede as they revolve about their center of mass. • Calculate the speed of a proton that is accelerated from rest through an electric potential difference of 120 V. (b) Calculate the speed of an electron that is accelerated through the same electric potential difference. • The mass of the Earth is 5.97×1024kg5.97×1024kg , and the mass of the Moon is 7.35×1022kg7.35×1022kg . The distance of separation, measured between their centers, is 3.84×108m.3.84×108m. Locate the center of mass of the Earth-Moon system as measured from the center of the Earth. • You may use the Rayleigh criterion for the limiting angle of resolution of an eye. The standard may be overly optimistic for human vision. • A standing wave is described by the wave function y=6sin(π2x)cos(100πt)y=6sin(π2x)cos(100πt) where xx and yy are in meters and tt is in seconds. (a) Prepare graphs showing yy as a function of xx for five instants: t=0t=0 ,5ms,10ms,15ms,5ms,10ms,15ms, and 20 msms . (b) From the graph, identify the wavelength of the wave and explain how to do so. (c) From the graph, identify the frequency of the wave and explain how to do so. (d) From the equation, directly identify the wavelength of the wave and explain how to do so. (e) From the equation, directly identify the frequency and explain how to do so. • An all-electric home uses 2000 kWh of electric energy per month. Assuming all energy released from fusion could be captured, how many fusion events described by the reaction 21H+31H→42He+10n21H+31H→42He+10n would be required to keep this home running for one year? • A block of mass M=0.450kgM=0.450kg is attached to one end of a cord of mass m=0.00320kgm=0.00320kg ; the other end of the cord is attached to a fixed point. The block rotates with constant angular speed ω=10.0rad/sω=10.0rad/s in a circle on a frictionless, horizontal table as shown in Figure P16.55P16.55 . What time interval is required for a transverse wave to travel along the string from the center of the circle to the block? • A wire is formed into a circle having a diameter of 10.0 cm and is placed in a uniform magnetic field of 3.00 mT. The wire carries a current of 5.00 A. Find (a) the maximum torque on the wire and (b) the range of potential energies of the wire–field system for different orientations of the circle. • The biggest stuffed animal in the world is a snake 420 m long, constructed by Norwegian children. Suppose the snake is laid out in a park as shown in Figure P 3.51, forming two straight sides of a 105° angle, with one side 240 m long. Olaf and Inge run a race they invent. Inge runs directly from the tail of the snake to its head, and Olaf starts from the same place at the same moment but runs along the snake. (a) If both children run steadily at 12.0 km/h, Inge reaches the head of the snake how much earlier than Olaf? (b) If Inge runs the race again at a constant speed of 12.0 km/h, at what constant speed must Olaf run to reach the end of the snake at the same time as Inge? • A hypodermic syringe contains a medicine with the density of water (Fig. Pl4.51). The barrel of the syringe has a cross-sectional area$A=2.50 \times 10^{-5} \mathrm{m}^{2},$and the needle has a cross-sectional area$a=1.00 \times 10^{-8} \mathrm{m}^{2} .$In the absence of a force on the plunger, the pressure everywhere is 1.00 atm. A force$\overrightarrow{\mathbf{F}}$of magnitude 2.00$\mathrm{N}acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle’s tip. • Suppose the ionization energy of an atom is 4.10 eV. In the spectrum of this same atom, we observe emission lines with wavelengths 310nm,400nm,310nm,400nm, and 1377.8nm.1377.8nm. Use this information to construct the energy-level diagram with the fewest levels. Assume the higher levels are closer together. • The switch in Figure P32.21P32.21 is open for t<0t<0 and is then thrown closed at time t=0.t=0. Assume R=4.00Ω,L=nR=4.00Ω,L=n 1.00H,1.00H, and E=10.0VE=10.0V . Find (a)(a) the current in the inductor and (b) the current in the switch as functions of time • A plane electromagnetic wave of intensity 6.00 W/m2W/m2 moving in the xx direction, strikes a small perfectly reflecting pocket mirror, of area 40.0cm2,40.0cm2, held in the yy lane. (a) What momentum does the wave transfer to the mirror each second? (b) Find the force the wave exerts on the mirror. (c) Explain the relationship between the answers to parts (a) and (b). • During the power stroke in a four-stroke automobile engine, the piston is forced down as the mixture of combustion products and air undergoes an adiabatic expansion. Assume ( 1 ) the engine is running at 2500 cycles/min; (2) the gauge pressure immediately before the expansion is 20.0 atm; (3)(3) the volumes of the mixture immediately before and after the expansion are 50.0 cm3cm3 and 400 cm3cm3 , respectively (Fig. P21.23); (4) the time interval for the expansion is one-fourth that of the total cycle; and (5) the mixture behaves like an ideal gas with specific heat ratio 1.40.1.40. Find the average power generated during the power stroke. • Green light (λ=546nm)(λ=546nm) illuminates a pair of narrow, parallel slits separated by 0.250 mmmm . Make a graph of I/ImaxI/Imax as a function of θθ for the interference pattern observed on a screen 1.20 mm away from the plane of the parallel slits. Let θθ range over the interval from −0.3∘−0.3∘ to +0.3∘.+0.3∘. • A skateboarder with his board can be modeled as a particle of mass 76.0 kg, located at his center of mass, 0.500 m above the ground. As shown in Figure P 11.48, the skateboarder starts from rest in a crouching position at one lip of a half-pipe (point A). The half-pipe forms one half of a cylinder of radius 6.80 m with its axis horizontal. On his descent, the skateboarder moves without friction and maintains his crouch so that his center of mass moves through one quarter of a circle. (a) Find his speed at the bottom of the half-pipe (point B). (b) Find his angular momentum about the center of curvature at this point. (c) Immediately after passing point B, he stands up and raises his arms, lifting his center of gravity to 0.950 m above the concrete (point C ). Explain why his angular momentum is constant in this maneuver, whereas the kinetic energy of his body is not constant. (d) Find his speed immediately after he stands up. (e) How much chemical energy in the skate-boarder’s legs was converted into mechanical energy in the skateboarder–Earth system when he stood up? • A beam of 580 -nm light passes through two closely spaced glass plates at close to normal incidence as shown in Figure P37.33. For what minimum nonzero value of the plate separation dd is the transmitted light bright? • A vertical cylinder of cross-sectional area AA is fitted with a tight-fitting, frictionless piston of mass mm (Fig. PI9. 52)) . The piston is not restricted in its motion in any way and is supported by the gas at pressure PP below it. Atmospheric pressure is P0P0 . We wish to find the height hh in Figure P19.52P19.52 . (a) What analysis model is appropriate to describe the piston? (b) Write an appropriate force equation for the piston from this analysis model in terms of P,P0,m,A,P,P0,m,A, and gg . (c) Suppose nn moles of an ideal gas are in the cylinder at a temperature of T. Substitute for PP in your answer to part (b)(b) to find the height hh of the piston above the bottom of the cylinder. • A standing-wave pattern is observed in a thin wire with a length of 3.00 m. The wave function is y=0.00200sin(πx)cos(100πt)y=0.00200sin(πx)cos(100πt) where xx and yy are in meters and t is in seconds. (a) How many loops does this pattern exhibit? (b) What is the fundamental frequency of vibration of the wire? (c) What If? If the original frequency is held constant and the tension in the wire is increased by a factor of 9, how many loops are present in the new pattern? • An automobile has a mass of 1 500 kg, and its aluminum brakes have an overall mass of 6.00 kg. (a) Assume all the mechanical energy that transforms into internal energy when the car stops is deposited in the brakes and no energy is transferred out of the brakes by heat. The brakes are originally at 20.0°C. How many times can the car be stopped from 25.0 m/s before the brakes start to melt? (b) Identify some effects ignored in part (a) that are important in a more realistic assessment of the warming of the brakes. • The gravitational force exerted on a baseball is 2.21 N down. A pitcher throws the ball horizontally with velocity 18.0 m/s by uniformly accelerating it along a straight horizontal line for a time interval of 170 ms . The ball starts from rest. (a) Through what distance does it move before its release? (b) What are the magnitude and direction of the force the pitcher exerts on the ball? • In Figure P 29.42, the cube is 40.0 cm on each edge. Four straight segments of wire- ab,bc,cd,ab,bc,cd, and da−da− form a closed loop that carries a current I=5.00I=5.00 A in the direction shown. A uniform magnetic field of magnitude B=0.0200B=0.0200 TT is in the positive yy direction. Determine the magnetic force vector on (a) a b,(b) b c,(c) c d, and (d) da. (e) Explain how you could find the force exerted on the fourth of these segments from the forces on the other three, without further calculation involving the magnetic field. • An airplane maintains a speed of 630 km/hkm/h relative to the air it is flying through as it makes a trip to a city 750 kmkm away to the north. ( a) What time interval is required for the trip if the plane flies through a headwind blowing at 35.0 km/hkm/h toward the south? (b) What time interval is required if there is a tailwind with the same speed? (c) What time interval is required if there is a crosswind blowing at 35.0 km/hkm/h to the east relative to the ground? • Consider a thin, spherical shell of radius 14.0 cm with a total charge of 32.0μCμC distributed uniformly on its surface. Find the electric field (a) 10.0 cm and (b) 20.0 cm from the center of the charge distribution. • The current in a coil changes from 3.50 A to 2.00 A in the same direction in 0.500 s. If the average emf induced in the coil is 12.0 mV, what is the inductance of the coil? • In Figure P28.71,P28.71, suppose the switch has been closed for a time interval sufficiently long for the capacitor to become fully charged. Find (a) the steady-state current in each resistor and (b) the charge QQ on the capacitor. (c) The switch is now opened at t=0.t=0. Write an equation for the current in R2R2 as a function of time and (d) find the time interval required for the charge on the capacitor to fall to one-fifth its initial value. • An electron is trapped in an infinitely deep potential well 0.300 nm in length. (a) If the electron is in its ground state, what is the probability of finding it within 0.100 nm of the left-hand wall? (b) Identify the classical probability of finding the electron in this interval and state how it compares with the answer to part (a). (c) Repeat parts (a) and (b) assuming the particle is in the 99 th energy state. • Consider the roller coaster described in Problem 52. Because of some friction between the coaster and the track, the coaster enters the circular section at a speed of 15.0 m/s rather than the 22.0 m/s in Problem 52. Is this situation more or less dangerous for the passengers than that in Problem 52? Assume the circular section is still frictionless. • The spring constant of an automotive suspension spring increases with increasing load due to a spring coil that is widest at the bottom, smoothly tapering to a smaller diameter near the top. The result is a softer ride on normal road surfaces from the wider coils, but the car does not bottom out on bumps because when the lower coils collapse, the stiffer coils near the top absorb the load. For such springs, the force exerted by the spring can be empirically found to be given by F=axb . For a tapered spiral spring that compresses 12.9 cm with a 1000−N load and 31.5 cm with a 5000−N load, (a) evaluate the constants a and b in the empirical equation for F and (b) find the work needed to compress the spring 25.0cm. • A basin surrounding a drain has the shape of a circular cone opening upward, having everywhere an angle of 35.0∘0∘ with the horizontal. A 25.0-g ice cube is set sliding around the cone without friction in a horizontal circle of radius R. (a) Find the speed the ice cube must have as a function of R. (b) Is any piece of data unnecessary for the solution? Suppose R is made two times larger. (c) Will the required speed increase, decrease, or stay constant? If it changes, by what factor? (d) Will the time required for each revolution increase, decrease, or stay constant? If it changes, by what factor? (e) Do the answers to parts (c) and (d) seem contradictory? Explain. • You may use the Rayleigh criterion for the limiting angle of resolution of an eye. The standard may be overly optimistic for human vision. • Prepare a table like Table 22.1 for the following occurrence. You toss four coins into the air simultancously and then record the results of your tosses in terms of the numbers of heads (H) and tails (T) that result. For example, HHTH and HTHH are two possible ways in which three heads and one tail can be achieved. (b) On the basis of your table, what is the most probable result recorded for a toss? In terms of entropy, (c) what is the most ordered macrostate, and (d) what is the most disordered? • An elementary theorem in statistics states that the root-mean-square uncertainty in a quantity rr is given by Δr=√⟨r2⟩−⟨r⟩2Δr=⟨r2⟩−⟨r⟩2−−−−−−−−−√ . Determine the uncertainty in the radial position of the electron in the ground state of the hydrogen atom. Use the average value of rr found in Example 3:⟨r⟩=3a0/242.3:⟨r⟩=3a0/2 . The average value of the squared dis- tance between the electron and the proton is given by ⟨r2⟩=∫all space|ψ|2r2dV=∫∞0P(r)r2dr⟨r2⟩=∫all space|ψ|2r2dV=∫∞0P(r)r2dr all space • John is pushing his daughter Rachel in a wheelbarrow when it is stopped by a brick of height hh (Fig. PI2.21). The handles make an angle of θθ with the ground. Due to the weight of Rachel and the wheelbarrow, a downward force mgmg is exerted at the center of the wheel, which has a radius RR . (a) What force FF must John apply along the handles to just start the wheel over the brick? (b) What are the components of the force that the brick exerts on the wheel just as the wheel begins to lift over the brick? In both parts, assume the brick remains fixed and does not slide along the ground. Also assume the force applied by John is directed exactly toward the center of the wheel. • A 30.0 -g metal ball having net charge Q=5.00μCQ=5.00μC is thrown out of a window horizontally north at a speed v=20.0m/sv=20.0m/s . The window is at a height h=h= 20.0 mm above the ground. A uniform, horizontal magnetic field of magnitude B=0.0100TB=0.0100T is perpendicular to the plane of the ball’s trajectory and directed toward the west. (a) Assuming the ball follows the same trajectory as it would in the absence of the magnetic field, find the magnetic force acting on the ball just before it hits the ground. (b) Based on the result of part (a), is it justified for three-significant-digit precision to assume the trajectory is unaffected by the magnetic field? Explain. • The rotor in a certain electric motor is a flat, rectangular coil with 80 turns of wire and dimensions 2.50 cm by 4.00 cm. The rotor rotates in a uniform magnetic field of 0.800 T. When the plane of the rotor is perpendicular to the direction of the magnetic field, the rotor carries a current of 10.0 mA. In this orientation, the magnetic moment of the rotor is directed opposite the magnetic field. The rotor then turns through one-half revolution. This process is repeated to cause the rotor to turn steadily at an angular speed of 3.60×1033.60×103 rev/min. (a) Find the maximum torque acting on the rotor. (b) Find the peak power output of the motor. (c) Determine the amount of work performed by the magnetic field on the rotor in every full revolution. (d) What is the average power of the motor? • A triangular glass prism with apex angle 60.0∘0∘ has an index of refraction of 1.50.1.50. (a) Show that if its angle of incidence on the first surface is θ1=48.6∘,θ1=48.6∘, light will pass symmetrically through the prism as shown in Figure 35.17 . (b) Find the angle of deviation δminδmin for θ1=48.6∘θ1=48.6∘. (c) What If? Find the angle of deviation if the angle of incidence on the first surface is 45.6∘.45.6∘. (d) Find the angle of deviation if θ1=θ1= 51.6∘.51.6∘. • A theory of nuclear astrophysics proposes that all the elements heavier than iron are formed in supernova explosions ending the lives of massive stars. Assume equal amounts of 235 U and 238 U were created at the time of the explosion and the present 235 U/238U ratio on the Earth is 0.00725. The half-lives of 235U and 238U are 0.704×109yr and 4.47×109 yr, respectively. How long ago did the star(s) explode that released the elements that formed the Earth? • A machine part rotates at an angular speed of 0.060rad/s;0.060rad/s; its speed is then increased to 2.2 rad/srad/s at an angular acceleration of 0.70rad/s2.0.70rad/s2. (a) Find the angle through which the part rotates before reaching this final speed. (b) If both the initial and final angular speeds are doubled and the angular acceleration remains the same, by what factor is the angular displacement changed? Why? • A time-varying current II is sent through a 50.0−mH50.0−mH inductor from a source as shown in Figure P32.63aP32.63a . The current is constant at I=−1.00mAI=−1.00mA until t=0t=0 and then varies with time afterward as shown in Figure P32.63bP32.63b . Make a graph of the emf across the inductor as a function of time. • A multicylinder gasoline engine in an airplane, operating at 2.50×1032.50×103 rev/min, takes in energy 7.89×103J7.89×103J and exhausts 4.58×103J4.58×103J for each revolution of the crankshaft. (a) How many liters of fuel does it consume in 1.00 hh of operation if the heat of combustion of the fuel is equal to 4.03×107J/L24.03×107J/L2 (b) What is the mechanical power output of the engine? Ignore friction and express the answer in horsepower. (c) What is the torque exerted by the crank-shaft on the load? (d) What power must the exhaust and cooling system transfer out of the engine? • Find the equivalent capacitance between points aa and bb in the combination of capacitors shown in Figure P26.29P26.29 . • A certain nuclear plant generates internal energy at a rate of 3.065 GW and transfers energy out of the plant by electrical transmission at a rate of 1.000 GW . Of the waste energy, 0% is ejected to the atmosphere and the remainder is passed into a river. A state law requires that the river water be warmed by no more than 3.50∘C when it is returned to the river. (a) Determine the amount of cooling water necessary (in kilograms per hour and cubic meters per hour) to cool the plant. (b) Assume fission generates 7.80×1010J/g of 235U . Determine the rate of fuel burning (in kilograms per hour) of 255U. • A block of mass m1=20.0kgm1=20.0kg m2=30.0kgm2=30.0kg by a massless string that passes over a light, frictionless pulley. The 30.0−kg30.0−kg block is connected to a spring that has negli- gible mass and a force constant of k=250N/mk=250N/m as shown in Figure 64. The spring is unstretched when the system is as shown in the figure, and the incline is frictionless. The 20.0 -kg block is pulled a distance h=20.0cmh=20.0cm down the incline of angle θ=40.0∘θ=40.0∘ and released from rest. Find the speed of each block when the spring is again unstretched. • A person is working near the secondary of a transformer as shown in Figure P33.52P33.52 . The primary voltage is 120 VV at 60.0 HzHz . The secondary voltage is 5000 VV . The capacitance Cs,Cs, which is the stray capacitance between the hand and the secondary winding, is 20.0 pFpF . Assuming the person has a body resistance to ground of Rb=50.0kΩ,Rb=50.0kΩ, determine the rms voltage across the body. Suggestion: Model the secondary of the transformer as an AC source. • the technique known as electron spin resonance (ESR), a sample containing unpaired electrons is placed in a magnetic field. Consider a situation in which a single electron (not contained in an atom) is immersed in a magnetic field. In this simple situation, only two energy states are possible, corresponding to ms=±12.ms=±12. In ESR, the absorption of a photon causes the electron’s spin magnetic moment to flip from the lower energy state to the higher energy state. According to Section 29.5,29.5, the change in energy is 2μBμB . (The lower energy state corresponds to the case in which the zz component of the magnetic moment →μspinμ→spin is aligned with the magnetic field, and the higher energy state corresponds to the case in which the zz component of →μspinμ→spin is aligned opposite to the field. )) What is the photon frequency required to excite an ESR transition in a 0.350−T0.350−T magnetic field? • A large, flat, horizontal sheet of charge has a charge per unit area of 9.00μC/m29.00μC/m2. Find the electric field just above the middle of the sheet. • A U-tube open at both ends is partially filled with water (Fig. Pl4.73a). Oil having a density 750\mathrm{kg} / \mathrm{m}^{3}$is then poured into the right arm and forms a column$L=5.00 \mathrm{cm}$high (Fig. Pl4.73b). (a) Determine the difference$h$in the heights of the two liquid surfaces. (b) The right arm is then shielded from any air motion while air is blown across the top of the left arm until the surfaces of the two liquids are at the same height (Fig. Pl4.73c). Determine the speed of the air being blown across the left arm. Take the density of air as constant at 1.20$\mathrm{kg} / \mathrm{m}^{3}$. • Lightbulb A is marked “25 W 120 V,” and lightbulb B is marked ” 100 W120VW120V ” These labels mean that each lightbulb has its respective power delivered to it when it is connected to a constant 120−V120−V source. (a) Find the resistance of each lightbulb. (b) During what time interval does 1.00 CC pass into lightbulb A? (c) Is this charge different upon its exit versus its entry into the lightbulb? Explain. (d) In what time interval does 1.00 JJ pass into lightbulb A? (e) By what mechanisms does this energy enter and exit the lightbulb? Explain. (f) Find the cost of running lightbulb A continuously for 30.0 days, assuming the electric company sells its product at$0.110$0.110 per kWhkWh . • The most common isotope of radon is 222Rn,222Rn, which has half-life 3.82 days. (a) What fraction of the nuclei that were on the Earth one week ago are now undecayed? (b) Of those that existed one year ago? (C) In view of these results, explain why radon remains a problem, contributing significantly to our background radiation exposure. • An aluminum wire is held between two clamps under zero tension at room temperature. Reducing the temperature, which results in a decrease in the wire’s equilibrium length, increases the tension in the wire. Taking the cross-sectional area of the wire to be 5.00×10−6m25.00×10−6m2 , the density to be 2.70×103kg/m3,2.70×103kg/m3, and Young’s modulus to be 7.00×1010N/m27.00×1010N/m2 , what strain (ΔL/L)(ΔL/L) results in a transverse wave speed of 100 m/sm/s ? • Motion-picture film is projected at a frequency of 24.0 frames per second. Each photograph on the film is the same height of 19.0 mmmm , just like each oscillation in a wave is the same length. Model the height of a frame as the wave-length of a wave. At what constant speed does the film pass into the projector? • Why is the following situation impossible? On their 40 th birthday, twins Speedo and Goslo say good-bye as Speedo takes off for a planet that is 50 ly away. He travels at a constant speed of 0.85cc and immediately turns around and comes back to the Earth after arriving at the planet. Upon arriving back at the Earth, Speedo has a joyous reunion with Goslo. • A simple pendulum is 5.00 mm long. What is the period of small oscillations for this pendulum if it is located in an elevator (a) accelerating upward at 5.00 m/s2m/s2 (b) Accelerating downward at 5.00 m/s2m/s2 (c) What is the period of this pendulum if it is placed in a truck that is accelerating horizontally at 5.00 m/s2?m/s2? • In the particle under constant acceleration model, we identify the variables and parameters$v_{x i}, v_{x f}, a_{x}, t,$and$x_{f}-x_{i}$. Of the equations in Table 2.2 , the first does not involve$x_{f}-x_{i},$the second does not contain$a_{x}$, the third omits$v_{x y},$and the last leaves out$t .$So, to complete the set, there should be an equation not involving$v_{x x}$(a) Derive it from the others. (b) Use the equation in part (a) to solve Problem 29 in one step. • Within the green dashed circle shown in Figure P31.33, the magnetic field changes with time according to the expression B=2.00t3−4.00t2+0.800 , where B is in teslas, t is in seconds, and R=2.50cm. When t=2.00s , calculate (a) the magnitude and (b) the direction of the force exerted on an electron located at point P1, which is at a distance r1=5.00cm from the center of the circular field region. (c) At what instant is this force equal to zero? • Assume a certain liquid, with density 1 230$\mathrm{kg} / \mathrm{m}^{3}$, exerts no friction force on spherical objects. A ball of mass 2.10$\mathrm{kg}$and radius 9.00$\mathrm{cm}$is dropped from rest into a deep tank of this liquid from a height of 3.30$\mathrm{m}$above the surface. (a) Find the speed at which the ball enters the liquid. (b) Evaluate the magnitudes of the two forces that are exerted on the ball as it moves through the liquid. (c) Explain why the ball moves down only a limited distance into the liquid and calculate this distance. (d) With what speed will the ball pop up out of the liquid? (e) How does the time interval$\Delta t_{\text { down }},$during which the ball moves from the surface down to its lowest point, compare with the time interval$\Delta t_{\text { up }}$for the return trip between the same two points? (f) What If? Now modify the model to suppose the liquid exerts a small friction force on the ball, opposite in direction to its motion. In this case, how do the time intervals$\Delta t_{\text { down }}$and$\Delta t_{\text { up }}$compare? Explain your answer with a conceptual argument rather than a numerical calculation. • A conducting bar of length ℓ moves to the right on two frictionless rails as shown in Figure P31.28 . A uniform magnetic field directed into the page has a magnitude of 0.300 T. Assume R=9.00Ω and ℓ=0.350m. (a) At what constant speed should the bar move to produce an 8.50−mA current in the resistor? (b) What is the direction of the induced current? (c) At what rate is energy delivered to the resistor? (d) Explain the origin of the energy being delivered to the resistor. • Calculate the mass of a solid gold rectangular bar that has dimensions of$4.50 \mathrm{cm} \times 11.0 \mathrm{cm} \times 26.0 \mathrm{cm} .• Express the units of the force constant of a spring in SI fundamental units. • A red light flashes at position xR=3.00mxR=3.00m and time tR=tR= 1.00×10−9s1.00×10−9s and a blue light flashes at xB=5.00mxB=5.00m and tB=9.00×10−9stB=9.00×10−9s , all measured in the S reference frame. Reference frame S′S′ moves uniformly to the right and has its origin at the same point as SS at t=t′=0.t=t′=0. Both flashes are observed to occur at the same place in S′S′ . (a) Find the relative speed between SS and S′S′ . (b) Find the location of the two flashes in frame S′S′ (c) At what time does the red flash occur in the S′S′ frame? • Consider the particles in a gas centrifuge, a device used to separate particles of different mass by whirling them in a circular path of radius rr at angular speed ωω . The force acting on a gas molecule toward the center of the centrifuge is m0ω2r.m0ω2r. (a) Discuss how a gas centrifuge can be used to separate particles of different mass. (b) Suppose the centrifuge contains a gas of particles of identical mass. Show that the density of the particles as a function of rr is n(r)=n0em0r2ω2/2kBTn(r)=n0em0r2ω2/2kBT • Three charged particles are aligned along the xx axis as shown in Figure P23.61.P23.61. Find the electric field at (a) the position (2.00m,0)(2.00m,0) and (b) the position (0,2.00m).(0,2.00m). • As shown in Figure P 18.41, water is pumped into a tall, vertical cylinder at a volume flow rate R.R. The radius of the cylinder is r,r, and at the open top of the cylinder a tuning fork is vibrating with a frequency f.f. As the water rises, what time interval elapses between successive resonances? • Find the equivalent capacitance of a 4.20−μF4.20−μF capacitor and an 8.50−μF8.50−μF capacitor when they are connected (a) in series and (b)(b) in parallel. • An ice cube whose edges measure 20.0\mathrm{mm}$is floating in a glass of ice-cold water, and one of the ice cube’s faces is parallel to the water’s surface. (a) How far below the water surface is the bottom face of the block? (b) Ice-cold ethyl alcohol is gently poured onto the water surface to form a layer 5.00$\mathrm{mm}$thick above the water. The alcohol does not mix with the water. When the ice cube again attains hydrostatic equilibrium, what is the distance from the top of the water to the bottom face of the block? (c) Additional cold ethyl alcohol is poured onto the water’s surface until the top surface of the alcohol coincides with the top surface of the ice cube (in hydrostatic equilibrium). How thick is the required layer of ethyl alcohol? • On a marimba (Fig. P 18.63), the wooden bar that sounds a tone when struck vibrates in a transverse standing wave having three antinodes and two nodes. The lowest-frequency note is 87.0 Hz, produced by a bar 40.0 cm long. (a) Find the speed of transverse waves on the bar. (b) A resonant pipe suspended vertically below the center of the bar enhances the loudness of the emitted sound. If the pipe is open at the top end only, what length of the pipe is required to resonate with the bar in part (a)? • Why is the following situation impossible? It is early on a Saturday morning, and much to your displeasure your next-door neighbor starts mowing his lawn. As you try to get back to sleep, your next-door neighbor on the other side of your house also begins to mow the lawn with an identical mower the same distance away. This situation annoys you greatly because the total sound now has twice the loudness it had when only one neighbor was mowing. • Two traveling sinusoidal waves are described by the wave functions y1=5.00sin[π(4.00x−1200t)]y1=5.00sin[π(4.00x−1200t)] y2=5.00sin[π(4.00x−1200t−0.250)]y2=5.00sin[π(4.00x−1200t−0.250)] where x,y1,x,y1, and y2y2 are in meters and tt is in seconds. (a) What is the amplitude of the resultant wave function y1+y2?y1+y2? (b) What is the frequency of the resultant wave function? • At one instant, a 17.5−kg17.5−kg sled is moving over a horizontal surface of snow at 3.50 m/sm/s . After 8.75 s has elapsed, the sled stops. Use a momentum approach to find the average friction force acting on the sled while it was moving. • Potassium chloride is an ionically bonded molecule that is sold as a salt substitute for use in a low-sodium diet. The electron affinity of chlorine is 3.6 eV. An energy input of 0.70 eV is required to form separate K+K+ and Cl−Cl− ions from separate KK and ClCl atoms. What is the ionization energy of KK ? • An N -turn square coil with side ℓ and resistance R is pulled to the right at constant speed v in the presence of a uniform magnetic field B acting perpendicular to the coil as shown in Figure P31.64 . At t=0 , the right side of the coil has just departed the right edge of the field. At time t, the left side of the coil enters the region where B=0. In terms of the quantities N,B,ℓ,v, and R, find symbolic expressions for (a) the magnitude of the induced emf in the loop during the time interval from t=0 to t,(b) the magnitude of the induced current in the coil, (c) the power delivered to the coil, and (d) the force required to remove the coil from the field. (e) What is the direction of the induced current in the loop? (f) What is the direction of the magnetic force on the loop while it is being pulled out of the field? • What is the equivalent resistance of the combination of identical resistors between points aa and bb in Figure P28.5?P28.5? • A 1.00−μF1.00−μF capacitor is charged by a 40.0−V40.0−V power supply. The fully charged capacitor is then discharged through a 10.0−mH10.0−mH inductor. Find the maximum current in the resulting oscillations. • The speed of a water wave is described by v=√gdv=gd−−√ , where dd is the water depth, assumed to be small compared to the wavelength. Because their speed changes, water waves refract when moving into a region of different depth. (a) Sketch a map of an ocean beach on the eastern side of a landmass. Show contour lines of constant depth under water, assuming a reasonably uniform slope. (b) Suppose waves approach the coast from a storm far away to the north-northeast. Demonstrate that the waves move nearly perpendicular to the shoreline when they reach the beach. (c) Sketch a map of a coastline with alternating bays and headlands as suggested in Figure P35.32P35.32 . Again make a reasonable guess about the shape of contour lines of constant depth. (d) Suppose waves approach the coast, carrying energy with uniform density along originally straight wave fronts. Show that the energy reaching the coast is concentrated at the headlands and has lower intensity in the bays. • When an automobile moves with constant speed down a highway, most of the power developed by the engine is used to compensate for the energy transformations due to friction forces exerted on the car by the air and the road. If the power developed by an engine is 175 hp, estimate the total friction force acting on the car when it is moving at a speed of 29 m/sm/s . One horsepower equals 746 WW . • The resistor in Figure P33.77 on page 982 represents the midrange speaker in a three-speaker system. Assume its resistance to be constant at 8.00Ω. The source represents an audio amplifier producing signals of uniform amplitude ΔVmax=10.0V at all audio frequencies. The inductor and capacitor are to function as a band-pass filter with ΔVout/ΔVin=12 at 200 Hz and at 4.00×103Hz . Determine the required values of (a) L and (b) C. Find (c) the maximum value of the ratio ΔVout/ΔVin; (d) the frequency f0 at which the ratio has its maximum value; (e) the phase shift between Δvin and Δvout at 200 Hz , at f0 , and at 4.00×103 Hz; and (f) the average power transferred to the speaker at 200Hz, at f0, and at 4.00×103Hz . (g) Treating the filter as a resonant circuit, find its quality factor. • The wave function of a quantum particle of mass m is ψ(x)=Acos(kx)+Bsin(kx) where A,B, and k are constants. (a) Assuming the particle is free (U=0), show that ψ(x) is a solution of the Schrödinger equation (Eq.41.15). (b) Find the corresponding energy E of the particle. • In Example 5.8, we investigated the apparent weight of a fish in an elevator. Now consider a 72.0−kg man standing on a spring scale in an elevator. Starting from rest, the elevator ascends, attaining its maximum speed of 1.20 m/s in 0.800 s. It travels with this constant speed for the next 5.00 s. The elevator then undergoes a uniform acceleration in the negative y direction for 1.50 s and comes to rest. What does the spring scale register (a) before the elevator starts to move, (b) during the first 0.800s, (c) while the elevator is traveling at constant speed, and (d) during the time interval it is slowing down? • For a quantum particle of mass m in the ground state of a square well with length L and infinitely high walls, the uncertainty in position is Δx≈L. (a) Use the uncertainty principle to estimate the uncertainty in its momentum. (b) Because the particle stays inside the box, its average momentum must be zero. Its average squared momentum is then ⟨p2⟩≈(Δp)2 . Estimate the energy of the particle. (c) State how the result of part (b) compares with the actual ground-state energy. • The record number of boat lifts, including the boat and its ten crew members, was achieved by Sami Heinonen and Juha Rasanen of Sweden in 2000 . They lifted a total mass of 653.2 kg approximately 4 in. off the ground a total of 24 times. Estimate the total work done by the two men on the boat in this record lift, ignoring the negative work done by the men when they lowered the boat back to the ground. • Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37.08 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk. • For the configuration shown in Figure P 24.51, suppose a=5.00cm,b=20.0cm,a=5.00cm,b=20.0cm, and c=25.0cm.c=25.0cm. Furthermore, suppose the electric field at a point 10.0 cmcm from the center is measured to be 3.60×103N/C3.60×103N/C radially inward and the electric field at a point 50.0 cm from the center is of magnitude 200 N/C and points radially outward. From this information, find (a) the charge on the insulating sphere, (b) the net charge on the hollow conducting sphere, (c) the charge on the inner surface of the hollow conducting sphere, and (d) the charge on the outer surface of the hollow conducting sphere. • A 65.0-kg bungee jumper steps off a bridge with a light bungee cord tied to her body and to the bridge. The unstretched length of the cord is 11.0 m. The jumper reaches the bottom of her motion 36.0 m below the bridge before bouncing back. We wish to find the time interval between her leaving the bridge and her arriving at the bottom of her motion. Her overall motion can be separated into an 11.0-m free fall and a 25.0-m section of simple harmonic oscillation. (a) For the free-fall part, what is the appropriate analysis model to describe her motion? (b) For what time interval is she in free fall? (c) For the simple harmonic oscillation part of the plunge, is the system of the bungee jumper, the spring, and the Earth isolated or non-isolated? (d) From your response in part (c) find the spring constant of the bungee cord. (e) What is the location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper? (f) What is the angular frequency of the oscillation? (g) What time interval is required for the cord to stretch by 25.0 m? (h) What is the total time interval for the entire 36.0-m drop? • Two vectors are given by →A=−3ˆi+7ˆj−4ˆkA→=−3i^+7j^−4k^ and →B=6ˆi−10ˆj+9ˆk.B→=6i^−10j^+9k^. Evaluate the quantities (a) cos−1[→A⋅→B/AB]cos−1[A→⋅B→/AB] and (b) sin−1[|→A×→B|/AB].sin−1[|A→×B→|/AB]. (c) Which give(s) the angle between the vectors? • A long, straight wire carries a current I.I. A right-angle bend is made in the middle of the wire. The bend forms an arc of a circle of radius rr as shown in Figure P30.11P30.11 . Determine the magnetic field at point P,P, the center of the arc. • Pat builds a track for his model car out of solid wood as shown in Figure PI2.5. The track is 5.00 cmcm wide, 1.00 mm high, and 3.00 mm long. The runway is cut so that it forms a parabola with the equation y=(x−3)2/9.y=(x−3)2/9. Locate the horizontal coordinate of the center of gravity of this track. • An electron is contained in a one-dimensional box of length 0.100 nmnm (a) Draw an energy-level diagram for the electron for levels up to n=4.n=4. (b) Photons are emitted by the electron making downward transitions that could eventually carry it from the n=4n=4 state to the n=1n=1 state. Find the wavelengths of all such photons. • A child rolls a marble on a bent track that is 100$\mathrm{cm}$long as shown in Figure$\mathrm{P} 2.14$. We use$x$to represent the position of the marble along the track. On the horizontal sections from$x=0$to$x=20 \mathrm{cm}$and from$x=40 \mathrm{cm}$to$x=60 \mathrm{cm},$the marble rolls with constant speed. On the sloping sections, the marble’s speed changes steadily. At the places where the slope changes, the marble stays on the track and does not undergo any sudden changes in speed. The child gives the marble some initial speed at$x=0$and$t=0$and then watches it roll to$x=90 \mathrm{cm},$where it turns around, eventually returning to$x=0$with the same speed with which the child released it. Prepare graphs of$x$versus$t, v_{x}$versus$t,$and$a_{x}$versus$t,vertically aligned with their time axes identical, to show the motion of the marble. You will not be able to place numbers other than zero on the horizontal axis or on the velocity or acceleration axes, but show the correct graph shapes. • Two conductors having net charges of +10.0μC+10.0μC and −10.0μC−10.0μC have a potential difference of 10.0 VV between them. (a) Determine the capacitance of the system. (b) What is the potential difference between the two conductors if the charges on each are increased to +100μC+100μC and −100μC−100μC ? • An aluminum rod 1.60 m long is held at its center. It is stroked with a rosin-coated cloth to set up a longitudinal vibration. The speed of sound in a thin rod of aluminum is 5 100 m/s. (a) What is the fundamental frequency of the waves established in the rod? (b) What harmonics are set up in the rod held in this manner? (c) What If? What would be the fundamental frequency if the rod were copper, in which the speed of sound is 3 560 m/s? • Another series of nuclear reactions that can produce energy in the interior of stars is the carbon cycle first proposed by Hans Bethe in 1939, leading to his Nobel Prize in Physics in 1967. This cycle is most efficient when the central temperature in a star is above 1.6×107K1.6×107K . Because the temperature at the center of the Sun is only 1.5×107K1.5×107K , the following cycle produces less than 10%% of the Sun’s energy. (a) A high-energy proton is absorbed by 12C12C . Another nucleus, A,A, is produced in the reaction, along with a gamma ray. Identify nucleus A.A. (b) Nucleus AA decays through positron emission to form nucleus BB . Identify nucleus BB (c) Nucleus BB absorbs a proton to produce nucleus C and a gamma ray. Identify nucleus CC . (d) Nucleus C absorbs a proton to produce nucleus DD and a gamma ray. Identify nucleus D.D. (e) Nucleus DD decays through positron emission to produce nucleus E.E. Identify nucleus E.E. (f) Nucleus EE absorbs a proton to produce nucleus FF plus an alpha particle. Identify nucleus FF . (g) What is the significance of the final nucleus in the last step of the cycle outlined in part (f)? • Why is the following situation impossible? An ideal gas undergoes a process with the following parameters: Q=10.0JQ=10.0J , W=12.0J,W=12.0J, and ΔT=−2.00∘ΔT=−2.00∘C. • Three charged particles are located at the corners of an equilateral triangle as shown in Figure P23.13. Calculate the total electric force on the 7.00−μC7.00−μC charge. • Two capacitors give an equivalent capacitance of 9.00 pFpF when connected in parallel and an equivalent capacitance of 2.00 pFpF when connected in series. What is the capacitance of each capacitor? • In Example 6.5, we investigated the forces a child experiences on a Ferris wheel. Assume the data in that example applies to this problem. What force (magnitude and direction) does the seat exert on a 40.0-kg child when the child is halfway between top and bottom? • Gold is the most ductile of all metals. For example, one gram of gold can be drawn into a wire 2.40 kmkm long. The density of gold is 19.3×103kg/m319.3×103kg/m3 , and its resistivity is 2.44×10−8Ω⋅2.44×10−8Ω⋅m. What is the resistance of such a wire at 20.0∘C20.0∘C ? • A baseball is hit so that it travels straight upward after being struck by the bat. A fan observes that it takes 3.00 s for the ball to reach its maximum height. Find (a) the ball’s initial velocity and (b) the height it reaches. • The net nuclear fusion reaction inside the Sun can be written as 41H→4He+E.41H→4He+E. The rest energy of each hydrogen atom is 938.78MeV,938.78MeV, and the rest energy of the helium- 4 atom is 3728.4 MeV. Calculate the percentage of the starting mass that is transformed to other forms of energy. • A light-emitting diode (LED) made of the semiconductor GaAsP emits red light (λ=650nm).(λ=650nm). Determine the energy-band gap EgEg for this semiconductor. • A plano-convex lens having a radius of curvature of r=r= 4.00 mm is placed on a concave glass surface whose radius of curvature is R=12.0mR=12.0m as shown in Figure P37.72P37.72 . Assuming 500 -nm light is incident normal to the flat surface of the lens, determine the radius of the 100 th bright ring. • A 10.0−V10.0−V battery, a 5.00−Ω5.00−Ω resistor, and a 10.0−H10.0−H inductor are connected in series. After the current in the circuit has reached its maximum value, calculate (a) the power being supplied by the battery, (b) the power being delivered to the resistor, (c) the power being delivered to the inductor, and (d) the energy stored in the magnetic field of the inductor. • Consider a car moving at highway speed u.u. Is its actual kinetic energy larger or smaller than 12mu2?12mu2? Make an order- of-magnitude estimate of the amount by which its actual kinetic energy differs from 12mu2.12mu2. In your solution, state the quantities you take as data and the values you measure or estimate for them. You may find Appendix B. 5 useful. • A helium–neon laser emits light that has a wavelength of 632.8 nm. The circular aperture through which the beam emerges has a diameter of 0.500 cm. Estimate the diameter of the beam 10.0 km from the laser. • Light from an argon laser strikes a diffraction grating that has 5 310 grooves per centimeter. The central and first-order principal maxima are separated by 0.488 m on a wall 1.72 m from the grating. Determine the wavelength of the laser light. • A photon produces a proton-antiproton pair according to the reaction γ→p+p¯¯¯γ→p+p¯ . (a) What is the minimum possible frequency of the photon? (b) What is its wavelength? • Let →B=5.00m at 60.0∘. Let the vector →C have the same magnitude as →A and a direction angle greater than that of →A by 25.0∘. Let →A⋅→B=30.0m2 and →B⋅→C=35.0m2. Find the magnitude and direction of →A. • A more general definition of the temperature coefficient of resistivity is α=1ρdρdTα=1ρdρdT where ρρ is the resistivity at temperature T.T. (a) Assuming αα is constant, show that ρ=ρ0eα(T−T0)ρ=ρ0eα(T−T0) where ρ0ρ0 is the resistivity at temperature T0.T0. (b) Using the series expansion ex≈1+xex≈1+x for x<<1x<<1 , show that the resistivity is given approximately by the expression ρ=ρ0[1+α(T−T0)]forα(T−T0)<<1ρ=ρ0[1+α(T−T0)]forα(T−T0)<<1 • Two 1.50−V1.50−V batteries – with their positive terminals in the same direction-are inserted in series into a flashlight. One battery has an internal resistance of 0.255Ω,0.255Ω, and the other has an internal resistance of 0.153 . When the switch is closed, the bulb carries a current of 600 mAmA . • This problem complements Problem 84 in Chapter 10. In the operation of a single cylinder internal combustion piston engine, one charge of fuel explodes to drive the piston outward in the power stroke. Part of its energy output is stored in a turning flywheel. This energy is then used to push the piston inward to compress the next charge of fuel and air. In this compression process, assume an original volume of 0.120 L of a diatomic ideal gas at atmospheric pressure is compressed adiabatically to one-eighth of its original volume. (a) Find the work input required to compress the gas. (b) Assume the flywheel is a solid disk of mass 5.10 kg and radius 8.50 cm, turning freely without friction between the power stroke and the compression stroke. How fast must the flywheel turn immediately after the power stroke? This situation represents the minimum angular speed at which the engine can operate without stalling. (c) When the engine’s operation is well above the point of stalling, assume the flywheel puts 5.00% of its maximum energy into compressing the next charge of fuel and air. Find its maximum angular speed in this case. • The dissociation energy of ground-state molecular hydrogen is 4.48 eV, but it only takes 3.96 eV to dissociate it when it starts in the first excited vibrational state with J=0.J=0. Using this information, determine the depth of the H2H2 molecular potential-energy function. • An AC source has an output rms voltage of 78.0 VV at a frequency of 80.0 HzHz . If the source is connected across a 25.0−mH25.0−mH inductor, what are (a) the inductive reactance of the circuit, (b) the rms current in the circuit, and (c) the maximum current in the circuit? • Why is the following situation impossible? An apparatus is designed so that steam initially at T=150∘C,P=1.00T=150∘C,P=1.00 atm, and V=0.500m3V=0.500m3 in a piston-cylinder apparatus undergoes a process in which (1) the volume remains constant and the pressure drops to 0.870 atm, followed by ( 2 ) an expansion in which the pressure remains constant and the volume increases to 1.00m3,1.00m3, followed by (3)(3) a return to the initial conditions. It is important that the pressure of the gas never fall below 0.850 atm so that the piston will support a delicate and very expensive part of the apparatus. Without such support, the delicate apparatus can be severely damaged and rendered useless. When the design is turned into a working prototype, it operates perfectly. • A heat engine takes in 360 JJ of energy from a hot reservoir and performs 25.0 JJ of work in each cycle. Find (a) the efficiency of the engine and (b)(b) the energy expelled to the cold reservoir in each cycle. • A flint glass plate rests on the bottom of an aquarium tank. The plate is 8.00\mathrm{cm}$thick (vertical dimension) and is covered with a layer of water 12.0$\mathrm{cm}$deep. Calculate the apparent thickness of the plate as viewed from straight above the water. • Figure P31.38 is a graph of the induced emf versus time for a coil of N turns rotating with angular speed ω in a uniform magnetic field directed perpendicular to the coil’s axis of rotation. What If? Copy this sketch (on a larger scale) and on the same set of axes show the graph of emf versus t (a) if the number of turns in the coil is doubled, (b) if instead the angular speed is doubled, and (c) if the angular speed is doubled while the number of turns in the coil is halved. • What value of nini is associated with the 94.96 -nm spectral line in the Lyman series of hydrogen? (b) What If? Could this wavelength be associated with the Paschen series? (c) Could this wavelength be associated with the Balmer series? • A helium-neon laser can produce a green laser beam instead of a red one. Figure P42.54P42.54 shows the transitions involved to form the red beam and the green beam. After a population inversion is established, neon atoms make a variety of downward transitions in falling from the state labeled E4∗E4∗ down eventually to level E1E1 (arbitrarily assigned the energy E1=0E1=0 ). The atoms emit both red light with a wavelength of 632.8 nmnm in a transition E∗4−E3E∗4−E3 and green light with a wavelength of 543 nmnm in a competing transition E∗4−E2.E∗4−E2. (a) What is the energy E22E22 . Assume the atoms are in a cavity between mirrors designed to reflect the green light with high efficiency but to allow the red light to leave the cavity immediately. Then stimulated emission can lead to the buildup of a collimated beam of green light between the mirrors having a greater intensity than that of the red light. To constitute the radiated laser beam, a small fraction of the green light is permitted to escape by transmission through one mirror. The mirrors forming the resonant cavity can be made of layers of silicon dioxide (index of refraction n=1.458n=1.458 ) and titanium dioxide (index of refraction varies between 1.9 and 2.6).2.6). (b) How thick a layer of silicon dioxide, between layers of titanium dioxide, would minimize reflection of the red light? (c) What should be the thickness of a similar but separate layer of silicon dioxide to maximize reflection of the green light? • A sound wave propagates in air at 27∘C27∘C with frequency 4.00 kHzkHz . It passes through a region where the temperature gradually changes and then moves through air at 0∘C0∘ Give numerical answers to the following questions to the extent possible and state your reasoning about what happens to the wave physically. (a) What happens to the speed of the wave? (b) What happens to its frequency? (c) What happens to its wavelength? • An object 2.00$\mathrm{cm}$high is placed 40.0$\mathrm{cm}$to the left of a converging lens having a focal length of$30.0 \mathrm{cm} .$A diverging lens with a focal length of$-20.0 \mathrm{cm}$is placed 110$\mathrm{cm}$to the right of the converging lens. Determine (a) the position and$(b)$the magnification of the final image. (c) Is the image upright or inverted? (d) What If? Repeat parts (a) through (c) for the case in which the second lens is a converging lens having a focal length of$20.0 \mathrm{cm} .$• A 150 -glider moves at v1=2.30m/sv1=2.30m/s on an air track toward an originally stationary 200-g glider as shown in Figure P17.51. The gliders undergo a completely inelastic collision and latch together over a time interval of 7.00 ms. A student suggests roughly half the decrease in mechanical energy of the two-glider system is transferred to the environment by sound. Is this suggestion reasonable? To evaluate the idea, find the implied sound level at a position 0.800 m from the gliders. If the student’s idea is unreasonable, suggest a better idea. • For the circuit shown in Figure P28.32P28.32 , we wish to find the currents I1,I2,I1,I2, and I3I3 . Use Kirchhoff’s rules to obtain equations for (a)(a) the upper loop, (b) the lower loop, and (c) the junction on the left side. In each case, suppress units for clarity and simplify, combining the terms. (d) Solve the junction equation for I3I3 . (e) Using the equation found in part (d), eliminate I3I3 from the equation found in part (b). (f). (f). (f) Solve the equations found in parts (a) and (e) simultaneously for the two unknowns I1I1 and I2I2 . (g) Substitute the answers found in part (f) into the junction equation found in part (d), solving for I3.I3. (h) What is the significance of the negative answer for I2?I2? • A student taking a quiz finds on a reference sheet the two equations f=1Tandv=Tμ−−√f=1Tandv=Tμ She has forgotten what TT represents in each equation. (a) Use dimensional analysis to determine the units required for TT in each equation. (b) Explain how you can identify the physical quantity each TT represents from the units. • The rms speed of an oxygen molecule (O2)(O2) in a container of oxygen gas is 625 m/sm/s . What is the temperature of the gas? • A well-insulated electric water heater warms 109 kgkg of water from 20.0∘C20.0∘C to 49.0∘C49.0∘C in 25.0 minmin . Find the resistance of its heating element, which is connected across a 240−V240−V potential difference. • What mass of water at 25.0∘0∘C must be allowed to come to thermal equilibrium with a 1.85 -kg cube of aluminum initially at 150∘C150∘C to lower the temperature of the aluminum to 65.0∘C65.0∘C ? Assume any water turned to steam subsequently condenses. • Figure P44.8P44.8 shows the potential energy for two protons as a function of separation distance. In the text, it was claimed that, to be visible on such a graph, the peak in the curve is exaggerated by a factor of ten. (a) Find the electric potential energy of a pair of protons separated by 4.00fm4.00fm. (b) Verify that the peak in Figure P44.8P44.8 is exaggerated by a factor of ten. • An AM radio station broadcasts isotropically (equally in all directions) with an average power of 4.00 kW. A receiving antenna 65.0 cm long is at a location 4.00 mi from the transmitter. Compute the amplitude of the emf that is induced by this signal between the ends of the receiving antenna. • What is the approximate size of the smallest object on the Earth that astronauts can resolve by eye when they are orbiting 250 km above the Earth? Assume λ=500nmλ=500nm and a pupil diameter of 5.00 mm. • Consider the double-slit arrangement shown in Figure P37.56, where the slit separation is dd and the distance from the slit to the screen is L.L. A sheet of transparent plastic having an index of refraction nn and thickness tt is placed over the upper slit. As a result, the central maximum of the interference pattern moves upward a distance y′.y′. Find y′.y′. • Monochromatic coherent light of amplitude E0E0 and angular frequency ωω passes through three parallel slits, each separated by a distance dd from its neighbor. (a) Show that the time-averaged intensity as a function of the angle θθ is I(θ)=Imax[1+2cos(2πdsinθλ)]2I(θ)=Imax[1+2cos(2πdsinθλ)]2 (b) Explain how this expression describes both the primary and the secondary maxima. (c) Determine the ratio of the intensities of the primary and secondary maxima. • A single-turn square loop of wire, 2.00cm2.00cm on each edge, carries a clockwise current of 0.200 A. The loop is inside a solenoid, with the plane of the loop perpendicular to the magnetic field of the solenoid. The solenoid has 30.0 turns/cm and carries a clockwise current of 15.0A15.0A Find (a) the force on each side of the loop and (b) the torque acting on the loop. • A supermarket sells rolls of aluminum foil, plastic wrap, and waxed paper. (a) Describe a capacitor made from such materials. Compute order-of-magnitude estimates for (b) its capacitance and (c) its breakdown voltage. • A 15.0 -m uniform ladder weighing 500 NN rests against a frictionless wall. The ladder makes a 60.0∘0∘ angle with the horizontal. (a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when an 800-N firefighter has climbed 4.00 mm along the ladder from the bottom. (b) If the ladder is just on the verge of slipping when the firefighter is 9.00 mm from the bottom, what is the coefficient of static friction between ladder and ground? • (Table Cant Copy) • The United States possesses the ten largest warships in the world, aircraft carriers of the Nimitz class. Suppose one of the ships bobs up to float 11.0$\mathrm{cm}$higher in the ocean water when 50 fighters take off from it in a time interval of 25$\mathrm{min}$, at a location where the free-fall acceleration is 9.78$\mathrm{m} / \mathrm{s}^{2}$. The planes have an average laden mass of 29000$\mathrm{kg}$. Find the horizontal area enclosed by the waterline of the ship. • A Young’s interference experiment is performed with blue green argon laser light. The separation between the slits is 0.500 mm, and the screen is located 3.30 m from the slits. The first bright fringe is located 3.40 mm from the center of the interference pattern. What is the wavelength of the argon laser light? • Assume the average density of the Universe is equal to the critical density. (a) Prove that the age of the Universe is given by 2/(3H).2/(3H). (b) Calculate 2/(3H)/(3H) and express it in years. • A regular tetrahedron is a pyramid with a triangular base and triangular sides as shown in Figure P28.69. Imagine the six straight lines in Figure P28.69 are each 10.0-V resistors, with junctions at the four vertices. A 12.0-V battery is connected to any two of the vertices. Find (a) the equivalent resistance of the tetrahedron between these vertices and (b) the current in the battery. • Show that the amount of work required to assemble four identical charged particles of magnitude QQ at the corners of a square of side ss is 5.41keQ2/s.5.41keQ2/s. • A satellite of mass m,m, originally on the surface of the Earth, is placed into Earth orbit at an altitude h.h. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite’s speed? (c) What is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet’s daily rotation. Represent the mass and radius of the Earth as MEME and RE,RE, respectively. • A ball is thrown directly downward with an initial speed of 8.00$\mathrm{m} / \mathrm{s}$from a height of$30.0 \mathrm{m} .$After what time interval does it strike the ground? • The free-fall acceleration on the surface of the Moon is about one-sixth that on the surface of the Earth. The radius of the Moon is about 0.250RE(RE=Earth’s radius=RE(RE=Earth’s radius= 6.37×106m6.37×106m ). Find the ratio of their average densities, ρMoon/ρEarthρMoon/ρEarth • A flow calorimeter is an apparatus used to measure the specific heat of a liquid. The technique of flow calorimetry involves measuring the temperature difference between the input and output points of a flowing stream of the liquid while energy is added by heat at a known rate. A liquid of density 900 kg/m3kg/m3 flows through the calorimeter with volume flow rate of 2.00 L/minL/min . At steady state, a temperature difference 3.50∘50∘C is established between the input and output points when energy is supplied at the rate of 200 WW .What is the specific heat of the liquid? • A charge of 170μCμC is at the center of a cube of edge 80.0 cm. No other charges are nearby. (a) Find the flux through each face of the cube. (b) Find the flux through the whole surface of the cube. (c) What If? Would your answers to either part (a) or part (b) change if the charge were not at the center? Explain. • An electron moves through a uniform electric field →E=(2.50ˆi+5.00ˆj)V/mE→=(2.50i^+5.00j^)V/m and a uniform magnetic field →B=0.400ˆkT.B→=0.400k^T. Determine the acceleration of the electron when it has a velocity →v=10.0ˆim/sv→=10.0i^m/s • An unpolarized beam of light is incident on a stack of ideal polarizing filters. The axis of the first filter is perpendicular to the axis of the last filter in the stack. Find the fraction by which the transmitted beam’s intensity is reduced in the three following cases. (a) Three filters are in the stack, each with its transmission axis at 45.0° relative to the preceding filter. (b) Four filters are in the stack, each with its transmission axis at 30.0° relative to the preceding filter. (c) Seven filters are in the stack, each with its transmission axis at 15.0° relative to the preceding filter. (d) Comment on comparing the answers to parts (a), (b), and (c). • A clock with a brass pendulum has a period of 1.000 s at 20.0∘C20.0∘C . If the temperature increases to 30.0∘C,30.0∘C, (a) by how much does the period change and (b)(b) how much time does the clock gain or lose in one week? • As shown in Figure P10.71, two blocks are connected by a string of negligible mass passing over a pulley of radius r=0.250mr=0.250m and moment of inertia I.I. The block on the frictionless incline is moving with a constant acceleration of magnitude a=2.00m/s2.a=2.00m/s2. From this information, we wish to find the moment of inertia of the pulley. (a) What analysis model is appropriate for the blocks? (b) What analysis model is appropriate for the pulley? (c) From the analysis model in part (a), find the tension T1⋅(d)T1⋅(d) Similarly, find the tension T2.T2. (e) From the analysis model in part (b), find a symbolic expression for the moment of inertia of the pulley in terms of the tensions T1T1 and T2,T2, the pulley radius r,r, and the acceleration a.(f)a.(f) Find the numerical value of the moment of inertia of the pulley. • The temperature at the surface of the Sun is approximately 5800 KK , and the temperature at the surface of the Earth is approximately 290 KK . What entropy change of the Universe occurs when 1.00×103J1.00×103J of energy is transferred by radiation from the Sun to the Earth? • Consider a horizontal interface between air above and glass of index of refraction 1.55 below. (a) Draw a light ray incident from the air at angle of incidence 30.0∘.30.0∘. Determine the angles of the reflected and refracted rays and show them on the diagram. (b) What If? Now suppose the light ray is incident from the glass at an angle of 30.0∘.30.0∘. Determine the angles of the reflected and refracted rays and show all three rays on a new diagram. (c) For rays incident from the air onto the air-glass surface, determine and tabulate the angles of reflection and refraction for all the angles of inci- dence at 10.0∘0∘ intervals from 0∘0∘ to 90.0∘.90.0∘. (d) Do the same for light rays coming up to the interface through the glass. • A parallel-plate capacitor is constructed using a dielectric material whose dielectric constant is 3.00 and whose dielectric strength is 2.00×108V/m2.00×108V/m . The desired capacitance is 0.250μFμF , and the capacitor must withstand a maximum potential difference of 4.00 kVkV . Find the minimum area of the capacitor plates. • The Acela is an electric train on the Washington–New York-Boston run, carrying passengers at 170$\mathrm{mi} / \mathrm{h}$. A velocity-time graph for the Acela is shown in Figure P2.57. (a) Describe the train’s motion in each successive time interval. (b) Find the train’s peak positive acceleration in the motion graphed. (c) Find the train’s displacement in miles between$t=0$and$t=200 \mathrm{s}$. • Hooke’s law describes a certain light spring of unstretched length 35.0cm. When one end is attached to the top of a doorframe and a 7.50 -kg object is hung from the other end, the length of the spring is 41.5 cm . (a) Find its spring constant. (b) The load and the spring are taken down. Two people pull in opposite directions on the ends of the spring, each with a force of 190 N . Find the length of the spring in this situation. • A tuning fork vibrating at 512 Hz falls from rest and accelerates at 9.80 m/s2m/s2 . How far below the point of release is the tuning fork when waves of frequency 485 HzHz reach the release point? • A diffraction grating has 4 200 rulings/cm. On a screen 2.00 mm from the grating, it is found that for a particular order m, the maxima corresponding to two closely spaced wavelengths of sodium (589.0 nm and 589.6 nm) are separated by 1.54 mm. Determine the value of m.m. • High-frequency sound can be used to produce standing-wave vibrations in a wine glass. A standing-wave vibration in a wine glass is observed to have four nodes and four antinodes equally spaced around the 20.0-cm circumference of the rim of the glass. If transverse waves move around the glass at 900 m/s, an opera singer would have to produce a high harmonic with what frequency to shatter the glass with a resonant vibration as shown in Figure P 18.32? • Consider a closed triangular box resting within a horizontal electric field of magnitude E=7.80×104N/CE=7.80×104N/C as shown in Figure P 24.4. Calculate the electric flux through ( a the vertical rectangular surface, (b) the slanted surface, and (c) the entire surface of the box. • Two coils, held in fixed positions, have a mutual inductance of 100μHμH . What is the peak emf in one coil when the current in the other coil is I(t)=10.0sin(1.00×103t),I(t)=10.0sin⁡(1.00×103t), where II is in amperes and tt is in seconds? • A particle of mass m1m1 is fired at a stationary particle of mass m2,m2, and a reaction takes place in which new particles are created out of the incident kinetic energy. Taken together, the product particles have total mass m3.m3. The minimum kinetic energy the bombarding particle must have so as to induce the reaction is called the threshold energy. At this energy, the kinetic energy of the products is a minimum, so the fraction of the incident kinetic energy that is available to create new particles is a maximum. This condition is met when all the product particles have the same velocity and the particles have no kinetic energy of motion relative to one another. (a) By using conservation of relativistic energy and momentum and the relativistic energy-momentum relation, show that the threshold kinetic energy is Kmin=[m23−(m1+m2)2]c22m2Kmin=[m32−(m1+m2)2]c22m2 Calculate the threshold kinetic energy for each of the following reactions: (b) p+p→p+p+p+p¯¯¯p+p→p+p+p+p¯ (one of the initial protons is at rest, and antiprotons are produced); (c) π−+p→K0+Λ0π−+p→K0+Λ0 (the proton is at rest, and strange particles are produced); (d) p+p+p+p+π0p+p+p+p+π0 (one of the initial protons is at rest, and pions are produced; and (e) p+p¯¯¯→Z0p+p¯→Z0 (one of the initial particles is at rest, and Z0Z0 particles of mass 91.2 GeV/c2GeV/c2 are produced). • A converging lens has a focal length of 10.0 cm. Construct accurate ray diagrams for object distances of (i) 20.0 cm and (ii) 5.00 cm. (a) From your ray diagrams, determine the location of each image. (b) Is the image real or virtual? (c) Is the image upright or inverted? (d) What is the magnification of the image? (e) Compare your results with the values found algebraically. (f) Comment on difficulties in constructing the graph that could lead to differences between the graphical and algebraic answers. • In a student experiment, a constant-volume gas thermometer is calibrated in dry ice (−78.5∘C)(−78.5∘C) and in boiling ethyl alcohol (78.0∘C).(78.0∘C). The separate pressures are 0.900 atmatm and 1.635 atmatm . (a) What value of absolute zero in degrees Celsius does the calibration yield? What pressures would be found at (b) the freezing and (c) the boiling points of water? Hint: Use the linear relationship P=P= A+BT,A+BT, where AA and BB are constants. • Find the nuclear radii of (a) 21H,21H, (b) 6027Co,(c)19779Au,6027Co,(c)19779Au, and (d) 29994Pu.29994Pu. • A star ending its life with a mass of four to eight times the Sun’s mass is expected to collapse and then undergo a a supernova event. In the remnant that is not carried away by the supernova explosion, protons and electrons combine to form a neutron star with approximately twice the mass of the Sun. Such a star can be thought of as a gigantic atomic nucleus. Assume r=aA1/3(Eq.44.1).r=aA1/3(Eq.44.1). If a star of mass 3.98×1030kg3.98×1030kg is composed entirely of neutrons (mn=(mn= 67×10−27kg),1.67×10−27kg), what would its radius be? • The magnetic flux through a metal ring varies with time t according to ΦB=at3−bt2, where ΦB is in webers, a=6.00s−3,b=18.0s−2, and t is in seconds. The resistance of the ring is 3.00Ω . For the interval from t=0 to t=2.00s determine the maximum current induced in the ring. • A velocity-time graph for an object moving along the$x$axis is shown in Figure$\mathrm{P} 2.13 .$(a) Plot a graph of the acceleration versus time. Determine the average acceleration of the object$(b)$in the time interval$t=5.00 \mathrm{s}$to$t=15.0 \mathrm{s}$and (c) in the time interval$t=0$to$t=20.0 \mathrm{s}$• Two automobiles of equal mass approach an intersection. One vehicle is traveling with speed 13.0 m/sm/s toward the east, and the other is traveling north with speed v2i⋅v2i⋅ Neither driver sees the other. The vehicles collide in the intersection and stick together, leaving parallel skid marks at an angle of 55.0∘0∘ north of east. The speed limit for both roads is 35 mi/hmi/h , and the driver of the northward-moving vehicle claims he was within the speed limit when the collision occurred. Is he telling the truth? Explain your reasoning. • An emf of 24.0mV24.0mV is induced in a 500 -turn coil when the current is changing at the rate of 10.0A/s10.0A/s. What is the magnetic flux through each turn of the coil at an instant when the current is 4.00 A? • The following reactions are observed: 94Be+n→104Be+γQ=6.812MeV 94Be+γ→8Be+n4Q=−1.665MeV Calculate the masses of 8Be and 10 Be in unified mass units to four decimal places from these data. • The vector position of a particle varies in time according to the expression →r=3.00ˆi−6.00t2ˆj, where →r is in meters and t is in seconds. (a) Find an expression for the velocity of the particle as a function of time. (b) Determine the acceleration of the particle as a function of time. (c) Calculate the particle’s position and velocity at t=1.00 s. • Find the magnitude of the electric force between a Na+Na+ ion and a Cl−Cl− ion separated by 0.50nm.0.50nm. (b) Would the answer change if the sodium ion were replaced by Li ++ and the chloride ion by Br−−? Explain. • The “Vomit Comet.” In microgravity astronaut training and equipment testing, NASA flies a KC135A aircraft along a parabolic flight path. As shown in Figure P4.47,P4.47, the aircraft climbs from 24000ft24000ft to 31000ft,31000ft, where it enters a parabolic path with a velocity of 143m/s143m/s nose high at 45.0∘0∘ and exits with velocity 143m/s143m/s at 45.0∘45.0∘ nose low. During this portion of the flight, the aircraft and objects inside its padded cabin are in free fall; astronauts and equipment float freely as if there were no gravity. What are the aircraft’s (a) speed and (b) altitude at the top of the maneuver? (c) What is the time interval spent in microgravity? • An ideal refrigerator or ideal heat pump is equivalent to a Carnot engine running in reverse. That is, energy |Qd||Qd| is taken in from a cold reservoir and energy |Qh||Qh| is rejected to a hot reservoir. (a) Show that the work that must be sup- plied to run the refrigerator or heat pump is W=Th−TcTc|Qc|W=Th−TcTc|Qc| (b) Show that the coefficient of performance (COP) of the ideal refrigerator is COP=TcTh−TcCOP=TcTh−Tc • An object is at$x=0$at$t=0$and moves along the$x$axis according to the velocity-time graph in Figure$\mathrm{P} 2.50 .$(a) What is the object’s acceleration between 0 and 4.0$\mathrm{s}$? (b) What is the object’s acceleration between 4.0$\mathrm{s}$and 9.0$\mathrm{s}$? (c) What is the object’s acceleration between 13.0$\mathrm{s}$and 18.0$\mathrm{s}$? (d) At what time(s) is the object moving with the lowest speed? (e) At what time is the object farthest from$x=0$? (f) What is the final position$x$of the object at$t=18.0 \mathrm{s}$? (g) Through what total distance has the object moved between$t=0$and$t=18.0 \mathrm{s} ?$• A sample of an ideal gas expands isothermally, doubling in volume. (a) Show that the work done on the gas in expanding is W=−nRTW=−nRT in 2.2. (b) Because the internal energy EintEint of an ideal gas depends solely on its temperature, the change in internal energy is zero during the expansion. It follows from the first law that the energy input to the gas by heat during the expansion is equal to the energy output by work. Does this process have 100%% efficiency in converting energy input by heat into work output? (C) Does this conversion violate the second law? Explain. • A 3.00 -g lead bullet at 30.0∘0∘C is fired at a speed of 240 m/sm/s into a large block of ice at 0∘C0∘C , in which it becomes embedded. What quantity of ice melts? • Consider a light ray traveling between air and a diamond cut in the shape shown in Figure P35.40. (a) Find the critical angle for total internal reflection for light in the diamond incident on the interface between the diamond and the outside air. (b) Consider the light ray incident normally on the top surface of the diamond as shown in Figure P35.40. Show that the light traveling toward point P in the diamond is totally reflected. What If? Suppose the diamond is immersed in water. (c) What is the critical angle at the diamond–water interface? (d) When the diamond is immersed in water, does the light ray entering the top surface in Figure P35.40 undergo total internal reflection at P? Explain. (e) If the light ray entering the diamond remains vertical as shown in Figure P35.40, which way should the diamond in the water be rotated about an axis perpendicular to the page through OO so that light will exit the diamond at P?P? (f) At what angle of rotation in part (e) will light first exit the diamond at point P?P? • How much work is done on the steam when 1.00 mol of water at 100∘C100∘C boils and becomes 1.00 molmol of steam at 100∘C100∘C at 1.00 atm pressure? Assume the steam to behave as an ideal gas. (b) Determine the change in internal energy of the system of the water and steam as the water vaporizes. • An electric drill with a steel drill bit of mass m 527.0 g and diameter 0.635 cm is used to drill into a cubical steel block of mass M 5 240 g. Assume steel has the same properties as iron. The cutting process can be modeled as happening at one point on the circumference of the bit. This point moves in a helix at constant tangential speed 40.0 m/s and exerts a force of constant magnitude 3.20 N on the block. As shown in Figure P20.10, a groove in the bit carries the chips up to the top of the block, where they form a pile around the hole. The drill is turned on and drills into the block for a time interval of 15.0 s. Let’s assume this time interval is long enough for conduction within the steel to bring it all to a uniform temperature. Furthermore, assume the steel objects lose a negligible amount of energy by conduction, convection, and radiation into their environment. (a) Suppose the drill bit cuts three-quarters of the way through the block during 15.0 s. Find the temperature change of the whole quantity of steel. (b) What If? Now suppose the drill bit is dull and cuts only one-eighth of the way through the block in 15.0 s. Identify the temperature change of the whole quantity of steel in this case. (c) What pieces of data, if any, are unnecessary for the solution? Explain. • A 2.00 -m length of wire is held in an east-west direction and moves horizontally to the north with a speed of 0.500 m/s . The Earth’s magnetic field in this region is of magnitude 50.0μT and is directed northward and 53.0∘ below the horizontal. (a) Calculate the magnitude of the induced emf between the ends of the wire and (b) determine which end is positive. • Supernova Shelton 1987 AA , located approximately 170000 ly from the Earth, is estimated to have emitted a burst of neutrinos carrying energy ∼1046J∼1046J (Fig. P46.53). Suppose the average neutrino energy was 6 MeVMeV and your mother’s body presented cross-sectional area 5000 cm2cm2 . To an order of magnitude, how many of these neutrinos passed through her? • Construct a diagram like that of Figure 44.19 for the cases when I equals (a)52 and (b)4. • An opaque cylindrical tank with an open top has a diameter of 3.00 m and is completely filled with water. When the afternoon Sun reaches an angle of 28.0∘0∘ above the horizon, sunlight ceases to illuminate any part of the bottom of the tank. How deep is the tank? • Scientific work is currently underway to determine whether weak oscillating magnetic fields can affect human health. For example, one study found that drivers of trains had a higher incidence of blood cancer than other railway workers, possibly due to long exposure to mechanical devices in the train engine cab. Consider a magnetic field of magnitude 1.00×10−3T1.00×10−3T , oscillating sinusoidally at 60.0 HzHz . If the diameter of a red blood cell is 8.00μm,8.00μm, determine the maximum emf that can be generated around the perimeter of a cell in this field. • A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are 4.00 m to the north, 2.00 m northeast, and 1.00 m at 30.0° west of south (Fig. P 3.27). Starting at the same initial point, an expert golfer could make the hole in what single displacement? • For what value of vv does γ=1.0100γ=1.0100 ? Observe that for speeds lower than this value, time dilation and length contraction are effects amounting to less than 1%.1%. • X-rays having an energy of 300 keV undergo Compton scattering from a target. The scattered rays are detected at 37.0∘0∘ relative to the incident rays. Find (a) the Compton shift at this angle, (b) the energy of the scattered x-ray, and (c) the energy of the recoiling electron. • A basketball player is standing on the floor 10.0 mm from the basket as in Figure P4.50.P4.50. The height of the basket is 3.05m,3.05m, and he shoots the ball at a 40.0∘0∘ angle with the horizontal from a height of 2.00 mm . (a) What is the acceleration of the basketball at the highest point in its trajectory? (b) At what speed must the player throw the basketball so that the ball goes through the hoop without striking the backboard? • An 80.0−Ω resistor and a 200−mH inductor are connected in parallel across a 100−V (rms), 60.0−Hz source. (a) What is the rms current in the resistor? (b) By what angle does the total current lead or lag behind the voltage? • Light of wavelength 700 nmnm is incident on the face of a fused quartz prism (n=1.458at700nm)(n=1.458at700nm) at an incidence angle of 75.0∘.75.0∘. The apex angle of the prism is 60.0∘.60.0∘. Calculate the angle (a) of refraction at the first surface, (b) of incidence at the second surface, (c) of refraction at the second surface, and (d) between the incident and emerging rays. • One description of the potential energy of a diatomic molecule is given by the Lennard–Jones potential, U=Ar12−Br6U=Ar12−Br6 where AA and BB are constants and rr is the separation distance between the atoms. For the H2H2 molecule, take A=0.124×10−120eV⋅m12A=0.124×10−120eV⋅m12 and B=1.488×10−60eV⋅B=1.488×10−60eV⋅m6. Find (a) the separation distance r0r0 at which the energy of the molecule is a minimum and (b) the energy EE required to break up the H2H2 molecule. • Suppose a duck lives in a universe in which h=2πJ⋅sh=2πJ⋅s The duck has a mass of 2.00 kgkg and is initially known to be within a pond 1.00 mm wide. (a) What is the minimum uncertainty in the component of the duck’s velocity parallel to the pond’s width? (b) Assuming this uncertainty in speed prevails for 5.00 s, determine the uncertainty in the duck’s position after this time interval. • Two particles each with charge +2.00μC+2.00μC are located on the xx axis. One is at x=1.00m,x=1.00m, and the other is at x=−1.00mx=−1.00m (a) Determine the electric potential on the yy axis at y=0.500m.y=0.500m. (b) Calculate the change in electric potential energy of the system as a third charged particle of −3.00μC−3.00μC is brought from infinitely far away to a position on the yy axis at y=0.500m.y=0.500m. • Identify the unknown nuclides and particles X and X′ in the nuclear reactions (a) X+42He→i412Mg+10n,(b)25jU+ 10n→9038Sr+X+2(10n), and (c)2(11H)→21H+X+X′ • The unit of magnetic flux is named for Wilhelm Weber. A practical-size unit of magnetic field is named for Johann Karl Friedrich Gauss. Along with their individual accomplishments, Weber and Gauss built a telegraph in 1833 that consisted of a battery and switch, at one end of a transmission line 3 km long, operating an electromagnet at the other end. Suppose their transmission line was as diagrammed in Figure P30.27. Two long, parallel wires, each having a mass per unit length of 40.0 g/m, are supported in a horizontal plane by strings ℓ=6.00cmℓ=6.00cm long. When both wires carry the same current, the wires repel each other so that the angle between the supporting strings is θ=16.0∘.θ=16.0∘. (a) Are the currents in the same direction or in opposite directions? (b) Find the magnitude of the current. (C) If this transmission line were taken to Mars, would the current required to separate the wires by the same angle be larger or smaller than that required on the Earth? Why? • Consider a radioactive sample. Determine the ratio of the number of nuclei decaying during the first half of its half-life to the number of nuclei decaying during the second half of its half-life. • Assuming the antenna of a 10.0-kW radio station radiates spherical electromagnetic waves, (a) compute the maximum value of the magnetic field 5.00 km from the antenna and (b) state how this value compares with the surface magnetic field of the Earth. • Calculate the work that must be done on charges brought from infinity to charge a spherical shell of radius R=R= 0.100 mm to a total charge Q=125μCQ=125μC . • More than 2 300 years ago, the Greek teacher Aristotle wrote the first book called Physics. Put into more precise terminology, this passage is from the end of its Section Eta: Let PP be the power of an agent causing motion; w,w, the load moved; dd , the distance covered; and Δt,Δt, the time interval required. Then (1)(1) a power equal to PP will in an interval of time equal to ΔtΔt move w/2w/2 a distance 2d;2d; or (2)(2) it will move w/2w/2 the given distance dd in the time interval Δt/2.Δt/2. Also, if (3)(3) the given power PP moves the given load ww a distance d/2d/2 in time interval Δt/2,Δt/2, then (4)P/2(4)P/2 will move w/2w/2 the given distance dd in the given time interval ΔtΔt (a) Show that Aristotle’s proportions are included in the equation PΔt=bwd,PΔt=bwd, where bb is a proportionality constant. (b) Show that our theory of motion includes this part of Aristotle’s theory as one special case. In particular, describe a situation in which it is true, derive the equation representing Aristotle’s proportions, and determine the proportionality constant. • Capacitors C1=6.00μFC1=6.00μF and C2=2.00μFC2=2.00μF are charged as a parallel combination across a 250−V250−V battery. The capacitors are disconnected from the battery and from each other. They are then connected positive plate to negative plate and negative plate to positive plate. Calculate the resulting charge on each capacitor. • An automobile tire is inflated with air originally at 10.0∘0∘C and normal atmospheric pressure. During the process, the air is compressed to 28.0%% of its original volume and the temperature is increased to 40.0∘C.40.0∘C. (a) What is the tire pressure? (b) After the car is driven at high speed, the tire’s air temperature rises to 85.0∘C85.0∘C and the tire’s interior volume increases by 2.00%% . What is the new tire pressure (absolute)? • A 1200 -kg car traveling initially at vCi=25.0m/svCi=25.0m/s in an easterly direction crashes into the back of a 9000−kg9000−kg truck moving in the same direction at vTi=20.0m/svTi=20.0m/s (Fig. P9.18). The velocity of the car immediately after the collision is vCf=18.0m/svCf=18.0m/s to the east. ( a) What is the velocity of the truck immediately after the collision? (b) What is the change in mechanical energy of the car-truck system in the collision? (c) Account for this change in mechanical energy. • When a wire carries an AC current with a known frequency, you can use a Rogowskicoil to determine the amplitude Imax of the current without disconnecting the wire to shunt the current through a meter. The Rogowski coil, shown in Figure P31.16 (page 918), simply clips around the wire. It consists of a toroidal conductor wrapped around a circular return cord. Let n represent the number of turns in the toroid per unit distance along it. Let A represent the cross- sectional area of the toroid. Let I(t)=Imax sin ωt represent the current to be measured. (a) Show that the amplitude of the emf induced in the Rogowski coil is Emax=μ0nAωImax (b) Explain why the wire carrying the unknown current need not be at the center of the Rogowski coil and why the coil will not respond to nearby currents that it does not enclose. • A beaker of mass$m_{b}$containing oil of mass$m_{o}$and density$\rho_{o}$rests on a scale. A block of iron of mass$m_{\mathrm{Fe}}$suspended from a spring scale is completely submerged in the oil as shown in Figure$\mathrm{P} 14.65 .$Determine the equilibrium readings of both scales. • Figure$P 36.81$shows a thin converging lens for which the radii of curvature of its surfaces have magnitudes of 9.00$\mathrm{cm}$and$11.0 \mathrm{cm} .$The lens is in front of a concave spherical mirror with the radius of curvature$R=8.00 \mathrm{cm} .$Assume the focal points$F_{1}$and$F_{2}$of the lens are 5.00$\mathrm{cm}$from the center of the lens. (a) Determine the index of refraction of the lens material. The lens and mirror are 20.0$\mathrm{cm}$apart, and an object is placed 8.00$\mathrm{cm}$to the left of the lens. Determine (b) the position of the final image and (c) its magnification as seen by the eye in the figure. (d) Is the final image inverted or upright? Explain. • When gamma rays are incident on matter, the intensity of the gamma rays passing through the material varies with depth xx as I(x)=I0e−μxI(x)=I0e−μx , where I0I0 is the intensity of the radiation at the surface of the material (at x=0)x=0) and μμ is the linear absorption coefficient. (a) Determine the “half-thickness” for a material with linear absorption coefficient μ,μ, that is, the thickness of the material that would absorb half the incident gamma rays. (b) What thickness changes the radiation by a factor of f?f? • Verify by direct substitution that the wave function for a standing wave given in Equation 18.1, y=(2Asinkx)cosωty=(2Asinkx)cosωt is a solution of the general linear wave equation, Equation 16.27: ∂2y∂x2=1v2∂2y∂t2∂2y∂x2=1v2∂2y∂t2 • An infinitely long, cylindrical, insulating shell of inner radius aa and outer radius bb has a uniform volume charge density ρ.ρ. A line of uniform linear charge density λλ is placed along the axis of the shell. Determine the electric field for (a) r<a,(b)a<r<b,r<a,(b)a<r<b, and (c)r>b.(c)r>b. • The spring of the pressure gauge shown in Figure P14.7 has a force constant of 1 250 N/m, and the piston has a diameter of 1.20 cm. As the gauge is lowered into water in a lake, what change in depth causes the piston to move in by 0.750 cm? • A person walks first at a constant speed of 5.00$\mathrm{m} / \mathrm{s}$along a straight line from point$\mathbb{Q}$to point$\mathbb{B}$and then back along the line from$\mathbb{B}$to$@$at a constant speed of 3.00$\mathrm{m} / \mathrm{s}$. (a) What is her average speed over the entire trip? (b) What is her average velocity over the entire trip? • The following charges are located inside a submarine: 5.00μC,−9.00μC,27.0μC,5.00μC,−9.00μC,27.0μC, and −84.0μC−84.0μC . (a) Calculate the net electric flux through the hull of the submarine. (b) Is the number of electric field lines leaving the submarine greater than, equal to, or less than the number entering it? • Occasionally, high-energy muons collide with electrons and produce two neutrinos according to the reaction μ++e−→2νμ++e−→2ν . What kind of neutrinos are they? • A model airplane with mass 0.750 kg is tethered to the ground by a wire so that it flies in a horizontal circle 30.0 m in radius. The airplane engine provides a net thrust of 0.800 N perpendicular to the tethering wire. (a) Find the torque the net thrust produces about the center of the circle. (b) Find the angular acceleration of the airplane. (c) Find the translational acceleration of the airplane tangent to its flight path. • Massive stars ending their lives in supernova explosions produce the nuclei of all the atoms in the bottom half of the periodic table by fusion of smaller nuclei. This problem roughly models that process. A particle of mass m=m= 1.99×10−26kg1.99×10−26kg kg moving with a velocity →u=0.500ciu→=0.500ci collides head-on and sticks to a particle of mass m′=m/3m′=m/3 moving with the velocity →u=−0.500ci.u→=−0.500ci. What is the mass of the resulting particle? • A uniform ladder of length LL and mass m1m1 rests against a frictionless wall. The ladder makes an angle θθ with the horizontal. (a) Find the horizontal and vertical forces the ground exerts on the base of the ladder when a firefighter of mass m2m2 has climbed a distance xx along the ladder from the bottom. (b) If the ladder is just on the verge of slipping when the firefighter is a distance dd along the ladder from the bottom, what is the coefficient of static friction between ladder and ground? • A particle of mass mm moves in a circle of radius RR at a constant speed vv as shown in Figure P 11.17. The motion begins at point QQ at time t=0.t=0. Determine the angular momentum of the particle about the axis perpendicular to the page through point PP as a function of time. • A heat engine operates between two reservoirs at T2=T2= 600 KK and T1=350KT1=350K . It takes in 1.00×103J1.00×103J of energy from the higher-temperature reservoir and performs 250 JJ of work. Find (a) the entropy change of the Universe ΔSUΔSU for this process and (b) the work WW that could have been done by an ideal Carnot engine operating between these two reservoirs. (c) Show that the difference between the amounts of work done in parts (a) and (b) is T1ΔSUT1ΔSU. • Determine (a) the capacitance and (b) the maximum potential difference that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of 1.75 cm2cm2 and a plate separation of 0.0400mm.0.0400mm. • A car traveling on a flat (unbanked), circular track accelerates uniformly from rest with a tangential acceleration of a. The car makes it one quarter of the way around the circle before it skids off the track. From these data, determine the coefficient of static friction between the car and the track. • How much charge is on each plate of a 4.00−μF4.00−μF capacitor when it is connected to a 12.0−V12.0−V battery? (b) If this same capacitor is connected to a 1.50−V1.50−V battery, what charge is stored? • The index of refraction for red light in water is 1.331 and that for blue light is 1.340.1.340. If a ray of white light enters the water at an angle of incidence of 83.0∘,83.0∘, what are the underwater angles of refraction for the (a) red and (b) blue components of the light? • A particle with charge QQ is located on the axis of a circle of radius RR at a distance bb from the plane of the circle (Fig. P 24.64). Show that if one- fourth of the electric flux from the charge passes through the circle, then R=3–√b.R=3b. • The temperature coefficients of resistivity αα in Table 27.2 are based on a reference temperature T0T0 of 20.0∘0∘C . Suppose the coefficients were given the symbol α′α′ and were based on a T0T0 of 0∘C0∘C . What would the coefficient α′α′ for silver be? Note: The coefficient αα satisfies ρ=ρ= ρ0[1+α(T−T0)],ρ0[1+α(T−T0)], where ρ0ρ0 is the resistivity of the material at T0=20.0∘C.T0=20.0∘C. The coefficient α′α′ must satisfy the expression ρ=ρ′0[1+α′T],ρ=ρ′0[1+α′T], where ρ′0ρ′0 is the resistivity of the material at 0∘C.0∘C. • Find the equivalent resistance between points aa and bb in Figure P28.44.P28.44. • Consider a fusion generator built to create 3.00 GW of power. Determine the rate of fuel burning in grams per hour if the D–T reaction is used. (b) Do the same for the D–D reaction, assuming the reaction products are split evenly between (n,3He)(n,3He) and (p,3H)(p,3H) • One mole of an ideal gas does 3000 JJ of work on its surroundings as it expands isothermally to a final pressure of 1.00 atmatm and volume of 25.0 L. Determine (a) the initial volume and (b) the temperature of the gas. • Figure P25.34P25.34 represents a graph of the electric potential in a region of space versus position xx where the electric field is parallel to the xx axis. Draw a graph of the xx component of the electric field versus xx in this region. • When a person stands on tiptoe on one foot (a strenuous position), the position of the foot is as shown in Figure P12.42aP12.42a . The total gravitational force F→εF→ε on the body is supported by the normal force n→n→ exerted by the floor on the toes of one foot. A mechanical model of the situation is shown in Figure P12.42bP12.42b , where T→T→ is the force exerted on the foot by the Achilles tendon and R→R→ is the force exerted on the foot by the tibia. Find the values of T,R,T,R, and θθ when Fg=700N.Fg=700N. • A 1.50 -kg iron horseshoe initially at 600∘C600∘C is dropped into a bucket containing 20.0 kgkg of water at 25.0∘0∘C . What is the final temperature of the water-horseshoe system? Ignore the heat capacity of the container and assume a negligible amount of water boils away. • The 64.0-cm-long string of a guitar has a fundamental frequency of 330 Hz when it vibrates freely along its entire length. A fret is provided for limiting vibration to just the lower two-thirds of the string. (a) If the string is pressed down at this fret and plucked, what is the new fundamental frequency? (b) What If? The guitarist can play a “natural harmonic” by gently touching the string at the location of this fret and plucking the string at about one-sixth of the way along its length from the other end. What frequency will be heard then? • An object is originally at the$x_{i}=0 \mathrm{cm}$position of a meterstick located on the$x$axis. A converging lens of focal length 26.0$\mathrm{cm}$is fixed at the position$32.0 \mathrm{cm} .$Then we gradually slide the object to the position$x_{f}=12.0 \mathrm{cm} .$(a) Find the location$x^{\prime}$of the object’s image as a function of the object position$x$(b) Describe the pattern of the image’s motion with reference to a graph or a table of values. (c) As the object moves 12.0$\mathrm{cm}$to the right, how far does the image move? (d) In what direction or directions? • As we go down the periodic table, which subshell is filled first, the 3dd or the 4 s subshell? (b) Which electronic configuration has a lower energy, [Ar] 3d44s2d44s2 or [Ar] d54s1?d54s1? Note: The notation [Ar] represents the filled configuration for argon. Suggestion: Which has the greater number of unpaired spins? (c) Identify the element with the electronic configuration in part (b). • Calculate the sound level (in decibels) of a sound wave that has an intensity of 4.00μW/m2.4.00μW/m2. • Figure P5.19 shows the horizontal forces acting on a sailboat moving north at constant velocity, seen from at point straight above its mast. At the particular speed of the sailboat, the water exerts a 220 -N drag force on its hull and θ=40.0∘. For each of the situations (a) and (b) described below, write two component equations representing Newton’s second law. Then solve the equations for P (the force exerted by the wind on the sail) and for n (the force exerted by the water on the keel). (a) Choose the x direction as east and the y direction as north. (b) Now choose the x direction as θ= 40.0∘ north of east and the y direction as θ=40.0∘ west of (c) Compare your solutions to parts (a) and (b). Do the results agree? Is one method significantly easier? • A speedboat travels in a straight line and increases in speed uniformly from$v_{i}=20.0 \mathrm{m} / \mathrm{s}$to$v_{f}=30.0 \mathrm{m} / \mathrm{s}$in a displacement$\Delta x$of 200$\mathrm{m}$. We wish to find the time interval required for the boat to move through this displacement. (a) Draw a coordinate system for this situation. (b) What analysis model is most appropriate for describing this situation? (c) From the analysis model, what equation is most appropriate for finding the acceleration of the speedboat? (d) Solve the equation selected in part (c) symbolically for the boat’s acceleration in terms of$v_{i}, v_{f},$and$\Delta x$. (e) Substitute numerical values to obtain the acceleration numerically. (f) Find the time interval mentioned above. • The two wires shown in Figure P30.19P30.19 are separated by d=10.0cmd=10.0cm and carry currents of I=5.00AI=5.00A in opposite directions. Find the magnitude and direction of the net magnetic field (a) at a point midway between the wires;(a) at a point midway between the wires; (b) at point P1,10.0cmP1,10.0cm to the right of the wire on the right; and (c)(c) at point P2,2d=20.0cmP2,2d=20.0cm to the left of the wire on the left. • Consider the circuit in Figure P32.16,P32.16, taking E=6.00V,E=6.00V, L=8.00mH,L=8.00mH, and R=4.00Ω.R=4.00Ω. (a) What is the inductive time constant of the circuit? (b) Calculate the current in the circuit 250μμ s after the switch is closed. (c) What is the value of the final steady-state current? (d) After what time interval does the current reach 80.0%% of its maximum value? • A steel cable 3.00 cm2cm2 in cross-sectional area has a mass of 2.40 kgkg per meter of length. If 500 mm of the cable is hung over a vertical cliff, how much does the cable stretch under its own weight? Take Ysteel=2.00×1011N/m2Ysteel=2.00×1011N/m2 . • Figure P5.29P5.29 shows the speed of a person’s body as he does a chin-up. Assume the motion is vertical and the mass of the person’s body is 64.0kg.64.0kg. Determine the force exerted by the chin-up bar on his body at (a) t=0,t=0, (b) t=0.5st=0.5s (c) t=1.1s,t=1.1s, and (d)t=1.6s(d)t=1.6s • A fellow astronaut passes by you in a spacecraft traveling at a high speed. The astronaut tells you that his craft is 20.0 mm long and that the identical craft you are sitting in is 19.0 mm long. According to your observations, (a) how long is yourcraft, (b) how long is the astronaut’s craft, and (c) what is the speed of the astronaut’s craft relative to your craft? • Niobium metal becomes a superconductor when cooled below 9 KK . Its superconductivity is destroyed when the surface magnetic field exceeds 0.100 TT . In the absence of any external magnetic field, determine the maximum current a 2.00 -mm-diameter niobium wire can carry and remain superconducting. • Under pressure, liquid helium can solidify as each atom bonds with four others, and each bond has an average energy of 1.74×10−23J1.74×10−23J . Find the latent heat of fusion for helium in joules per gram. (The molar mass of He is 4.00 g/mol.) • Consider the lens–mirror arrangement shown in Figure P36.72. There are two final image positions to the left of the lens of focal length$f_{\mathrm{L}}$. One image position is due to light traveling from the object to the left and passing through the lens. The other image position is due to light traveling to the right from the object, reflecting from the mirror of focal length$f_{\mathrm{M}}$and then passing through the lens. For a given object position$p$between the lens and the mirror and measured with respect to the lens, there are two separation distances$d$between the lens and mirror that will cause the two images described above to be at the same location. Find both positions. • A 400−Ω400−Ω resistor, an inductor, and a capacitor are in series with an AC source. The reactance of the inductor is 700Ω,700Ω, and the circuit impedance is 760Ω.760Ω. (a) What are the possible values of the reactance of the capacitor? (b) If you find that the power delivered to the circuit decreases as you raise the frequency, what is the capacitive reactance in the original circuit? (C) Repeat part (a) assuming the resistance is 200ΩΩ instead of 400ΩΩ and the circuit impedance continues to be 760ΩΩ . • On a day that the temperature is 20.0∘C,20.0∘C, a concrete walk is poured in such a way that the ends of the walk are unable to move. Take Young’s modulus for concrete to be 7.00×109N/m27.00×109N/m2 and the compressive strength to be 2.00×109N/m22.00×109N/m2 (a) What is the stress in the cement on a hot day of 50.0∘C50.0∘C ? (b) Does the concrete fracture? • An airtight freezer holds nn moles of air at 25.0∘0∘C and 1.00atm.1.00atm. The air is then cooled to −18.0∘C−18.0∘C . (a) What is the change in entropy of the air if the volume is held constant? (b) What would the entropy change be if the pressure were maintained at 1.00 atm during the cooling? • The nuclei of the O2O2 molecule are separated by a distance 1.20×10−10m.1.20×10−10m. The mass of each oxygen atom in the molecule is 2.66×10−26kg.2.66×10−26kg. (a) Determine the rotational energies of an oxygen molecule in electron volts for the levels corresponding to J=0,1,J=0,1, and 2.2. (b) The effective force constant kk between the atoms in the oxygen molecule is 177N/m.177N/m. Determine the vibrational energies (in electron volts) corresponding to v=0,1,v=0,1, and 2.2. • Calculate the momentum of a photon whose wavelength is 4.00×10−7m.4.00×10−7m. (b) Find the speed of an electron having the same momentum as the photon in part (a). • Determine the magnitude of the electric field at the surface of a lead-208 nucleus, which contains 82 protons and 126 neutrons. Assume the lead nucleus has a volume 208 times that of one proton and consider a proton to be a sphere of radius 1.20×10−15m1.20×10−15m. • When photons pass through matter, the intensity I of the beam (measured in watts per square meter) decreases exponentially according to I=I0e−μx where I is the intensity of the beam that just passed through a thickness x of material and I0 is the intensity of the incident beam. The constant μ is known as the linear absorption coefficient, and its value depends on the absorbing material and the wavelength of the photon beam. This wavelength (or energy) dependence allows us to filter out unwanted wavelengths from a broad-spectrum x-ray beam. (a) Two x-ray beams of wavelengths λ1 and λ2 and equal incident intensities pass through the same metal plate. Show that the ratio of the emergent beam intensities is I2I1=e−(μ2−μ1)x (b) Compute the ratio of intensities emerging from an aluminum plate 1.00 mm thick if the incident beam contains equal intensities of 50 pm and 100 pm x-rays. The values of μ for aluminum at these two wavelengths are μ1=5.40cm−1 at 50 pm and μ2=41.0cm−1 at 100 pm . (c) Repeat part (b) for an aluminum plate 10.0 mm thick. • The switch in Figure P28.78aP28.78a closes when ΔVc>23ΔVΔVc>23ΔV and opens when ΔVc<13ΔV.ΔVc<13ΔV. The ideal voltmeter reads a potential difference as plotted in Figure P28.78bP28.78b . What is the period TT of the waveform in terms of R1,R2,R1,R2, and C?C? • Figure P35.24P35.24 shows a light ray incident on a series of slabs having different refractive indices, where n1<n2<n3<n4n1<n2<n3<n4 . Notice that the path of the ray steadily bends toward the normal. If the variation in nn were continuous, the path would form a smooth curve. Use this idea and a ray diagram to explain why you can see the Sun at sunset after it has fallen below the horizon. • Show that at long wavelengths, Planck’s radiation law (Eq. 40.6)) reduces to the Rayleigh-Jeans law (Eq. 40.3).40.3). • Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest. A 0.500−kg sphere moving with a velocity given by (2.00ˆi−3.00ˆj+1.00ˆk)m/s strikes another sphere of mass 1.50 kg moving with an initial velocity of (−1.00ˆi+2.00ˆj−3.00ˆk)m/s. (a) The velocity of the 0.500− kg sphere after the collision is (−1.00ˆi+3.00ˆj−8.00ˆk) m/s . Find the final velocity of the 1.50−kg sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic). (b) Now assume the velocity of the 0.500 -kg sphere after the collision is (−0.250ˆi+0.750ˆj−2.00ˆk)m/s . Find the final velocity of the 1.50 -kg sphere and identify the kind of collision. (c) What If? Take the velocity of the 0.500−kg sphere after the collision as (−1.00i+3.00j+ak)m/s . Find the value of a and the velocity of the 1.50−kg sphere after an elastic collision. • The liquid-drop model of the atomic nucleus suggests high-energy oscillations of certain nuclei can split the nucleus into two unequal fragments plus a few neutrons. The fission products acquire kinetic energy from their mutual Coulomb repulsion. Assume the charge is distributed uniformly throughout the volume of each spherical fragment and, immediately before separating, each fragment is at rest and their surfaces are in contact. The electrons surrounding the nucleus can be ignored. Calculate the electric potential energy (in electron volts) of two spherical fragments from a uranium nucleus having the following charges and radii: 38ee and 5.50×10−15m,5.50×10−15m, and 54ee and 6.20×10−15m.6.20×10−15m. • In the 1968 Olympic games, University of Oregon jumper Dick Fosbury introduced a new technique of high jumping called the “Fosbury flop.” It contributed to raising the world record by about 30 cm and is currently used by nearly every world-class jumper. In this technique, the jumper goes over the bar face up while arching her back as much as possible as shown in Figure P9.76a. This action places her center of mass outside her body, below her back. As her body goes over the bar, her center of mass passes below the bar. Because a given energy input implies a certain elevation for her center of mass, the action of arching her back means that her body is higher than if her back were straight. As a model, consider the jumper as a thin uniform rod of length L. When the rod is straight, its center of mass is at its center. Now bend the rod in a circular arc so that it subtends an angle of 90.08 at the center of the arc as shown in Figure P9.76b. In this configuration, how far outside the rod is the center of mass? • A grinding wheel is in the form of a uniform solid disk of radius 7.00 cmcm and mass 2.00 kgkg . It starts from rest and accelerates uniformly under the action of the constant torque of 0.600 N⋅mN⋅m that the motor exerts on the wheel. (a) How long does the wheel take to reach its final operating speed of 1200 rev/minrev/min ? (b) Through how many revolutions does it turn while accelerating? • In a Compton scattering experiment, an x-ray photon scatters through an angle of 17.4∘4∘ from a free electron that is initially at rest. The electron recoils with a speed of ffff 2180 km/skm/s . Calculate (a) the wavelength of the incident photon and (b)(b) the angle through which the electron scatters. • The homopolar generator, also called the Faraday disk, is a low-voltage, high-current electric generator. It consists of a rotating conducting disk with one stationary brush (a sliding electrical contact) at its axle and another at a point on its circumference as shown in Figure P31.27. A uniform magnetic field is applied perpendicular to the plane of the disk. Assume the field is 0.900 T, the angular speed is 3.20×103rev/min , and the radius of the disk is 0.400m. Find the emf generated between the brushes. When superconducting coils are used to produce a large magnetic field, a homopolar generator can have a power output of several megawatts. Such a generator is useful, for example, in purifying metals by electrolysis. If a voltage is applied to the output terminals of the generator, it runs in reverse as a homopolar motor capable of providing great torque, useful in ship propulsion. • An aluminum rod is clamped one-fourth of the way along its length and set into longitudinal vibration by a variable-frequency driving source. The lowest frequency that produces resonance is 4 400 Hz. The speed of sound in an aluminum rod is 5 100 m/s. Determine the length of the rod. • A student uses an audio oscillator of adjustable frequency to measure the depth of a water well. The student reports hearing two successive resonances at 51.87 Hz and 59.85 Hz. (a) How deep is the well? (b) How many antinodes are in the standing wave at 51.87 Hz? • Two train whistles have identical frequencies of 180 Hz. When one train is at rest in the station and the other is moving nearby, a commuter standing on the station plat-form hears beats with a frequency of 2.00 beats/s when the whistles operate together. What are the two possible speeds and directions the moving train can have? • At what temperature would the average speed of helium atoms equal (a) the escape speed from the Earth, 12×104m/s1.12×104m/s , and (b) the escape speed from the Moon, 2.37×103m/s2.37×103m/s ? Note: The mass of a helium atom is 6.64×6.64× 10−27kg.10−27kg. • What is the electrical charge of the baryons with the quark compositions (a) u¯¯¯u¯¯¯d¯¯¯u¯u¯d¯ and (b)u¯¯¯d¯¯¯d¯¯¯(c)(b)u¯d¯d¯(c) What are these baryons called? • The acoustical system shown in Figure OQ18.1 is driven by a speaker emitting sound of frequency 756 Hz. (a) If constructive interference occurs at a particular location of the sliding section, by what minimum amount should the sliding section be moved upward so that destructive interference occurs instead? (b) What minimum distance from the original position of the sliding section will again result in constructive interference? • A sinusoidal wave traveling in the negative xx direction (to the left) has an amplitude of 20.0cm,20.0cm, a wavelength of 35.0cm,35.0cm, and a frequency of 12.0 HzHz . The transverse position of an element of the medium at t=0,x=0t=0,x=0 is y=y= −3.00cm,−3.00cm, and the element has a positive velocity here. We wish to find an expression for the wave function describing this wave. (a) Sketch the wave at t=0.t=0. (b) Find the angular wave number kk from the wavelength. (c) Find the period TT from the frequency. Find (d) the angular frequency ωω and (e) the wave speed v.v. ( ff ) From the information about t=0t=0 find the phase constant ϕ.ϕ. (g) Write an expression for the wave function y(x,t).y(x,t). • A house roof is a perfectly flat plane that makes an angle θθ with the horizontal. When its temperature changes, between TcTc before dawn each day and ThTh in the middle of each afternoon, the roof expands and contracts uniformly with a coefficient of thermal expansion α1α1 . Resting on the roof is a flat, rectangular metal plate with expansion coefficient α2,α2, greater than α1α1 . The length of the plate is L,L, measured along the slope of the roof. The component of the plate’s weight perpendicular to the roof is supported by a normal force uniformly distributed over the area of the plate. The coefficient of kinetic friction between the plate and the roof is μkμk . The plate is always at the same temperature as the roof, so we assume its temperature is continuously changing. Because of the difference in expansion coefficients, each bit of the plate is moving relative to the roof below it, except for points along a certain horizontal line running across the plate called the stationary line. If the temperature is rising, parts of the plate below the stationary line are moving down relative to the roof and feel a force of kinetic friction acting up the roof. Elements of area above the stationary line are sliding up the roof, and on them kinetic friction acts downward parallel to the roof. The stationary line occupies no area, so we assume no force of static friction acts on the plate while the temperature is changing. The plate as a whole is very nearly in equilibrium, so the net friction force on it must be equal to the component of its weight acting down the incline. (a) Prove that the stationary line is at a distance of L2(1−tanθμk)L2(1−tanθμk) below the top edge of the plate. (b) Analyze the forces that act on the plate when the temperature is falling and prove that the stationary line is at that same distance above the bottom edge of the plate. (c) Show that the plate steps down the roof like an inchworm, moving each day by the distance Lμk(α2−α1)(Th−Tc)tanθLμk(α2−α1)(Th−Tc)tanθ (d) Evaluate the distance an aluminum plate moves each day if its length is 1.20 mm , the temperature cycles between 4.00∘C4.00∘C and 36.0∘C36.0∘C , and if the roof has slope 18.5∘,18.5∘, coefficient of linear expansion 1.50×10−5(C)−11.50×10−5(C)−1 , and coefficient of friction 0.420 with the plate. (e) What If? What if the expansion coefficient of the plate is less than that of the roof? Will the plate creep up the roof? • A straight wire carrying a 3.00-A current is placed in a uniform magnetic field of magnitude 0.280 T directed perpendicular to the wire. (a) Find the magnitude of the magnetic force on a section of the wire having a length of 14.0 cm. (b) Explain why you can’t determine the direction of the magnetic force from the information given in the problem. • A liquid with a coefficient of volume expansion ββ just fills a spherical shell of volume VV (Fig. Pl9. 41). The shell and the open capillary of area AA projecting from the top of the sphere are made of a material with an average coefficient of linear expansion α.α. The liquid is free to expand into the capillary. Assuming the temperature increases by ΔT,ΔT, find the distance ΔhΔh the liquid rises in the capillary. • Three identical thin rods, each of length LL and mass m,m, are welded perpendicular to one another as shown in Figure P10.33P10.33 . The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. Determine the moment of inertia of this structure about this axis. • A ferry transports tourists between three islands. It sails from the first island to the second island, 4.76 km away, in a direction 37.0° north of east. It then sails from the second island to the third island in a direction 69.0° west of north. Finally it returns to the first island, sailing in a direction 28.0° east of south. Calculate the distance between (a) the second and third islands and (b) the first and third islands. • An electron and a proton are each placed at rest in a uniform electric field of magnitude 520 N/C. Calculate the speed of each particle 48.0 ns after being released. • A proton moves at 0.950 . Calculate its (a) rest energy, (b) total energy, and (c) kinetic energy. • Why is the following situation impossible? The object of mass m=4.00kgm=4.00kg in Figure P6.10P6.10 is attached to a vertical rod by two strings of length ℓ=2.00m.ℓ=2.00m. The strings are attached to the rod at points a distance d=3.00md=3.00m apart. The object rotates in a horizontal circle at a constant speed of v=3.00m/sv=3.00m/s , and the strings remain taut. The rod rotates along with the object so that the strings do not wrap onto the rod. What If? Could this situation be possible on another planet? • A 55.0-kg woman cheats on her diet and eats a 540 Calorie (540 kcal) jelly doughnut for breakfast. (a) How many joules of energy are the equivalent of one jelly doughnut? (b) How many steps must the woman climb on a very tall stairway to change the gravitational potential energy of the woman–Earth system by a value equivalent to the food energy in one jelly doughnut? Assume the height of a single stair is 15.0 cm. (c) If the human body is only 25.0% efficient in converting chemical potential energy to mechanical energy, how many steps must the woman climb to work off her breakfast? • An oceanographer is studying how the ion concentration in seawater depends on depth. She makes a measurement by lowering into the water a pair of concentric metallic cylinders (Fig. 63) at the end of a cable and taking data to determine the resistance between these electrodes as a function of depth. The water between the two cylinders forms a cylindrical shell of inner radius ra,ra, outer radius rb,rb, and length LL much larger than rb.rb. The scientist applies a potential difference ΔVΔV between the inner and outer surfaces, producing an outward radial current II . Let ρρ represent the resistivity of the water. (a) Find the resistance of the water between the cylinders in terms of L,ρ,ra,L,ρ,ra, and rb.rb. (b) Express the resistivity of the water in terms of the measured quantities L,raL,ra rb,ΔV,rb,ΔV, and I.I. • A light rope passes over a light, friction less pulley. One end is fastened to a bunch of bananas of mass M, and a monkey of mass MM clings to the other end (Fig. P 11.44). The monkey climbs the rope in an attempt to reach the bananas. (a) Treating the system as consisting of the monkey, bananas, rope, and pulley, find the net torque on the system about the pulley axis. (b) Using the result of part (a), determine the total angular momentum about the pulley axis and describe the motion of the system. (c) Will the monkey reach the bananas? • A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in Figure P6.6. The length of the arc ABC is 235 m, and the car completes the turn in 36.0 s. (a) What is the acceleration when the car is at B located at an angle of 35.0∘?35.0∘? Express your answer in terms of the unit vectors i and ˆjj^. Determine (b) the car’s average speed and (c) its average acceleration during the 36.0-s interval. • Two long, parallel wires are attracted to each other by a force per unit length of 320μN/mμN/m . One wire carries a current of 20.0 AA to the right and is located along the line y=y= 0.500 mm . The second wire lies along the xx axis. Determine the value of yy for the line in the plane of the two wires along which the total magnetic field is zero. • Why is the following situation impossible? An automobile has a vertical radio antenna of length ℓ=1.20m . The automobile travels on a curvy, horizontal road where the Earth’s magnetic field has a magnitude of B=50.0μT and is directed toward the north and downward at an angle of θ=65.0∘ below the horizontal. The motional emf devel- oped between the top and bottom of the antenna varies with the speed and direction of the automobile’s travel and has a maximum value of 4.50 mV . • Approximately 1 of every 3 300 water molecules contains one deuterium atom. (a) If all the deuterium nuclei in 1 L of water are fused in pairs according to the D−D fusion reaction 2H+2H→3He+n+3.27MeV , how much energy in joules is liberated? (b) What If? Burning gasoline produces approximately 3.40×107J/L . State how the energy obtainable from the fusion of the deuterium in 1 L of water compares with the energy liberated from the burning of 1 L of gasoline. • A heat pump has a coefficient of performance equal to 4.20 and requires a power of 1.75 kWkW to operate. (a) How much energy does the heat pump add to a home in one hour? (b) If the heat pump is reversed so that it acts as an air conditioner in the summer, what would be its coefficient of performance? • A daredevil plans to bungee jump from a balloon 65.0 m above the ground. He will use a uniform elastic cord, tied to a harness around his body, to stop his fall at a point 10.0 m above the ground. Model his body as a particle and the cord as having negligible mass and obeying Hooke’s law. In a preliminary test he finds that when hanging at rest from a 5.00-m length of the cord, his body weight stretches it by 1.50 m. He will drop from rest at the point where the top end of a longer section of the cord is attached to the stationary balloon. (a) What length of cord should he use? (b) What maximum acceleration will he experience? • Show that whenever white light is passed through a diffraction grating of any spacing size, the violet end of the spectrum in the third order on a screen always overlaps the red end of the spectrum in the second order. • A very large parallel-plate capacitor has uniform charge per unit area +σ+σ on the upper plate and −σ−σ on the lower plate. The plates are horizontal, and both move horizontally with speed vv to the right. (a) What is the magnetic field between the plates? (b) What is the magnetic field just above or just below the plates? (c) What are the magnitude and direction of the magnetic force per unit area on the upper plate? (d) At what extrapolated speed vv will the plate? Suggestion: Use Ampere’s law and choose a path that closes between the plates of the capacitor. • In air at 0∘C,0∘C, a 1.60−kg1.60−kg copper block at 0∘C0∘C is set sliding at 2.50 m/sm/s over a sheet of ice at 0∘C0∘C . Friction brings the block to rest. Find the mass of the ice that melts. (b) As the block slows down, identify its energy input QQ , its change in internal energy ΔEint,ΔEint, and the change in mechanical energy for the block-ice system. (c) For the ice as a system, identify its energy input QQ and its change in internal energy ΔEintΔEint (d) A1.60−kgA1.60−kg block of ice at 0∘C0∘C is set sliding at 2.50 m/sm/s over a sheet of copper at 0∘C0∘C . Friction brings the block to rest. Find the mass of the ice that melts. (e) Evaluate QQ and ΔEintΔEint for the block of ice as a system and ΔEmechΔEmech for the sheet as a system. (g) A thin, 1.60-kg slab of copper at 20∘C20∘C is set sliding at 2.50 m/sm/s over an identical stationary slab at the same temperature. Friction quickly stops the motion. Assuming no energy is transferred to the environment by heat, find the change in temperature of both objects. (h) Evaluate QQ and ΔEintΔEint for the sliding slab and ΔEmech forΔEmech for the two-slab system. (i) Evaluate QQ and ΔEintΔEint for the station- ary slab. • A student holds a laser that emits light of wavelength l. The laser beam passes though a pair of slits separated by a distance d, in a glass plate attached to the front of the laser. The beam then falls perpendicularly on a screen, creating an interference pattern on it. The student begins to walk directly toward the screen at speed v. The central maximum on the screen is stationary. Find the speed of the mth-order maxima on the screen, where m can be very large. • The picture tube in an old black-and-white television uses magnetic deflection coils rather than electric deflection plates. Suppose an electron beam is accelerated through a 50.0-kV potential difference and then through a region of uniform magnetic field 1.00 cm wide. The screen is located 10.0 cm from the center of the coils and is 50.0 cm wide. When the field is turned off, the electron beam hits the center of the screen. Ignoring relativistic corrections, what field magnitude is necessary to deflect the beam to the side of the screen? • A particle with a charge of −60.0nC−60.0nC is placed at the center of a nonconducting spherical shell of inner radius 20.0 cm and outer radius 25.0 cm. The spherical shell carries charge with a uniform density of −1.33μC/m3−1.33μC/m3 . A proton moves in a circular orbit just outside the spherical shell. Calculate the speed of the proton. • Why is the following situation impossible? In a new casino, a supersized pinball machine is introduced. Casino advertising boasts that a professional basketball player can lie on top of the machine and his head and feet will not hang off the edge! The ball launcher in the machine sends metal balls up one side of the machine and then into play. The spring in the launcher (Fig. P7.60) has a force constant of 1.20N/cm. The surface on which the ball moves is inclined θ=10.0∘ with respect to the horizontal. The spring is initially compressed its maximum distance d=5.00cm. ball of mass 100 g is projected into play by releasing the plunger. Casino visitors find the play of the giant machine quite exciting. • Do not hurt yourself; do not strike your hand against anything. Within these limitations, describe what you do to give your hand a large acceleration. Compute an order-of-magnitude estimate of this acceleration, stating the quantities you measure or estimate and their values. • A force platform is a tool used to analyze the performance of athletes by measuring the vertical force the athlete exerts on the ground as a function of time. Starting from rest, a 65.0−kg65.0−kg athlete jumps down onto the platform from a height of 0.600 mm . While she is in contact with the platform during the time interval 0<t<0.800s0<t<0.800s , the force she exerts on it is described by the function F=9200t−11500t2F=9200t−11500t2 where FF is in newtons and tt is in seconds. (a) What impulse did the athlete receive from the platform? (b) With what speed did she reach the platform? (c) With what speed did she leave it? (d) To what height did she jump upon leaving the platform? • Calculate the absolute pressure at the bottom of a freshwater lake at a point whose depth is$27.5 \mathrm{m} .$Assume the density of the water is$1.00 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$and that the air above is at a pressure of 101.3$\mathrm{kPa}$. (b) What force is exerted by the water on the window of an underwater vehicle at this depth if the window is circular and has a diameter of 35.0$\mathrm{cm}$? • Two small metallic spheres, each of mass m=0.200gm=0.200g , are suspended as pendulums by light strings of length LL as shown in Figure P23.10.P23.10. The spheres are given the same electric charge of 7.2nC,7.2nC, and they come to equilibrium when each string is at an angle of θ=θ= 5.00∘00∘ with the vertical. How long are the strings? • The toroid in Figure P32.67P32.67 consists of NN turns and has a rectangular cross section. Its inner and outer radii are a and bb , respectively. The figure shows half of the toroid to allow us to see its cross-section. Compute the inductance of a 500 -turn toroid for which a=10.0cm,b=12.0cm,a=10.0cm,b=12.0cm, and h=1.00cm.h=1.00cm. • A rotating wheel requires 3.00 s to rotate through 37.0 revolutions. Its angular speed at the end of the 3.00 -s interval is 98.0 rad/srad/s . What is the constant angular acceleration of the wheel? • A baseball approaches home plate at a speed of 45.0 m/sm/s , moving horizontally just before being hit by a bat. The batter hits a pop-up such that after hitting the bat, the baseball is moving at 55.0 m/sm/s straight up. The ball has a mass of 145 gg and is in contact with the bat for 2.00 msms . What is the average vector force the ball exerts on the bat during their interaction? • An atomic clock moves at 1000 km/hkm/h for 1.00 hh as measured by an identical clock on the Earth. At the end of the 1.00 -h interval, how many nanoseconds slow will the moving clock be compared with the Earth-based clock? • In a 250 -turn automobile alternator, the magnetic flux in each turn is ΦB=2.50×10−4 cos ωt, where ΦB is in webers, ω is the angular speed of the alternator, and t is in seconds. The alternator is geared to rotate three times for each engine revolution. When the engine is running at an angular speed of 1.00×103 rev/min, determine (a) the induced emf in the alternator as a function of time and (b) the maximum emf in the alternator. • You can feel a force of air drag on your hand if you stretch your arm out of the open window of a speeding car. Note: Do not endanger yourself. What is the order of magnitude of this force? In your solution, state the quantities you measure or estimate and their values. • The coefficient of friction between the block of mass m1=m1= 3.00 kgkg and the surface in Figure P8.22P8.22 is μk=0.400μk=0.400 . The system starts from rest. What is the speed of the ball of mass m2m2 =5.00kg=5.00kg when it has fallen a distance h=1.50m?h=1.50m? • A laser beam is incident on a 45∘−45∘−90∘45∘−45∘−90∘ prism perpendicular to one of its faces as shown in Figure P35. 27 The transmitted beam that exits the hypotenuse of the prism makes an angle of prism makes an angle of of the incident beam. Find the index of refraction of the prism. • Problems 56 and 70. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ball–pendulum combination swings up through a height h before coming to rest. George of the Jungle, with mass m, swings on a light vine hanging from a stationary tree branch. A second vine of equal length hangs from the same point, and a gorilla of larger mass M swings in the opposite direction on it. Both vines are horizontal when the primates start from rest at the same moment. George and the gorilla meet at the low- est point of their swings. Each is afraid that one vine will break, so they grab each other and hang on. They swing upward together, reaching a point where the vines make an angle of 35.0∘ with the vertical. Find the value of the ratio m/M. • Show that the function y(x,t)=x2+v2t2y(x,t)=x2+v2t2 is a solution to the wave equation. (b) Show that the function in part (a) can be written as f(x+vt)+g(x−vt)f(x+vt)+g(x−vt) and determine the functional forms for ff and g.g. (c) What If? Repeat parts (a)(a) and (b)(b) for the function y(x,t)=sin(x)cos(vt)y(x,t)=sin⁡(x)cos⁡(vt). • Why is the following situation impossible? A new diesel engine that increases fuel economy over previous models is designed. Automobiles fitted with this design become incredible best sellers. Two design features are responsible for the increased fuel economy: (1) the engine is made entirely of aluminum to reduce the weight of the automobile, and (2) the exhaust of the engine is used to prewarm the air to 508C before it enters the cylinder to increase the final temperature of the compressed gas. The engine has a compression ratio—that is, the ratio of the initial volume of the air to its final volume after compression—of 14.5. The compression process is adiabatic, and the air behaves as a diatomic ideal gas with g 5 1.40. • A piano string having a mass per unit length equal to 5.00×10−3kg/m5.00×10−3kg/m is under a tension of 1350 NN . Find the speed with which a wave travels on this string. • A 400-N child is in a swing that is attached to a pair of ropes 2.00 m long. Find the gravitational potential energy of the child-Earth system relative to the child’s lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.0∘ angle with the vertical, and (c) the child is at the bottom of the circular arc. • Equations 21.25 and 21.26 show that vrms>vavgvrms>vavg for a collection of gas particles, which turns out to be true whenever the particles have a distribution of speeds. Let us explore this inequality for a two-particle gas. Let the speed of one particle be v1=avavgv1=avavg and the other particle have speed v2=(2−a)vavgv2=(2−a)vavg (a) Show that the average of these two speeds is vavgvavg (b) Show that v2rms=v2avg(2−2a+a2)vrms2=vavg2(2−2a+a2) (c) Argue that the equation in part (b)(b) proves that, in general, vrms>vavg⋅(d)vrms>vavg⋅(d) Under what special condition will vrms=vavgvrms=vavg for the two-particle gas? • A fireworks rocket explodes at a height of 100 mm above the ground. An observer on the ground directly under the explosion experiences an average sound intensity of 7.00×10−2W/m27.00×10−2W/m2 for 0.200 ss . (a) What is the total amount of energy transferred away from the explosion by sound? (b) What is the sound level (in decibels) heard by the observer? • A 1.00-kg block of aluminum is warmed at atmospheric pressure so that its temperature increases from 22.0°C to 40.0°C. Find (a) the work done on the aluminum, (b) the energy added to it by heat, and (c) the change in its internal energy. • Explorers in the jungle find an ancient monument in the shape of a large isosceles triangle as shown in Figure P9.39. The monument is made from tens of thousands of small stone blocks of density 3800kg/m3.3800kg/m3. The monument is 15.7 mm high and 64.8 mm wide at its base and is everywhere 3.60 mm thick from front to back. Before the monument was built many years ago, all the stone blocks lay on the ground. How much work did laborers do on the blocks to put them in position while building the entire monument? Note: The gravitational potential energy of an object-Earth system is given by Ug=MgyCM,Ug=MgyCM, where MM is the total mass of the object and yCMyCM is the elevation of its center of mass above the chosen reference level. • What is the coefficient of performance of a refrigerator that operates with Carnot efficiency between temperatures −3.00∘C−3.00∘C and +27.0∘C?+27.0∘C? • Normalize the wave function for the ground state of a simple harmonic oscillator. That is, apply Equation 41.7 to Equation 41.26 and find the required value for the constant B in terms of m,ω, and fundamental constants. (b) Determine the probability of finding the oscillator in a narrow interval −δ/2<x<δ/2 around its equilibrium position. • A simple harmonic oscillator of amplitude A has a total energy E. Determine (a) the kinetic energy and (b) the potential energy when the position is one-third the amplitude. (c) For what values of the position does the kinetic energy equal one-half the potential energy? (d) Are there any values of the position where the kinetic energy is greater than the maximum potential energy? Explain. • A large nuclear power reactor produces approximately 3000MW3000MW of power in its core. Three months after a reactor is shut down, the core power from radioactive by-products is 10.0MW10.0MW. Assuming each emission delivers 1.00MeV1.00MeV of energy to the power, find the activity in becquerels three months after the reactor is shut down. • As soon as a traffic light turns green, a car speeds up from rest to 50.0$\mathrm{mi} / \mathrm{h}$with constant acceleration 9.00$\mathrm{mi} / \mathrm{h} / \mathrm{s}$. In the adjoining bicycle lane, a cyclist speeds up from rest to 20.0$\mathrm{mi} / \mathrm{h}$with constant acceleration 13.0$\mathrm{mi} / \mathrm{h} / \mathrm{s}$. Each vehicle maintains constant velocity after reaching its cruising speed. (a) For what time interval is the bicycle ahead of the car? (b) By what maximum distance does the bicycle lead the car? • Find (a) the equivalent resistance of the circuit in Figure P28.54, (b) the potential difference across each resistor, (c) each current indicated in Figure P28.54, and (d) the power delivered to each resistor. • A helium-filled balloon (whose envelope has a mass of$m_{b}=0.250 \mathrm{kg}$) is tied to a uniform string of length$\ell=$2.00$\mathrm{m}$and mass$m=0.0500 \mathrm{kg}$. The balloon is spherical with a radius of$r=0.400 \mathrm{m} .$When released in air of temperature$20^{\circ} \mathrm{C}$and density$\rho_{\text { air }}=1.20 \mathrm{kg} / \mathrm{m}^{3},$it lifts a length$h$of string and then remains stationary as shown in Figure$\mathrm{P} 14.56 .$We wish to find the length of string lifted by the balloon. (a) When the balloon remains stationary, what is the appropriate analysis model to describe it? (b) Write a force equation for the balloon from this model in terms of the buoyant force$B$, the weight$F_{b}$of the balloon, the weight$F_{\mathrm{He}}$of the helium, and the weight$F_{s}$of the segment of string of length$h$. (c) Make an appropriate substitution for each of these forces and solve symbolically for the mass$m_{s}$of the segment of string of length$h$in terms of$m_{b}, r, \rho_{\text { air }},$and the density of helium$\rho_{\mathrm{Hc}} .$(d) Find the numerical value of the mass$m_{s \cdot}$(e) Find the length$h$numerically. • You stand on the seat of a chair and then hop off. (a) During the time interval you are in flight down to the floor, the Earth moves toward you with an acceleration of what, order of magnitude? In your solution, explain your logic. Model the Earth as a perfectly solid object. (b) The Earth moves toward you through a distance of what order of magnitude? • The circuit in Figure P28.50a consists of three resistors and one battery with no internal resistance. (a) Find the current in the 5.00-V resistor. (b) Find the power delivered to the 5.00-V resistor. (c) In each of the circuits in Figures P28.50b, P28.50c, and P28.50d, an additional 15.0-V battery has been inserted into the circuit. Which diagram or diagrams represent a circuit that requires the use of Kirchhoff’s rules to find the currents? Explain why. (d) In which of these three new circuits is the smallest amount of power delivered to the 10.0-V resistor? (You need not • A beam of bright red light of wavelength 654 nm passes through a diffraction grating. Enclosing the space beyond the grating is a large semi cylindrical screen centered on the grating, with its axis parallel to the slits in the grating. Fifteen bright spots appear on the screen. Find (a) the maximum and (b) the minimum possible values for the slit separation in the diffraction grating. • In Figure P20.38, the change in internal energy of a gas that is taken from A to C along the blue path is 1800 J. The work done on the gas along the red path ABC is 2500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? (c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is 1500 J, how much energy must be added to the system by heat as it goes from point C to point D? • A small container of water is placed on a turntable inside a microwave oven, at a radius of 12.0 cm from the center. The turntable rotates steadily, turning one revolution in each 7.25 s. What angle does the water surface make with the horizontal? • A 3.00−μF3.00−μF capacitor is connected to a 12.0−V12.0−V battery. How much energy is stored in the capacitor? (b) Had the capacitor been connected to a 6.00−V6.00−V battery, how much energy would have been stored? • Plot yy versus tt at x=0x=0 for a sinusoidal wave of the form y=0.150cos(15.7x−50.3t),y=0.150cos⁡(15.7x−50.3t), where xx and yy are in meters and tt is in seconds. (b) Determine the period of vibration. (c) State how your result compares with the value found in Example 16.2 . • Assume the intensity of solar radiation incident on the cloud tops of the Earth is 1370 W/m2W/m2 (a) Taking the aver- age Earth-Sun separation to be 1.496×1011m,1.496×1011m, calculate the total power radiated by the Sun. Determine the maxi- mum values of (b) the electric field and (c) the magnetic field in the sunlight at the Earth’s location. • Two transverse sinusoidal waves combining in a medium are described by the wave functions y1=3.00sinπ(x+0.600t)y2=3.00sinπ(x−0.600t)y1=3.00sinπ(x+0.600t)y2=3.00sinπ(x−0.600t) where x,y1,x,y1, and y2y2 are in centimeters and tt is in seconds. Determine the maximum transverse position of an element of the medium at (a) x=0.250cmx=0.250cm,(b) x=0.500cm,x=0.500cm, and (c) x=1.50cm.x=1.50cm. (d) Find the three smallest values of xx corresponding to antinodes. • An attacker at the base of a castle wall 3.65 m high throws a rock straight up with speed 7.40$\mathrm{m} / \mathrm{s}$from a height of 1.55$\mathrm{m}$above the ground. (a) Will the rock reach the top of the wall? (b) If so, what is its speed at the top? If not, what initial speed must it have to reach the top? (c) Find the change in speed of a rock thrown straight down from the top of the wall at an initial speed of 7.40$\mathrm{m} / \mathrm{s}$and moving between the same two points. (d) Does the change in speed of the downward-moving rock agree with the magnitude of the speed change of the rock moving upward between the same elevations? (e) Explain physically why it does or does not agree. • An aluminum ring of radius r1=5.00cmr1=5.00cm and resistance 3.00×10−4Ω3.00×10−4Ω is placed around one end of a long air-core solenoid with 1000 turns per meter and radius r2=3.00cmr2=3.00cm as shown in Figure P31.9P31.9 . Assume the axial component of the field produced by the solenoid is one-half as strong over the area of the end of the solenoid as at the center of the solenoid. Also assume the solenoid produces negligible field outside its cross-sectional area. The current in the solenoid is increasing at a rate of 270 A/sA/s . (a) What is the induced current in the ring? At the center of the ring, what are (b) the magnitude and (c) the direction of the magnetic field produced by the induced current in the ring? • A 75.0 -g ice cube at 0∘C0∘C is placed in 825 gg of water at 25.0∘0∘C . What is the final temperature of the mixture? • A conducting rod moves with a constant velocity in a direction perpendicular to a long, straight wire carrying a current I as shown in Figure P31.59. Show that the magnitude of the emf generated between the ends of the rod is |ε|=μ0vIℓ2πr In this case, note that the emf decreases with increasing r as you might expect. • The angle of incidence of a light beam onto a reflecting surface is continuously variable. The reflected ray in air is completely polarized when the angle of incidence is 48.0°. What is the index of refraction of the reflecting material? • A 3.00−kg3.00−kg object undergoes an acceleration given by →a=(2.00ˆi+5.00ˆj)m/s2.a→=(2.00i^+5.00j^)m/s2. Find (a)(a) the resultant force acting on the object and (b)(b) the magnitude of the resultant force. • Two vectors A→A→ and B→B→ have precisely equal magnitudes. For the magnitude of A→+B→A→+B→ to be 100 times larger than the magnitude of A→−B→,A→−B→, what must be the angle between them? • The weight of a rectangular block of low-density material is 15.0 N. With a thin string, the center of the horizontal bottom face of the block is tied to the bottom of a beaker partly filled with water. When 25.0% of the block’s volume is submerged, the tension in the string is 10.0 N. (a) Find the buoyant force on the block. (b) Oil of density 800 kg/m3 is now steadily added to the beaker, forming a layer above the water and surrounding the block. The oil exerts forces on each of the four sidewalls of the block that the oil touches. What are the directions of these forces? (c) What happens to the string tension as the oil is added? Explain how the oil has this effect on the string tension. (d) The string breaks when its tension reaches 60.0 N. At this moment, 25.0% of the block’s volume is still below the water line. What additional fraction of the block’s volume is below the top surface of the oil? • A 12.0−V12.0−V battery is connected to a capacitor, resulting in 54.0μCμC of charge stored on the capacitor. How much energy is stored in the capacitor? • A soccer player kicks a rock horizontally off a 40.0 -m-high cliff into a pool of water. If the player hears the sound of the splash 3.00 s later, what was the initial speed given to the rock? Assume the speed of sound in air is 343 m/sm/s . • Prove that the exchange of a virtual particle of mass mm can be associated with a force with a range given by d≈12404πmc2=98.7mc2d≈12404πmc2=98.7mc2 where dd is in nanometers and mc2mc2 is in electron volts. (b) State the pattern of dependence of the range on the mass. (c) What is the range of the force that might be produced by the virtual exchange of a proton? • An air wedge is formed between two glass plates separated at one edge by a very fine wire of circular cross section as shown in Figure P37.35. When the wedge is illuminated from above by 600-nm light and viewed from above, 30 dark fringes are observed. Calculate the diameter d of the wire. • X-rays are scattered from a target at an angle of 55.0∘0∘ with the direction of the incident beam. Find the wavelength shift of the scattered x-rays. • The wave function for a quantum particle is ψ(x)=√aπ(x2+a2)ψ(x)=aπ(x2+a2)−−−−−−−−−−√ for a>0a>0 and −∞<x<+∞−∞<x<+∞ . Determine the probability that the particle is located somewhere between x=−ax=−a and x=+a. • Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt=−bv2dE/dt=−bv2 and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, E=12mv2+12kx2,E=12mv2+12kx2, and use Equation 15.31.15.31. • A beam of 6.61-MeV protons is incident on a target of 2713Al2713Al . Those that collide produce the reaction P+2713Al→2714Si+nP+2713Al→2714Si+n Ignoring any recoil of the product nucleus, determine the kinetic energy of the emerging neutrons. • Refer to Problem 34 and Figure P14.34. A hydrometer is to be constructed with a cylindrical floating rod. Nine fiduciary marks are to be placed along the rod to indicate densities of$0.98 \mathrm{g} / \mathrm{cm}^{3}, 1.00 \mathrm{g} / \mathrm{cm}^{3}, 1.02 \mathrm{g} / \mathrm{cm}^{3},1.04 \mathrm{g} / \mathrm{cm}^{3}, \ldots, 1.14 \mathrm{g} / \mathrm{cm}^{3} .$The row of marks is to start 0.200$\mathrm{cm}$from the top end of the rod and end 1.80$\mathrm{cm}$from the top end. (a) What is the required length of the rod? (b) What must be its average density? (c) Should the marks be equally spaced? Explain your answer. • Two capacitors, C1=5.00μFC1=5.00μF and C2=12.0μFC2=12.0μF , are connected in parallel, and the resulting combination is connected to a 9.00−V9.00−V battery. Find (a) the equivalent capacitance of the combination, (b) the potential difference across each capacitor, and (c)(c) the charge stored on each capacitor. • A wire of nonmagnetic material, with radius R, carries current uniformly distributed over its cross section. The total current carried by the wire is I. Show that the magnetic energy per unit length inside the wire is μ0I2/16πμ0I2/16π • The various spectral lines observed in the light from a distant quasar have longer wavelengths λ′nλn′ than the wave-lengths λnλn measured in light from a stationary source. Here-nn is an index taking different values for different spectral lines. The fractional change in wavelength toward the red is the same for all spectral lines. That is, the Doppler red-shift parameter ZZ defined by Z=λ′n−λnλnZ=λn′−λnλn is common to all spectral lines for one object. In terms of Z,Z, use Hubble’s law to determine (a) the speed of recession of the quasar and (b) the distance from the Earth to this quasar. • According to one estimate, there are 4.40×4.40× 106106 metric tons of world uranium reserves extractable at$130/kg$130/kg or less. We wish to determine if these reserves are sufficient to supply all the world’s energy needs. About 0.700%% of naturally occurring uranium is the fissionable isotope 235U.235U. (a) Calculate the mass of 235U235U in the reserve in grams. (b) Find the number of moles of 235U235U in the reserve. (c) Find the number of 235U235U nuclei in the reserve. (d) Assuming 200 MeV is obtained from each fission reaction and all this energy is captured, calculate the total energy in joules that can be extracted from the reserve. (e) Assuming the rate of world power consumption remains constant at 1.5×1013J/s1.5×1013J/s , how many years could the uranium reserve provide for all the world’s energy needs? (f) What conclusion can be drawn? • The immediate cause of many deaths is ventricular fibrillation, which is an uncoordinated quivering of the heart. An electric shock to the chest can cause momentary paralysis of the heart muscle, after which the heart sometimes resumes its proper beating. One type of defibrillator (chapter opening photo, page 740)) applies a strong electric shock- to the chest over a time interval of a few milliseconds. This device contains a capacitor of several microfarads, charged to several thousand volts. Electrodes called paddles are held against the chest on both sides of the heart, and the capacitor is discharged through the patient’s chest. Assume an energy of 300 JJ is to be delivered from a 30.0−μF30.0−μF capacitor. To what potential difference must it be charged? • An ideal voltmeter connected across a certain fresh 9-V battery reads 9.30 V, and an ideal ammeter briefly connected across the same battery reads 3.70 A. We say the battery has an open-circuit voltage of 9.30 V and a short-circuit current of 3.70 A. Model the battery as a source of emf e in series with an internal resistance r as in Active Figure 28.1a. Determine both (a) e and (b) r. An experimenter connects two of these identical batteries together as shown in Figure P28.68. Find (c) the open-circuit voltage and (d) the short-circuit current of the pair of connected batteries. (e) The experimenter connects a 12.0-V resistor between the exposed terminals of the connected batteries. Find the current in the resistor. (f) Find the power delivered to the resistor. (g) The experimenter connects a • Consider as a system the Sun with the Earth in a circular orbit around it. Find the magnitude of the change in the velocity of the Sun relative to the center of mass of the system over a six-month period. Ignore the influence of other celestial objects. You may obtain the necessary astronomical data from the endpapers of the book. • The most soaring vocal melody is in Johann Sebastian Bach’s Mass in B Minor. In one section, the basses, tenors, altos, and sopranos carry the melody from a low D to a high A. In concert pitch, these notes are now assigned frequencies of 146.8 Hz and 880.0 Hz. Find the wavelengths of (a) the initial note and (b) the final note. Assume the chorus sings the melody with a uniform sound level of 75.0 dB. Find the pressure amplitudes of (c) the initial note and (d) the final note. Find the displacement amplitudes of (e) the initial note and (f) the final note. • A straight ladder is leaning against the wall of a house. The ladder has rails 4.90 m long, joined by rungs 0.410 m long. Its bottom end is on solid but sloping ground so that the top of the ladder is 0.690 m to the left of where it should be, and the ladder is unsafe to climb. You want to put a flat rock under one foot of the ladder to compensate for the slope of the ground. (a) What should be the thickness of the rock? (b) Does using ideas from this chapter make it easier to explain the solution to part (a)? Explain your answer. • A ball of mass 0.200 kgkg with a velocity of 1.50 im/sim/s meets a ball of mass 0.300 kgkg with a velocity of −0.400ˆim/s−0.400i^m/s in a head-on, elastic collision. (a) Find their velocities after the collision. (b) Find the velocity of their center of mass before and after the collision. • Around 1965, engineers at the Toro Company invented a gasoline gauge for small engines diagrammed in Figure P35.44. The gauge has no moving parts. It consists of a flat slab of transparent plastic fitting vertically into a slot in the cap on the gas tank. None of the plastic has a reflective coating. The plastic projects from the horizontal top down nearly to the bottom of the opaque tank. Its lower edge is cut with facets making angles of 45∘45∘ with the horizontal. A lawn mower operator looks down from above and sees a boundary between bright and dark on the gauge. The location of the boundary, across the width of the plastic, indicates the quantity of gasoline in the tank. (a) Explain how the gauge works. (b) Explain the design requirements, if any, for the index of refraction of the plastic. • In the Rutherford scattering experiment, 4.00 -MeV alpha particles scatter off gold nuclei (containing 79 protons and 118 neutrons). Assume a particular alpha particle moves directly toward the gold nucleus and scatters backward at 180∘,180∘, and that the gold nucleus remains fixed throughout the entire process. Determine (a) the distance of closest approach of the alpha particle to the gold nucleus and (b) the maximum force exerted on the alpha particle. • A Marconi antenna, used by most AM radio stations, consists of the top half of a Hertz antenna (also known as a half-wave antenna because its length is λ/2).λ/2). The lower end of this Marconi (quarter-wave) antenna is connected to Earth ground, and the ground itself serves as the missing lower half. What are the heights of the Marconi antennas for radio stations broadcasting at (a) 560 kHz and (b) 1 600 kHz? • How much work does an ideal Carnot refrigerator require to remove 1.00 JJ of energy from liquid helium at 4.00 KandKand expel this energy to a room-temperature (293−K)(293−K) environment? • Two sinusoidal waves in a string are defined by the wave functions y1=2.00sin(20.0x−32.0t)y2=2.00sin(25.0x−40.0t)y1=2.00sin(20.0x−32.0t)y2=2.00sin(25.0x−40.0t) where x,y1,x,y1, and y2y2 are in centimeters and tt is in seconds. (a) What is the phase difference between these two waves at the point x=5.00cmx=5.00cm at t=2.00st=2.00s ? (b) What is the positive xx value closest to the origin for which the two phases differ by ±π±π at t=2.00s?t=2.00s? (At that location, the two waves add to zero.) • Two ships are moving along a line due east (Fig. P17.58). The trailing vessel has a speed relative to a land-based observation point of v1=64.0km/h,v1=64.0km/h, and the leading ship has a speed of v2=45.0km/hv2=45.0km/h relative to that point. The two ships are in a region of the ocean where the current is moving uniformly due west at vcurrent=10.0km/hvcurrent=10.0km/h . The trailing ship transmits a sonar signal at a frequency of 1200.0 Hz through the water. What frequency is monitored by the leading ship? • Light of wavelength 530 nm illuminates a pair of slits separated by 0.300 mm. If a screen is placed 2.00 m from the slits, determine the distance between the first and second dark fringes. • How many times will the incident beam shown in Figure P35.49 be reflected by each of the parallel mirrors? • A child’s pogo stick (Fig. P8.61P8.61 ) stores energy in a spring with a force constant of 2.50×104N/m.2.50×104N/m. At position A(Xa=−0.100m)A(Xa=−0.100m) the spring compression is a maximum and the child is momentarily at rest. At position B(Xb=0)B(Xb=0), the spring is relaxed and the child is moving upward.At position CC, the child is again momentarily at rest at the top of the jump. The combined mass of child and pogo stick is 25.0 kg. Although the boy must lean for- ward to remain balanced, the angle is small, so let’s assume the pogo stick is vertical. Also assume the boy does not bend his legs during the motion. (a) Calculate the total energy of the child-stick-Earth system, taking both gravitational and elastic potential energies as zero for x=0.x=0. (b) Determine XcXc (c) Calculate the speed of the child at x 5 0. (d) Determine the value of x for which the kinetic energy of the system is a maximum (e) Calculate the child’s maximum up ward speed. • When solid silver starts to melt, what is the approximate fraction of the conduction electrons that are thermally excited above the Fermi level? • The gravitational force exerted on a solid object is 5.00$\mathrm{N}$. When the object is suspended from a spring scale and sub- merged in water, the scale reads 3.50$\mathrm{N}$(Fig. Pl4.24). Find the density of the object. • A potter’s wheel moves uniformly from rest to an angular speed of 1.00 rev/s in 30.0 s. (a) Find its average angular acceleration in radians per second per second. (b) Would doubling the angular acceleration during the given period have doubled the final angular speed? • Find the maximum (Carnot) efficiency of an engine that absorbs energy from a hot reservoir at 545∘C545∘C and exhausts energy to a cold reservoir at 185∘C185∘C . • An object moving with uniform acceleration has a velocity of 12.0$\mathrm{cm} / \mathrm{s}$in the positive$x$direction when its$x$coordinate is$3.00 \mathrm{cm} .$If its$x$coordinate 2.00 s later is$-5.00 \mathrm{cm},$what is its acceleration? • Problems 69 through 72 apply ideas from this and earlier chapters to some properties of superconductors, which were introduced in Section 27.5. A fundamental property of a type I superconducting material is perfect diamagnetism, or demonstration of the Meissner effect, illustrated in Figure 30.27 in Section 30.6 and described as follows. If a sample of superconducting material is placed into an externally produced magnetic field or is cooled to become superconducting while it in a magnetic field, electric currents appear on the surface of the sample. The currents have precisely the strength and orientation required to make the total magnetic field be zero throughout the interior of the sample. This problem will help you understand the magnetic force that can then act on the sample. Compare this problem with Problem 63 in Chapter 26 , pertaining to the force attracting a perfect dielectric into a strong electric field. A vertical solenoid with a length of 120 cmcm and a diameter of 2.50 cmcm consists of 1400 turns of copper wire carrying a counterclockwise current (when viewed from above) of 2.00 AA as shown in Figure P32.72aP32.72a . (a) Find the magnetic field in the vacuum inside the solenoid. (b) Find the energy density of the magnetic field. Now a superconducting bar 2.20 cmcm in diameter is inserted partway into the solenoid. Its upper end is far outside the solenoid, where the magnetic field is negligible. The lower end of the bar is deep inside the solenoid. (c) Explain how you identify the direction required for the current on the curved surface of the bar so that the total magnetic field is zero within the bar. The field created by the supercurrents is sketched in Figure P32.72bP32.72b , and the total field is sketched in Figure P32.72c.P32.72c. (d) The field of the solenoid exerts a force on the current in the superconductor. Explain how you determine the direction of the force on the bar. (e) Noting that the units J/m3J/m3 of energy density are the same as the units N/m2N/m2 of pressure, calculate the magnitude of the force by multiplying the energy density of the solenoid field times the area of the bottom end of the superconducting bar. • Use a magnifying glass to look at the grains of table salt that come out of a salt shaker. Compare what you see with Figure 43.10a. The distance between a sodium ion and a nearest-neighbor chlorine ion is 0.261 nm. (a) Make an order-of-magnitude estimate of the number N of atoms in a typical grain of salt. (b) What If? Suppose you had a number of grains of salt equal to this number N. What would be the volume of this quantity of salt? • A table-tennis ball has a diameter of 3.80$\mathrm{cm}$and average density of$0.0840 \mathrm{g} / \mathrm{cm}^{3} .$What force is required to hold it completely submerged under water? • A long, cylindrical conductor of radius a has two cylindrical cavities each of diameter aa through its entire length as shown in the end view of Figure P30.75.P30.75. A current II is directed out of the page and is uniform through a cross section of the conducting material. Find the magnitude and direction of the magnetic field in terms of μ0,I,r,μ0,I,r, and aa at (a) point P1P1 and (b) point P2.P2. • Determine the temperature at which the resistance of an aluminum wire will be twice its value at 20.0∘0∘C . Assume its coefficient of resistivity remains constant. • To meet a U.S. Postal Service requirement, employees’ footwear must have a coefficient of static friction of 0.5 or more on a specified tile surface. A typical athletic shoe has a coefficient of static friction of 0.800. In an emergency, what is the minimum time interval in which a person starting from rest can move 3.00 m on the tile surface if she is wearing (a) footwear meeting the Postal Service minimum and (b) a typical athletic shoe? • Figure P31.60 shows a compact, circular coil with 220 turns and radius 12.0 cm immersed in a uniform magnetic field parallel to the axis of the coil. The rate of change of the field has the constant magnitude 20.0 mT/s. (a) What additional information is necessary to determine whether the coil is carrying clock-wise or counterclockwise current? (b) The coil overheats if more than 160 W of power is delivered to it. What resistance would the coil have at this critical point? (c) To run cooler, should it have lower resistance or higher resistance? • Consider the damped oscillator illustrated in Figure 15.20.15.20. The mass of the object is 375 gg , the spring constant is 100N/m,100N/m, and b=0.100N⋅s/m.b=0.100N⋅s/m. (a) Over what time interval does the amplitude drop to half its initial value? (b) What If? Over what time interval does the mechanical energy drop to half its initial value? (c) Show that, in general, the fractional rate at which the amplitude decreases in a damped harmonic oscillator is one-half the fractional rate at which the mechanical energy decreases. • A photon having wavelength λλ scatters off a free electron at A(Fig.P40.24)A(Fig.P40.24) , producing a second photon having wavelength λ′.λ′. This photon then scatters off another free electron at BB , producing a third photon having wavelength λ′′λ′′ and moving in a direction directly opposite the original photon as shown in the figure. Determine the value of Δλ=λ′′−λΔλ=λ′′−λ • An electric field of magnitude 3.50 kN/CkN/C is applied along the xx axis. Calculate the electric flux through a rectangular plane 0.350 mm wide and 0.700 mm long (a) if the plane is parallel to the yy z plane, (b) if the plane is parallel to the xyxy plane, and (c) if the plane contains the yy axis and its normal makes an angle of 40.0∘0∘ with the xx axis. • Two plane mirrors stand facing each other, 3.00 m apart, and a woman stands between them. The woman faces one of the mirrors from a distance of 1.00 m and holds her left arm out to the side of her body with the palm of her hand facing the closer mirror. (a) What is the apparent position of the closest image of her left hand, measured perpendicularly from the surface of the mirror in front of her? (b) Does it show the palm of her hand or the back of her hand? (c) What is the position of the next closest image? (d) Does it show the palm of her hand or the back of her hand? (e) What is the position of the third closest image? (f) Does it show the palm of her hand or the back of her hand? (g) Which of the images are real and which are virtual? • A sample consists of 1.00×1061.00×106 radioactive nuclei with a half-life of 10.0 h. No other nuclei are present at time t=0.t=0. The stable daughter nuclei accumulate in the sample as time goes on. (a) Derive an equation giving the number of daughter nuclei Nd as a function of time. (b) Sketch or describe a graph of the number of daughter nuclei as a function of time. (c) What are the maximum and minimum numbers of daughter nuclei, and when do they occur? (d) What are the maximum and minimum rates of change in the number of daughter nuclei, and when do they occur? • A proton is confined to move in a one-dimensional box of length 0.200 nm. (a) Find the lowest possible energy of the proton. (b) What If? What is the lowest possible energy of an electron confined to the same box? (c) How do you account for the great difference in your results for parts (a) and (b)? • Oxygen at pressures much greater than 1 atm is toxic to lung cells. Assume a deep-sea diver breathes a mixture of oxygen (O2)(O2) and helium (He). By weight, what ratio of helium to oxygen must be used if the diver is at an ocean depth of 50.0 mm ? • One leg of a Michelson interferometer contains an evacuated cylinder of length L,L, having glass plates on each end. A gas is slowly leaked into the cylinder until a pressure of 1 atm is reached. If NN bright fringes pass on the screen during this process when light of wavelength λλ is used, what is the index of refraction of the gas? • Obtain expressions in component form for the position vectors having the polar coordinates (a) 12.8 m, 150°; (b) 3.30 cm, 60.0°; and (c) 22.0 in., 215°. • One end of a cord is fixed and a small 0.500-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 2.00 m as shown in Figure P6.18P6.18 . When θ=20.0∘θ=20.0∘ the speed of the object is 8.00 m/sm/s . At this instant, find (a)(a) the tension in the string, (b) the tangential and radial components of acceleration, and (c)(c) the total acceleration. (d) Is your answer changed if the object is swinging down toward its lowest point instead of swinging up? (e) Explain your answer to part (d). • A line of charge with uniform density 35.0 nC/mnC/m lies along the line y=−15.0cmy=−15.0cm between the points with coordinates x=0x=0 and x=40.0cm.x=40.0cm. Find the electric field it creates at the origin. • Imagine that the entire Sun, of mass MS,MS, collapses to a sphere of radius RgRg such that the work required to remove a small mass mm from the surface would be equal to its rest energy mc2.mc2. This radius is called the gravitational radius for the Sun. (a) Use this approach to show that Rg=GMS/c2Rg=GMS/c2 (b) Find a numerical value for RgRg • Determine whether or not strangeness is conserved in the following decays and reactions. (a)Λ0→p+π−(c)p¯¯¯+p→πΛ0+Λ0(e)Ξ−→Λ0+π−(b)π−+p→Λ0+K0(d)π−+p→π−+Σ+(f)Ξ0→p+π−(a)Λ0→p+π−(b)π−+p→Λ0+K0(c)p¯+p→πΛ0+Λ0(d)π−+p→π−+Σ+(e)Ξ−→Λ0+π−(f)Ξ0→p+π− • Consider a uniformly charged, thin-walled, right circular cylindrical shell having total charge QQ , radius R,R, and length ℓ.ℓ. Determine the electric field at a point a distance dd from the right side of the cylinder as shown in Figure P23.38.P23.38. Suggestion: Use the result of Example 23.7 and treat the cylinder as a collection of ring charges. (b) What If? Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly distributed through its volume. Use the result of Example 23.8 to find the field it creates at the same point. • A thin copper bar of length ℓ=10.0cmℓ=10.0cm is supported horizontally by two (nonmagnetic) contacts at its ends. The bar carries a current of I1=100AI1=100A in the negative xx direction as shown in Figure P30.67P30.67 . At a distance h=0.500cmh=0.500cm below one end of the bar, a long, straight wire carries a current of I2=200AI2=200A in the positive zz direction. Determine the magnetic force exerted on the bar. • A firework charge is detonated many meters above the ground. At a distance of d1=500md1=500m from the explosion, the acoustic pressure reaches a maximum of ΔPmax=ΔPmax= 10.0 Pa (Fig. Pl7.31). Assume the speed of sound is constant at 343 m/sm/s throughout the atmosphere over the region considered, the ground absorbs all the sound falling on it, and the air absorbs sound energy as described by the rate 7.00dB/km.7.00dB/km. What is the sound level (in decibels) at a distance of d2=4.00×103md2=4.00×103m from the explosion? • Four resistors are connected to a battery as shown in Figure P28.16.P28.16. (a) Determine the potential difference across each resistor in terms of E.E. (b) Determine the current in each resistor in terms of I.I. (c) What If? If R3R3 is increased, explain what happens to the current in each of the resistors. (d) In the limit that R3→∞,R3→∞, what are the new values of the current in each resistor in terms of I,I, the original current in the battery? • A 4.00 -m-long pole stands vertically in a freshwater lake having a depth of 2.00 mm . The Sun is 40.0∘0∘ above the horizontal. Determine the length of the pole’s shadow on the bottom of the lake. • Pfund’s method for measuring the index of refraction of glass is illustrated in Figure P35.68. One face of a slab of thickness tt is painted white, and a small hole scraped clear at point PP serves as a source of diverging rays when the slab is illuminated from below. Ray PBB′PBB′ strikes the clear surface at the critical angle and is totally reflected, as are rays such as PCC′.PCC′. Rays such as PAA′PAA′ emerge from the clear surface. On the painted surface, there appears a dark circle of diameter dd surrounded by an illuminated region, or halo. (a) Derive an equation for nn in terms of the measured quantities dd and t.t. (b) What is the diameter of the dark circle if n=1.52n=1.52 for a slab 0.600 cmcm thick? (c) If white light is used, dispersion causes the critical angle to depend on color. Is the inner edge of the white halo tinged with red light or with violet light? Explain. • A ball of mass mm is connected to two rubber bands of length L,L, each under tension TT as shown in Figure P15.63P15.63 . The ball is displaced by a small distance yy perpendicular to the length of the rubber bands. Astaning the tension does not change, show that (a) the restoring force is −(2T/L)y−(2T/L)y and (b) the system exhibits simple harmonic motion with an angular frequency ω=2T/mL−−−−−−√ω=2T/mL . • A horizontal power line of length 58.0 m carries a current of 2.20 kA northward as shown in Figure P 29.39. The Earth’s magnetic field at this location has a magnitude of 5.00×10−5T5.00×10−5T . The field at this location is directed toward the north at an angle 65.0∘0∘ below the power line. Find (a) the magnitude and (b) the direction of the magnetic force on the power line. • Two protons approach each other with velocities of equal magnitude in opposite directions. What is the minimum kinetic energy of each proton if the two are to produce a π+π+ meson at rest in the reaction p+p→p+n+π+?p+p→p+n+π+? • When a high-energy proton or pion traveling near the speed of light collides with a nucleus, it travels an average distance of 3×10−15m3×10−15m before interacting. From this information, find the order of magnitude of the time interval required for the strong interaction to occur. • Given →M=2ˆi−3ˆj+ˆkM−→=2i^−3j^+k^ and →N=4ˆi+5ˆj−2ˆk,N→=4i^+5j^−2k^, calculate the vector product →M×→N.M−→×N→. • A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 250 km when traveling at 6.00 km/s. (a) What maximum distance from the Earth’s surface does it travel before falling back to the Earth? (b) Would its maximum distance from the surface be larger if the same rocket were fired with the same fuel load from a launch site on the equator? Why or why not? • Why is the following situation impossible? A technician is testing a circuit that contains a resistance RR . He realizes that a better design for the circuit would include a resistance 73R73R rather than R. He has three additional resistors, each with resistance R. By combining these additional resistors in a certain combination that is then placed in series with the original resistor, he achieves the desired resistance. • In Figure P 3.55, a spider is resting after starting to spin its web. The gravitational force on the spider makes it exert a downward force of 0.150 N on the junction of the three strands of silk. The junction is supported by different tension forces in the two strands above it so that the resultant force on the junction is zero. The two sloping strands are perpendicular, and we have chosen the xx and yy directions to be along them. The tension TxTx is 0.127 NN . Find (a) the tension TY,TY, (b) the angle the xx axis makes with the horizontal, and (c) the angle the yy axis makes with the horizontal. • Imagine that you stand tall and turn about a vertical axis through the top of your head and the point halfway between your ankles. Compute an order-of-magnitude estimate for the moment of inertia of your body for this rotation. In your solution, state the quantities you measure or estimate and their values. • Four identical charged particles (q=+10.0μC)(q=+10.0μC) are located on the corners of a rectangle as shown in Figure P25.23P25.23 . The dimensions of the rectangle are L=60.0cmL=60.0cm and W=15.0cm.W=15.0cm. Calculate the change in electric potential energy of the system as the particle at the lower left corner in Figure P25.23P25.23 is brought to this position from infinitely far away. Assume the other three particles in Figure P25.23P25.23 remain fixed in position. • An electron is confined to a one-dimensional region in which its ground-state (n=1) energy is 2.00 eV. (a) What is the length L of the region? (b) What energy input is required to promote the electron to its first excited state? • A block with a speaker bolted to it is connected to a spring having spring constant kk and oscillates as shown in Figure P17.39P17.39 . The total mass of the block and speaker is m,m, and the amplitude of this unit’s motion is AA . The speaker emits sound waves of frequency f.f. Determine nected to a spring having spring constant kk and oscillates as shown in Figure P17.39P17.39 . The total mass of the block and speaker is m,m, and the amplitude of this unit’s motion is AA . The speaker emits sound waves of frequency f.f. Determine • A 10000−N10000−N shark is supported by a rope attached to a 4.00 −m−m rod that can pivot at the base. (a) Calculate the tension in the cable between the rod and the wall, assuming the cable is holding the system in the position shown in Figure Pl2.49. Find (b) the horizontal force and (c) the vertical force exerted on the base of the rod. Ignore the weight of the rod. • The following equations are obtained from a force diagram of a rectangular farm gate, supported by two hinges on the left-hand side. A bucket of grain is hanging from the latch. −A+C=0−A+C=0 +B−392N−50.0N=0+B−392N−50.0N=0 A(0)+B(0)+C(1.80m)−392N(1.50m)A(0)+B(0)+C(1.80m)−392N(1.50m) −50.0N(3.00m)=0−50.0N(3.00m)=0 (a) Draw the force diagram and complete the statement of the problem, specifying the unknowns. (b) Determine the values of the unknowns and state the physical meaning of each. • The disk of the Sun subtends an angle of$0.533^{\circ}$at the Earth. What are (a) the position and (b) the diameter of the solar image formed by a concave spherical mirror with a radius of curvature of magnitude 3.00$\mathrm{m} ?$• An ideal gas is taken through a quasi-static process described by P=αV2P=αV2 with α=5.00atm/m6,α=5.00atm/m6, as shown in Figure P20.23P20.23 . The gas is expanded to 1.00 m3m3 . How much work is done on the expanding gas in this process? • Find to three significant digits the charge and the mass of the following particles. Suggestion: Begin by looking up the mass of a neutral atom on the periodic table of the elements in Appendix C. (a) an ionized hydrogen atom, represented as H+H+ (b) a singly ionized sodium atom, Na+(c)Na+(c) a chloride ion Cl−Cl− (d) a doubly ionized calcium atom, Ca++=Ca2+Ca++=Ca2+ (e) the center of an ammonia molecule, modeled as an N3−N3− ion (f) quadruply ionized nitrogen atoms, N4+N4+ , found in plasma in a hot star (g)(g) the nucleus of a nitrogen atom (h) the molecular ion H2O−H2O−. • AK+AK+ ion and a Cl−Cl− ion are separated by a distance of 5.00×10−10m.5.00×10−10m. Assuming the two ions act like charged particles, determine (a) the force each ion exerts on the other and (b) the potential energy of the two-ion system in electron volts. • The projection lens in a certain slide projector is a single thin lens. A slide$24.0 \mathrm{~mm}$high is to be projected so that its image fills a screen$1.80 \mathrm{~m}$high. The slide-to-screen distance is$3.00 \mathrm{~m}$. (a) Determine the focal length of the projection lens. (b) How far from the slide should the lens of the projector be placed so as to form the image on the screen? • The rotating loop in an AC generator is a square 10.0 cm on each side. It is rotated at 60.0 Hz in a uniform field of 0.800 T. Calculate (a) the flux through the loop as a function of time, (b) the emf induced in the loop, (c) the current induced in the loop for a loop resistance of 1.00Ω (d) the power delivered to the loop, and (e) the torque that must be exerted to rotate the loop. • Show that the integral ∫∞0e−2t/RCdt∫∞0e−2t/RCdt in Example 28.11 has the value 12RC.12RC. • A 1.00 -mH inductor and a 1.00−μF1.00−μF capacitor are connected in series. The current in the circuit increases linearly in time as I=20.0t,I=20.0t, where II is in amperes and tt is in seconds. The capacitor initially has no charge. Determine (a) the voltage across the inductor as a function of time, (b) the voltage across the capacitor as a function of time, and (c) the time when the energy stored in the capacitor first exceeds that in the inductor. • Figure P12.15P12.15 shows a claw hammer being used to pull a nail out of a horizontal board. The mass of the hammer is 1.00 kgkg . A force of 150 NN is exerted horizontally as shown, and the nail does not yet move relative to the board. Find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface on the point of contact with the hammer head. Assume the force the hammer exerts on the nail is parallel to the nail. • Three uniform spheres of masses m1=2.00kg,m2=m1=2.00kg,m2= 4.00kg,4.00kg, and m3=6.00kgm3=6.00kg are placed at the corners of a right triangle as shown in Figure P13.6. Calculate the resultant gravitational force on the object of mass m2,m2, assuming the spheres are isolated from the rest of the Universe. • A diatomic molecule consists of two atoms having masses m1m1 and m2m2 separated by a distance rr . Show that the moment of inertia about an axis through the center of mass of the molecule is given by Equation 43.3,I=μr2.43.3,I=μr2. • Assuming the cost of energy from the electric company is$0.110/kWh$0.110/kWh , compute the cost per day of operating a lamp that draws a current of 1.70 AA from a 110−V110−V line. • A sinusoidal wave is described by the wave function y=y= 0.25 sin(0.30x−40t)sin⁡(0.30x−40t) where xx and yy are in meters and tt is in seconds. Determine for this wave (a) the amplitude, (b) the angular frequency, (c) the angular wave number, (d) the wavelength, (e) the wave speed, and (f) the direction of motion. • A flexible chain weighing 40.0 NN hangs between two hooks located at the same height (Fig. Pl2.17). At each hook, the tangent to the chain makes an angle θ=42.0∘θ=42.0∘ with the horizontal. Find (a) the magnitude of the force each hook exerts on the chain and (b) the tension in the chain at its midpoint. Suggestion: For part (b), make a force diagram for half of the chain. • Show that the two waves with wave functions given by E1=6.00sin(100πt)E1=6.00sin(100πt) and E2=8.00sin(100πt+π/2)E2=8.00sin(100πt+π/2) add to give a wave with the wave function ERsin(100πt+ϕ).ERsin(100πt+ϕ). Find the required values for ERER and ϕ.ϕ. • The helicopter view in Fig. P 3.29 shows two people pulling on a stubborn mule. The person on the right pulls with a force →F1F→1 of magnitude 120 NN and direction of θ1=60.0∘.θ1=60.0∘. The person on the left pulls with a force →F2F→2 of magnitude 80.0 NN and direction of θ2=75.0∘.θ2=75.0∘. Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (symbolized N). • A 3.00−kg3.00−kg block starts from rest at the top of a 30.0∘30.0∘ incline and slides a distance of 2.00 mm down the incline in 1.50 s. Find (a) the magnitude of the acceleration of the block, (b) the coefficient of kinetic friction between block and plane, (c) the friction force acting on the block, and (d) the speed of the block after it has slid 2.00 mm . • A steel guitar string with a diameter of 1.00 mmmm is stretched between supports 80.0 cmcm apart. The temperature is 0.0∘C0.0∘C . (a) Find the mass per unit length of this string. (Use the value 7.86×103kg/m37.86×103kg/m3 for the density.) (b) The fundamental frequency of transverse oscillations of the string is 200 HzHz . What is the tension in the string? Next, the temperature is raised to 30.0∘C30.0∘C . Find the resulting values of (c)(c) the tension and (d)(d) the fundamental frequency. Assume both the Young’s modulus of 20.0×1010N/m220.0×1010N/m2 and the average coefficient of expansion α=11.0×10−6(∘C)−1α=11.0×10−6(∘C)−1 have constant values between 0.0∘C0.0∘C and 30.0∘C.30.0∘C. • A light source emitting radiation at frequency 7×10147×1014 HzHz is incapable of ejecting photoelectrons from a certain metal. In an attempt to use this source to eject photoelectrons from the metal, the source is given a velocity toward the metal. (a) Explain how this procedure can produce photoelectrons. (b) When the speed of the light source is equal to 0.280c,0.280c, photoelectrons just begin to be ejected from the metal. What is the work function of the metal? (c) When the speed of the light source is increased to 0.900 , determine the maximum kinetic energy of the photoelectrons. • Why is the following situation impossible? You set up an apparatus in your laboratory as follows. The xx axis is the symmetry axis of a stationary, uniformly charged ring of radius R=0.500mR=0.500m and charge Q=50.0μCQ=50.0μC and charge Q=50.0μCQ=50.0μC particle with charge Q=Q= 0.100 kgkg at the center of the ring and arrange for it to be constrained to move only along the xx axis. When it is displaced slightly, the particle is repelled by the ring and accelerates along the xx axis. The particle moves faster than you expected and strikes the opposite wall of your laboratory at 40.0 m/sm/s . • As thermonuclear fusion proceeds in its core, the Sun loses mass at a rate of 3.64×109kg/s3.64×109kg/s . During the 5000-yr period of recorded history, by how much has the length of the year changed due to the loss of mass from the Sun? Suggestions: Assume the Earth’s orbit is circular. No external torque acts on the Earth-Sun system, so the angular momentum of the Earth is constant. • A uniform solid disk of radius RR and mass MM is free to rotate on a frictionless pivot through a point on its rim (Fig. Pl0.53). If the disk is released from rest in the position shown by the copper-colored circle, (a) what is the speed of its center of mass when the disk reaches the position indicated by the dashed circle? (b) What is the speed of the lowest point on the disk in the dashed position? (c) What If? Repeat part (a) using a uniform hoop. • A photon with an energy Eγ=2.09Eγ=2.09 GeV creates a proton-antiproton pair in which the proton has a kinetic energy of 95.0 MeV. What is the kinetic energy of the antiproton? Note: mpc2=938.3MeVmpc2=938.3MeV . • An infinitely long insulating cylinder of radius RR has a volume charge density that varies with the radius as ρ=ρ0(a−rb)ρ=ρ0(a−rb) where ρ0,a,ρ0,a, and bb are positive constants and rr is the distance from the axis of the cylinder. Use Gauss’s law to determine the magnitude of the electric field at radial distances (a) r<Rr<R and (b) r>R.r>R. • A charged particle of mass 1.50 g is moving at a speed of 1.50×104m/s1.50×104m/s . Suddenly, a uniform magnetic field of magnitude 0.150 mT in a direction perpendicular to the particle’s velocity is turned on and then turned off in a time interval of 1.00 s. During this time interval, the magnitude and direction of the velocity of the particle undergo a negligible change, but the particle moves by a distance of 0.150 m in a direction perpendicular to the velocity. Find the charge on the particle. • In the air over a particular region at an altitude of 500 m above the ground, the electric field is 120 N/C directed downward. At 600 m above the ground, the electric field is 100 N/C downward. What is the average volume charge density in the layer of air between these two elevations? Is it positive or negative? • How fast are you personally making the entropy of the Universe increase right now? Compute an order-of-magnitude estimate, stating what quantities you take as data and the values you measure or estimate for them. • A proton (charge +e,+e, mass mp),mp), a deuteron (charge +e+e, mass 2mp),2mp), and an alpha particle (charge +2e,+2e, mass 4mp)mp) are accelerated from rest through a common potential difference ΔVΔV . Each of the particles enters a uniform magnetic field →B,B→, with its velocity in a direction perpendicular to →BB→. The proton moves in a circular path of radius rprp . In terms of rp,rp, determine (a) the radius rdrd of the circular orbit for the deuteron and (b) the radius rαrα for the alpha particle. • Two long, parallel wires carry currents of I1=3.00AI1=3.00A and I2=5.00AI2=5.00A in the directions indicated in Figure P30.18P30.18 . (a) Find the magnitude and direction of the magnetic field at a point midway between the wires. (b) Find the magnitude and direction of the magnetic field at point P,P, located d=20.0cmd=20.0cm above the wire carrying the 5.00−A5.00−A current. • Two motorcycles separated laterally by 2.30 m are approaching an observer wearing night-vision goggles sensitive to infrared light of wavelength 885 nm. (a) Assume the light propagates through perfectly steady and uniform air. What aperture diameter is required if the motorcycles’ headlights are to be resolved at a distance of 12.0 km? (b) Comment on how realistic the assumption in part (a) is. • An air rifle shoots a lead pellet by allowing high-pressure air to expand, propelling the pellet down the rifle barrel. Because this process happens very quickly, no appreciable thermal conduction occurs and the expansion is essentially adiabatic. Suppose the rifle starts with 12.0 cm3cm3 of compressed air, which behaves as an ideal gas with γ=1.40γ=1.40 .The expanding air pushes a 1.10 -g pellet as a piston with cross-sectional area 0.0300 cm2cm2 along the 50.0 -cm-long gun barrel. What initial pressure is required to eject the pellet with a muzzle speed of 120 m/sm/s ? Ignore the effects of the air in front of the bullet and friction with the inside walls of the barrel. • A wind turbine on a wind farm turns in response to a force of high-speed air resistance, R=12DρAv2.R=12DρAv2. The power available is P=Rv=12Dρπr2v3,P=Rv=12Dρπr2v3, where vv is the wind speed and we have assumed a circular face for the wind turbine of radius rr . Take the drag coefficient as D=1.00D=1.00 and the density of air from the front endpaper. For a wind turbine having r=1.50m,r=1.50m, calculate the power available with (a) v=v= 8.00 m/sm/s and (b)v=24.0m/s.(b)v=24.0m/s. The power delivered to the generator is limited by the efficiency of the system, about 25%.25%. For comparison, a large American home uses about 2 kWkW of electric power. • Transverse pulses travel with a speed of 200 m/sm/s along a taut copper wire whose diameter is 1.50mm.1.50mm. What is the tension in the wire? (The density of copper is 8.92g/cm3.)8.92g/cm3.) • Two objects are connected by a light string passing over a light, frictionless pulley as shown in Figure P8.7. The object of mass m1=5.00kgm1=5.00kg is released from rest at a height h=4.00mh=4.00m above the table. Using the isolated system model, (a) determine the speed of the object of mass m2=3.00kgm2=3.00kg just as the 5.00 -kg object hits the table and (b)(b) find the maximum height above the table to which the 3.00−kg3.00−kg object rises. • A 12.0-kg object hangs in equilibrium from a string with a total length of L=5.00mL=5.00m and a linear mass density of μ=0.00100kg/mμ=0.00100kg/m . The string is wrapped around two light, friction less pulleys that are separated by a distance of d=2.00md=2.00m (Fig. P l8.71 a). (a) Determine the tension in the string. (b) At what frequency must the string between the pulleys vibrate to form the standing-wave pattern shown in Figure P 18.71 b? • A 0.20 -kg stone is held 1.3 m above the top edge of a water well and then dropped into it. The well has a depth of 5.0 m . Relative to the configuration with the stone at the top edge of the well, what is the gravitational potential energy of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well? • A long, straight wire lies on a horizontal table and carries a current of 1.20μAμA . In a vacuum, a proton moves parallel to the wire (opposite the current) with a constant speed of 2.30×104m/s2.30×104m/s at a distance dd above the wire. Ignoring the magnetic field due to the Earth, determine the value of d.d. • You may use the Rayleigh criterion for the limiting angle of resolution of an eye. The standard may be overly optimistic for human vision. • Determine the type of neutrino or antineutrino involved in each of the following processes. (a)π+→π0+e++?(c)Λ0→p+μ−+?(b)?+p→μ−+p+π+(d)τ+→μ++?+?(a)π+→π0+e++?(b)?+p→μ−+p+π+(c)Λ0→p+μ−+?(d)τ+→μ++?+? • Write out the electronic configuration of the ground state for nitrogen (Z=7).(Z=7). (b) Write out the values for the possible set of quantum numbers n,ℓ,mℓ,n,ℓ,mℓ, and msms for the electrons in nitrogen. • An object of mass m1m1 hangs from a string that passes over a very light fixed pulley P1P1 as shown in Figure P5.34. The string connects to a second very light pulley P2.P2. A second string passes around this pulley with one end attached to a wall and the other to an object of mass m2m2 on a frictionless, horizontal table. (a) If a1 and a2 are the accelerations of m1 and m2, respectively, what is the relation between these accelerations? Find expressions for (b) the tensions of in the strings and (c) the accelerations a1 and a2 in terms of the masses m1 and m2, and g. • Jonathan is riding a bicycle and encounters a hill of height 7.30 m. At the base of the hill, he is traveling at 6.00 m/s. When he reaches the top of the hill, he is traveling at 1.00 m/s. Jonathan and his bicycle together have a mass of 85.0 kg. Ignore friction in the bicycle mechanism and between the bicycle tires and the road. (a) What is the total external work done on the system of Jonathan and the bicycle between the time he starts up the hill and the time he reaches the top? (b) What is the change in potential energy stored in Jonathan’s body during this process? (c) How much work does Jonathan do on the bicycle pedals within the Jonathan–bicycle–Earth system during this process? • A uniform electric field of magnitude 325 V/mV/m is directed in the negative yy direction in Figure P25.3P25.3 . The coordinates of point AA are (−0.200,−0.300)m,(−0.200,−0.300)m, and those of point BB are (0.400, 0.500) m. Calculate the electric potential difference VB−VAVB−VA using the dashed-line path. • A spacecraft is in empty space. It carries on board a gyroscope with a moment of inertia of Ig=20.0kg⋅m2Ig=20.0kg⋅m2 about the axis of the gyroscope. The moment of inertia of the spacecraft around the same axis is Is=5.00×105kg⋅m2Is=5.00×105kg⋅m2 . Neither the spacecraft nor the gyroscope is originally rotating. The gyroscope can be powered up in a negligible period of time to an angular speed of 100 rad/s. If the orientation of the spacecraft is to be changed by 30.0∘0∘, for what time interval should the gyroscope be operated? • When a potential difference of 150 VV is applied to the plates of a parallel-plate capacitor, the plates carry a surface charge density of 30.0nC/cm2.30.0nC/cm2. What is the spacing between the plates? • A uniform beam of length LL and mass mm shown in Figure PI2.16 is inclined at an angle θθ to the horizontal. Its upper end is connected to a wall by a rope, and its lower end rests on a rough, horizontal surface. The coefficient of static friction between the beam and surface is μxμx . Assume the angle θθ is such that the static friction force is at its maximum value. (a) Draw a force diagram for the beam. (b) Using the condition of rotational equilibrium, find an expression for the tension TT in the rope in terms of m,g,m,g, and θ.θ. (c) Using the condition of translational equilibrium, find a second expression for TT in terms of μs,m,μs,m, and gg . (d) Using the results from parts (a) through (c), obtain an expression for μsμs involving only the angle θ.θ. (e) What happens if the ladder is lifted upward and its base is placed back on the ground slightly to the left of its position in Figure Pl2. 16 ? Explain. • You may use the Rayleigh criterion for the limiting angle of resolution of an eye. The standard may be overly optimistic for human vision. • A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure$\mathrm{P} 14.34$. The bulb of a syringe is squeezed and released to let the atmosphere lift a sample of the liquid of interest into a tube containing a calibrated rod of known density. The rod, of length$L$and average density$\rho_{0}$, floats partially immersed in the liquid of density$\rho .$A length$h$of the rod protrudes above the surface of the liquid. Show that the density of the liquid is given by ρ=ρ0LL−hρ=ρ0LL−h • Suppose the Sun’s gravity were switched off. The planets would leave their orbits and fly away in straight lines as described by Newton’s first law. (a) Would Mercury ever be farther from the Sun than Pluto? (b) If so, find how long it would take Mercury to achieve this passage. If not, give a convincing argument that Pluto is always farther from the Sun than is Mercury. • At the Earth’s surface, a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth. • On a horizontal air track, a glider of mass m carries a G-shaped post. The post supports a small dense sphere, also of mass m, hanging just above the top of the glider on a cord of length L . The glider and sphere are initially at rest with the cord vertical. (Figure P9.47a shows a cart and a sphere similarly connected.) A constant horizontal force of magnitude F is applied to the glider, moving it through displacement x1; then the force is removed. During the time interval when the force is applied, the sphere moves through a displacement with horizontal component x2. (a) Find the horizontal component of the velocity of the center of mass of the glider-sphere system when the force is removed. (b) After the force is removed, the glider continues to move on the track and the sphere swings back and forth, both without friction. Find an expression for the largest angle the cord makes with the vertical. • Why is the following situation impossible? A proton is in an infinitely deep potential well of length 1.00nm.1.00nm. It absorbs a microwave photon of wavelength 6.06 mmmm and is excited into the next available quantum state. • Model the electric motor in a handheld electric mixer as a single flat, compact, circular coil carrying electric current in a region where a magnetic field is produced by an external permanent magnet. You need consider only one instant in the operation of the motor. (We will consider motors again in Chapter 31.) Make order-of-magnitude estimates of (a) the magnetic field, (b) the torque on the coil, (c) the current in the coil, (d) the coil’s area, and (e) the number of turns in the coil. The input power to the motor is electric, given by P=IΔV,P=IΔV, and the useful output power is mechanical, P=τω.P=τω. • What is the maximum possible coefficient of performance of a heat pump that brings energy from outdoors at −3.00∘C−3.00∘C into a 22.0∘0∘C house? Note: The work done to run the heat pump is also available to warm the house. • A wad of sticky clay with mass m and velocity v→i is fired at a solid cylinder of mass M and radius R (Fig. P 11.39). The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at a distance d<R from the center. (a) Find the angular speed of the system just after the clay strikes and sticks to the surface of the cylinder. (b) Is the mechanical energy of the clay–cylinder system constant in this process? Explain your answer. (c) Is the momentum of the clay–cylinder system constant in this process? Explain your answer. • Assume light of wavelength 650 nmnm passes through two slits 3.00μmμm wide, with their centers 9.00μmμm apart. Make a sketch of the combined diffraction and interference pattern in the form of a graph of intensity versus ϕ=(πasinθ)/λϕ=(πasin⁡θ)/λ . You may use Active Figure 38.7 as a starting point. • A cylindrical shell of radius 7.00 cm and length 2.40 m has its charge uniformly distributed on its curved surface. The magnitude of the electric field at a point 19.0 cm radially outward from its axis (measured from the midpoint of the shell) is 36.0 kN/C. Find (a) the net charge on the shell and (b) the electric field at a point 4.00 cm from the axis, measured radially outward from the midpoint of the shell. • An ideal gas is enclosed in a cylinder with a movable piston on top of it. The piston has a mass of 8000 gg and an area of 5.00 cm2cm2 and is free to slide up and down, keeping the pressure of the gas constant. How much work is done on the gas as the temperature of 0.200 molmol of the gas is raised from 20.0∘0∘C to 300∘C300∘C ? • Gas is contained in an 8.00−L8.00−L vessel at a temperature of 20.0∘0∘C and a pressure of 9.00atm.9.00atm. (a) Determine the number of moles of gas in the vessel. (b) How many molecules are in the vessel? • A continuous line of charge lies along the xx axis, extending from x=+x0x=+x0 to positive infinity. The line carries positive charge with a uniform linear charge density λ0.λ0. What are (a)(a) the magnitude and (b)(b) the direction of the electric field at the origin? • A river has a steady speed of 0.500 m/sm/s . A student swims upstream a distance of 1.00 kmkm and swims back to the starting point. ( a) If the student can swim at a speed of 1.20 m/sm/s in still water, how long does the trip take? (b) How mm much time is required in still water for the same length swim? (c) Intuitively, why does the swim take longer when there is a current? • Show that the wave function y=ln[b(x−vt)]y=ln⁡[b(x−vt)] is a solution to Equation 16.27, where bb is a constant. • One wire in a high-voltage transmission line carries 1000 AA starting at 700 kVkV for a distance of 100 mimi . If the resistance in the wire is 0.500Ω/miΩ/mi , what is the power loss due to the resistance of the wire? • An inductor is connected to an AC power supply having a maximum output voltage of 4.00 VV at a frequency of 300 HzHz . What inductance is needed to keep the rms current less than 2.00 mAmA ? • A 1.00-kg block of copper at 20.0°C is dropped into a large vessel of liquid nitrogen at 77.3 K. How many kilograms of nitrogen boil away by the time the copper reaches 77.3 K? (The specific heat of copper is 0.092 0 cal/g ? °C, and the latent heat of vaporization of nitrogen is 48.0 cal/g.) • The Sun radiates energy at the rate of 3.85×1026W . Suppose the net reaction 4(1H)+2(0∘e)→42He+2v+γ accounts for all the energy released. Calculate the number of protons fused per second. • A system consisting of nn moles of an ideal gas with molar specific heat at constant pressure CPCP undergoes two reversible processes. It starts with pressure PiPi and volume Vi,Vi, expands isothermally, and then contracts adiabatically to reach a final state with pressure PiPi and volume 3ViVi (a) Find its change in entropy in the isothermal process. (The entropy does not change in the adiabatic process.) (b) What If? Explain why the answer to part (a) must be the same as the answer to Problem 65 . (You do not need to solve Problem 65 to answer this question.) • Consider the apparatus shown in Figure P 18.73 a, where the hanging object has mass MM and the string is vibrating in its second harmonic. The vibrating blade at the left maintains a constant frequency. The wind begins to blow to the right, applying a constant horizontal force F→F→ on the hanging object. What is the magnitude of the force the wind must apply to the hanging object so that the string vibrates in its first harmonic as shown in Figure 18.73 b? • In Figure P30.68P30.68 , both currents in the infinitely long wires are 8.00 AA in the negative xx direction. The wires are separated by the distance 2a=6.00cm.2a=6.00cm. (a) Sketch the magnetic field pattern in the yzyz plane. (b) What is the value of the magnetic field at the origin? (c) At (y=0,z→∞)?(y=0,z→∞)? (d) Find the magnetic field at points along the zz axis as a function of z.z. (e) At what distance dd along the positive zz axis is the magnetic field a maximum? (f) What is this maximum value? • A particle with a mass of 0.500 kgkg is attached to a horizontal spring with a force constant of 50.0N/m.50.0N/m. At the moment t=0,t=0, the particle has its maximum speed of 20.0 m/sm/s and is moving to the left. (a) Determine the particle’s equation of motion, specifying its position as a function of time. (b) Where in the motion is the potential energy three val required for the particle to move from x=0x=0 to x=x= 1.00m.1.00m. (d) Find the length of a simple pendulum with the same period. • Members of a skydiving club were given the following data to use in planning their jumps. In the table, dd is the distance fallen from rest by a skydiver in a “free-fall stable spread position” versus the time of fall tt . (a) Convert the distances in feet into meters. (b) Graph dd (in meters) versus tt (c) Determine the value of the terminal speed vTvT by finding the slope of the straight portion of the curve. Use a least-squares fit to determine this slope. • Two wave pulses AA and BB are moving in opposite directions, each with a speed v=2.00cm/s.v=2.00cm/s. The amplitude of AA is twice the amplitude of BB. The pulses are shown in Figure P 18.2 at t=0.t=0. Sketch the resultant wave at t=1.00s,1.50s,t=1.00s,1.50s, 2.00s,2.50s,2.00s,2.50s, and 3.00s.3.00s. • A 10.6 -kg object oscillates at the end of a vertical spring that has a spring constant of 2.05×104N/m2.05×104N/m . The effect of air resistance is represented by the damping coefficient b=b= 3.00N⋅s/m.3.00N⋅s/m. (a) Calculate the frequency of the damped (b) By what percentage does the amplitude of the oscillation decrease in each cycle? (c) Find the time interval that elapses while the energy of the system drops to 5.00%% of its initial value. • A particle with positive charge q=3.20×10−19Cq=3.20×10−19C moves with a velocity →v=(2ˆi+3ˆj−ˆk)m/sv→=(2i^+3j^−k^)m/s through a region where both a uniform magnetic field and a uniform electric field exist. (a) Calculate the total force on the moving particle (in unit-vector notation), taking →B=(2ˆi+4ˆj+ˆk)TB→=(2i^+4j^+k^)T and →E=(4ˆi−ˆj−2ˆk)V/m.E→=(4i^−j^−2k^)V/m. (b) What angle does the force vector make with the positive xx axis? • Consider the rectangular cabinet of Problem 50 shown in Figure P12.50P12.50 , but with a force F→F→ applied horizontally at the upper edge. (a) What is the minimum force required to start to tip the cabinet? (b) What is the minimum coefficient of static friction required for the cabinet not to slide with the application of a force of this magnitude? (c) Find the magnitude and direction of the minimum force required to tip the cabinet if the point of application can be chosen anywhere on the cabinet. • Why is the following situation impossible? Two identical loudspeakers are driven by the same oscillator at frequency 200 Hz. They are located on the ground a distance d=4.00md=4.00m from each other. Starting far from the speakers, a man walks straight toward the right-hand speaker as shown in Figure P 18.10. After passing through three minima in sound intensity, he walks to the next maximum and stops. Ignore any sound reflection from the ground. • The charge per unit length on a long, straight filament is −90.0μC/m.−90.0μC/m. Find the electric field (a) 10.0 cm, (b) 20.0 cm, and (c) 100 cm from the filament, where distances are measured perpendicular to the length of the filament. • Liquid nitrogen has a boiling point of 77.3 KK and a latent heat of vaporization of 2.01×105J/kg2.01×105J/kg . A 25.0−W25.0−W electric heating element is immersed in an insulated vessel con- taining 25.0 LL of liquid nitrogen at its boiling point. How many kilograms of nitrogen are boiled away in a period of 4.00 hh ? • A free neutron beta decays by creating a proton, an electron, and an antineutrino according to the reaction n→p+e−+ν¯¯¯.n→p+e−+ν¯. What If? Imagine that a free neutron were to decay by creating a proton and electron according to the reaction n→p+e−n→p+e− and assume the neutron is initially at rest in the laboratory. (a) Determine the energy released in this reaction. (b) Energy and momentum are conserved in the reaction. Determine the speeds of the proton and the electron after the reaction. (C) Is either of these particles moving at a relativistic speed? Explain. • Find the mass density of a proton, modeling it as a solid sphere of radius 1.00×10−15m.1.00×10−15m. ( b) What If? Consider a classical model of an electron as a uniform solid sphere with the same density as the proton. Find its radius. (c) Imagine that this electron possesses spin angular momentum Iω=ℏ/2Iω=ℏ/2 because of classical rotation about the zz axis. Determine the speed of a point on the equator of the electron. (d) State how this speed compares with the speed of light. • Suppose you wish to fabricate a uniform wire from 1.00 gg of copper. If the wire is to have a resistance of R=R= 0.500ΩΩ and all the copper is to be used, what must be (a) the length and (b) the diameter of this wire? • Show that the maximum magnitude EmaxEmax of the electric field along the axis of a uniformly charged ring occurs at x=x= a/2–√(see Fig.23.16)a/2(see Fig.23.16) and has the value Q/(63–√πϵ0a2).Q/(63πϵ0a2). • A flowerpot is knocked off a window ledge from a height d=d= 20.0 mm above the sidewalk as shown in Figure P17.11. It falls toward an unsuspecting man of height h=1.75mh=1.75m who is standing below. Assume the man requires a time interval of Δt=0.300Δt=0.300 s to respond to the warning. How close to the sidewalk can the flowerpot fall before it is too late for a warning shouted from the balcony to reach the man in time? • An astronaut, stranded in space 10.0 m from her spacecraft and at rest relative to it, has a mass (including equipment) of 110 kg. Because she has a 100-W flashlight that forms a directed beam, she considers using the beam as a photon rocket to propel herself continuously toward the spacecraft. (a) Calculate the time interval required for her to reach the spacecraft by this method. (b) What If? Suppose she throws the 3.00-kg flashlight in the direction away from the spacecraft instead. After being thrown, the flashlight moves at 12.0 m/s relative to the recoiling astronaut. After what time interval will the astronaut reach the spacecraft? • Suppose an object has thickness$d p$so that it extends from object distance$p$to$p+d p .$(a) Prove that the thickness$d q$of its image is given by$\left(-q^{2} / p^{2}\right) d p$. (b) The longitudinal magnification of the object is$M_{\text { long }}=d q / d p$. How is the longitudinal magnification related to the lateral magnification$M ?$• After the Sun exhausts its nuclear fuel, its ultimate fate will be to collapse to a white dwarf state. In this state, it would have approximately the same mass as it has now, but its radius would be equal to the radius of the Earth. Calculate (a) the average density of the white dwarf, (b) the surface free-fall acceleration, and (c) the gravitational potential energy associated with a 1.00-kg object at the surface of the white dwarf. • A horizontal 800-N merry-go-round is a solid disk of radius 1.50 m and is started from rest by a constant horizontal force of 50.0 N applied tangentially to the edge of the disk. Find the kinetic energy of the disk after 3.00 s. • In an experiment on the transport of nutrients in a plant’s root structure, two radioactive nuclides X and Y are used. Initially, 2.50 times more nuclei of type X are present than of type Y. At a time 3.00 d later, there are 4.20 times more nuclei of type X than of type Y . Isotope Y has a half-life of 1.60 d. What is the half-life of isotope X ? • The wavelengths of the Lyman series for hydrogen are given by 1λ=RH(1−1n2)n=2,3,4,…1λ=RH(1−1n2)n=2,3,4,… (a) Calculate the wavelengths of the first three lines in this series. (b) Identify the region of the electromagnetic spectrum in which these lines appear. • Light in air (assume n=1)n=1) strikes the surface of a liquid of index of refraction nℓnℓ at the polarizing angle. The part of the beam refracted into the liquid strikes a submerged slab of material with refractive index nn as shown in Figure P 38.59. The light reflected from the upper surface of the slab is completely polarized. Find the angle θθ between the water surface and the surface of the slab as a function of nn and nℓnℓ . • A uniform piece of sheet metal is shaped as shown in Figure P9.38. Compute the xx and yy coordinates of the center of mass of the piece. • What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earth’s orbit? (b) Voyager 1 achieved a maximum speed of 125000 km/hkm/h on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system? • A 50.0-turn circular coil of radius 5.00 cm can be oriented in any direction in a uniform magnetic field having a magnitude of 0.500 T. If the coil carries a current of 25.0 mA, find the magnitude of the maximum possible torque exerted on the coil. • A tuning fork with a frequency of f=512Hzf=512Hz is placed near the top of the tube shown in Figure P 18.38. The water level is lowered so that the length LL slowly increases from an initial value of 20.0 cm. Determine the next two values of LL that correspond to resonant modes. • The temperature of an electric heating element is 150°C. At what wavelength does the radiation emitted from the heating element reach its peak? • A small object is attached to the end of a string to form a simple pendulum. The period of its harmonic motion is measured for small angular displacements and three lengths. For lengths of 1.000 m, 0.750 m, and 0.500 m, total time intervals for 50 oscillations of 99.8 s, 86.6 s, and 71.1 s are measured with a stopwatch. (a) Determine the period of motion for each length. (b) Determine the mean value of g obtained from these three independent measurements and compare it with the accepted value. (c) Plot T2T2 versus L and obtain a value for g from the slope of your best-fit straight-line graph. (d) Compare the value found in part (c) with that obtained in part (b). • A thin, cylindrical rod ℓ=24.0cmℓ=24.0cm long with mass m=1.20kgm=1.20kg has a ball of diameter d=8.00cmd=8.00cm and mass M=2.00kgM=2.00kg attached to one end. The arrangement is originally vertical and stationary, with the ball at the top as shown in Figure P10.50.P10.50. The combination is free to pivot about the bottom end of the rod after being given a slight nudge. (a) After the combination rotates through 90 degrees, what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed of the center of mass of the ball? (d) How does it compare with the speed had the ball fallen freely through the same distance of 28 cmcm ? • A “Fe nucleus at rest emits a 14.0 -keV photon. Use conservation of energy and momentum to find the kinetic energy of the recoiling nucleus in electron volts. Use Mc2=8.60×Mc2=8.60× 10−9J10−9J for the final state of the “Fe nucleus. • An airplane is cruising at altitude 10$\mathrm{km}$. The pressure outside the craft is 0.287 atm; within the passenger com- partment, the pressure is 1.00$\mathrm{atm}$and the temperature is$20^{\circ} \mathrm{C}$. A small leak occurs in one of the window seals in the passenger compartment. Model the air as an ideal fluid to estimate the speed of the airstream flowing through the leak. • An aging coyote cannot run fast enough to catch a roadrunner. He purchases on eBay a set of jet-powered roller skates, which provide a constant horizontal acceleration of 15.0 m/s2m/s2 (Fig. P4.62). The coyote starts at rest 70.0 m from the edge of a cliff at the instant the road-runner zips past in the direction of the cliff. (a) Determine the minimum constant speed the roadrunner must have to reach the cliff before the coyote. At the edge of the cliff, the roadrunner escapes by making a sudden turn, while the coyote continues straight ahead. The coyote’s skates remain horizontal and continue to operate while he is in flight, so his acceleration while in the air is (15.0ˆi−9.80ˆj)m/s2(15.0i^−9.80j^)m/s2 . (b) The cliff is 100 mm above the flat floor of the desert. Determine how far from the base of the vertical cliff the coyote lands. (c) Determine the components of the coyote’s impact velocity. • An AC source with ΔVmax=150VΔVmax=150V and f=50.0Hzf=50.0Hz is connected between points aa and dd in Figure P33.24P33.24 Calculate the maximum voltages between (a) points aa and b,b, (b) points bb and c,(c)c,(c) points cc and d,d, and (d) points bb and d.d. • The intensity of sunlight at the Earth’s distance from the Sun is 1370W/m2.1370W/m2. Assume the Earth absorbs all the sunlight incident upon it. (a) Find the total force the Sun exerts on the Earth due to radiation pressure. (b) Explain how this force compares with the Sun’s gravitational attraction. • A space station, in the form of a wheel 120 mm in diameter, rotates to provide an “artificial gravity” of 3.00 m/s2m/s2 for persons who walk around on the inner wall of the outer rim. Find the rate of the wheel’s rotation in revolutions per minute that will produce this effect. • As in Example 28.2, consider a power supply with fixed emf e and internal resistance r causing current in a load resistance R. In this problem, R is fixed and r is a variable. The efficiency is defined as the energy delivered to the load divided by the energy delivered by the emf. (a) When the internal resistance is adjusted for maximum power transfer, what is the efficiency? (b) What should be the internal resistance for maximum possible efficiency? (c) When the electric company sells energy to a customer, does it have a goal of high efficiency or of maximum power transfer? Explain. (d) When a student connects a loudspeaker to an amplifier, does she most want high efficiency or high power transfer? Explain. • A quantum simple harmonic oscillator consists of an electron bound by a restoring force proportional to its position relative to a certain equilibrium point. The proportionality constant is 8.99 N/m. What is the longest wavelength of light that can excite the oscillator? • This problem extends the reasoning of Section 26.4 , Problem 36 in Chapter 26,26, Problem 38 in Chapter 30,30, and Section 32.3 . (a) Consider a capacitor with vacuum between its large, closely spaced, oppositely charged parallel plates. Show that the force on one plate can be accounted for by thinking of the electric field between the plates as exerting a “negative pressure” equal to the energy density of the electric field. (b) Consider two infinite plane sheets carrying electric currents in opposite directions with equal linear current densities JsJs . Calculate the force per area acting on one sheet due to the magnetic field, of magnitude μ0Js/2,μ0Js/2, created by the other sheet. (c) Calculate the net magnetic field between the sheets and the field outside of the volume between them. (d) Calculate the energy density in the magnetic field between the sheets. (e) Show that the force on one sheet can be accounted for by thinking of the magnetic field between the sheets as exerting a positive pressure equal to its energy density. This result for magnetic pressure applies to all current configurations, not only to sheets of current. • Astronauts on a distant planet toss a rock into the air. With the aid of a camera that takes pictures at a steady rate, they record the rock’s height as a function of time as given in the following table. (a) Find the rock’s average velocity in the time interval between each measurement and the next. (b) Using these average velocities to approximate instantaneous velocities at the midpoints of the time intervals, make a graph of velocity as a function of time. (c) Does the rock move with constant acceleration? If so, plot a straight line of best fit on the graph and calculate its slope to find the acceleration. • A 33 H nucleus beta decays into 33 He by creating an electron and an antineutrino according to the reaction 31H→32He+e−+¯ν31H→32He+e−+ν¯¯¯ Determine the total energy released in this decay. • What is the order of magnitude of the number of protons in your body? (b) Of the number of neutrons? (c) Of the number of electrons? • The red light emitted by a helium-neon laser has a wave- length of 632.8nm.632.8nm. What is the frequency of the light waves? • A time-dependent force, →F=(8.00ˆi−4.00tˆj),F→=(8.00i^−4.00tj^), where →FF→ is in newtons and tt is in seconds, is exerted on a 2.00 -kg object initially at rest. (a) At what time will the object be moving with a speed of 15.0 m/sm/s (b) How far is the object from its initial position when its speed is 15.0 m/sm/s ? (c) Through what total displacement has the object traveled at this moment? • Why is the following situation impossible? A worker in a factory pulls a cabinet across the floor using a rope as shown in Figure P12.50a. The rope make an angle θ=37.0∘θ=37.0∘ with the floor and is tied h1=10.0cmh1=10.0cm from the bottom of the cabinet. The uniform rectangular cabinet has height ℓ=ℓ= 100 cmcm and width w=60.0cm,w=60.0cm, and it weighs 400 NN . The cabinet slides with constant speed when a force F=300NF=300N is applied through the rope. The worker tires of walking backward. He fastens the rope to a point on the cabinet h2=65.0cmh2=65.0cm off the floor and lays the rope over his shoulder so that he can walk forward and pull as shown in Figure P12.50bP12.50b . In this way, the rope again makes an angle of θ=37.0∘θ=37.0∘ with the horizontal and again has a tension of 300 NN . Using this technique, the worker is able to slide the cabinet over a long distance on the floor without tiring. • Why is the following situation impossible? A laser beam strikes one end of a slab of material of length L=42.0cmL=42.0cm and thickness t=3.10mmt=3.10mm as shown in Figure P35.54P35.54 (not to scale). It enters the material at the center of the left end, striking it at an angle of incidence of θ=50.0∘.θ=50.0∘. The index of refraction of the slab is n=1.48n=1.48 . The light makes 85 internal reflections from the top and bottom of the slab before exiting at the other end. • An ideal gas is enclosed in a cylinder that has a movable piston on top. The piston has a mass mm and an area AA and is free to slide up and down, keeping the pressure of the gas constant. How much work is done on the gas as the temperature of nn mol of the gas is raised from T1T1 to T2?T2? • Why is the following situation impossible? A technician is measuring the index of refraction of a solid material by observing the polarization of light reflected from its surface. She notices that when a light beam is projected from air onto the material surface, the reflected light is totally polarized parallel to the surface when the incident angle is 41.0°. • A copper wire has a circular cross section with a radius of 1.25 mmmm . ( a) If the wire carries a current of 3.70 AA , find the drift speed of the electrons in this wire. (b) All other things being equal, what happens to the drift speed in wires made of metal having a larger number of conduction electrons per atom than copper? Explain. • The accompanying table shows data obtained in a photo electric experiment. (a) Using these data, make a graph similar to Active Figure 40.11 that plots as a straight line. From the graph, determine (b) an experimental value for Planck’s constant (in joule-seconds) and (c) the work function (in electron volts) for the surface. (Two significant figures for each answer are sufficient.) Wavelength(nm)588505445399Maximum Kinetic Energyof Photoelectrons (eV)0.670.981.351.63WavelengthMaximum Kinetic Energy(nm)of Photoelectrons (eV)5880.675050.984451.353991.63 • Can the circuit shown in Figure P28.31P28.31 be reduced to a single resistor connected to the battery? Explain. Calculate the currents (b) I1,(c)I2,I1,(c)I2, and (d) I3.I3. • In a Newton’s-rings experiment, a plano-convex glass (n=(n= 1.52 ) lens having radius r=5.00cmr=5.00cm is placed on a flat plate as shown in Figure P37.59P37.59 . When light of wavelength λ=λ= 650 nmnm is incident normally, 55 bright rings are observed, with the last one precisely on the edge of the lens. (a) What is the radius RR of curvature of the convex surface of the lens? (b) What is the focal length of the lens? • An electron is represented by the time-independent wave function ψ(x)={Ae−axforx>0Ae+axforx<0 (a) Sketch the wave function as a function of x . (b) Sketch the probability density representing the likelihood that the electron is found between x and x+dx. (c) Only an infinite value of potential energy could produce the discontinuity in the derivative of the wave function at x=0 . Aside from this feature, argue that ψ(x) can be a physically reasonable wave function. (d) Normalize the wave function. (e) Determine the probability of finding the electron somewhere in the range −12α≤x≤12α • A series RLC circuit in which R=1.00Ω,L=1.00mH, and C=1.00nF is connected to an AC source delivering 1.00 V (rms). (a) Make a precise graph of the power delivered to the circuit as a function of the frequency and (b) verify that the full width of the resonance peak at half-maximum is R/2πL. • A jet airliner, moving initially at 300 mi/h to the east, suddenly enters a region where the wind is blowing at 100 mi/h toward the direction 30.0° north of east. What are the new speed and direction of the aircraft relative to the ground? • In a region far from the epicenter of an earthquake, a seismic wave can be modeled as transporting energy in a single direction without absorption, just as a string wave does. Suppose the seismic wave moves from granite into mudfill with similar density but with a much smaller bulk modulus. Assume the speed of the wave gradually drops by a factor of 25.0 , with negligible reflection of the wave. (a) Explain whether the amplitude of the ground shaking will increase or decrease. (b) Does it change by a predictable factor? (This phenomenon led to the collapse of part of the Nimitz Freeway in Oakland, California, during the Loma Prieta earthquake of 1989.1989. ) • Blaise Pascal duplicated Torricelli’s barometer using a red Bordeaux wine, of density$984 \mathrm{kg} / \mathrm{m}^{3},$as the working liquid (Fig. Pl4.16). (a) What was the height$h$of the wine column for normal atmospheric pressure? (b) Would you expect the vacuum above the column to be as good as for mercury? • A dive-bomber has a velocity of 280 m/sm/s at an angle θθ below the horizontal. When the altitude of the aircraft is 2.15km,2.15km, it releases a bomb, which subsequently hits a target on the ground. The magnitude of the displacement from the point of release of the bomb to the target is 3.25km.3.25km. Find the angle θ.θ. • Five particles with equal negative charges −q−q are placed symmetrically around a circle of radius R.R. Calculate the electric potential at the center of the circle. • An amusement park ride consists of a large vertical cylinder that spins about its axis fast enough that any person inside is held up against the wall when the floor drops away (Fig. P6.59). The coefficient of static friction between person and wall is μs,μs, and the radius of the cylinder is R.R. (a) Show that the maximum period of revolution necessary to keep the person from falling is T=(4π2Rμδ/g)1/2.T=(4π2Rμδ/g)1/2. (b) If the rate of revolution of the cylinder is made to be somewhat larger, what happens to the magnitude of each one of the forces acting on the person? What happens in the motion of the person? (c) If the rate of revolution of the cylinder is instead made to be somewhat smaller, what happens to the magnitude of each one of the forces acting on the person? How does the motion of the person change? • In a Geiger tube, the voltage between the electrodes is typically 1.00 kVkV and the current pulse discharges a 5.00−pF5.00−pF capacitor. (a) What is the energy amplification of this device for a 0.500 -MeV electron? (b) How many electrons participate in the avalanche caused by the single initial electron? • If A spherical shell with inner radius rara and outer radius rbrb is formed from a material of resistivity ρρ . It carries current radially, with uniform density in all directions. Show that its resistance is R=ρ4π(1ra−1rb)R=ρ4π(1ra−1rb) • When a metal bar is connected between a hot reservoir at TkTk and a cold reservoir at ToTo the energy transferred by heat from the hot reservoir to the cold reservoir is QQ . In this irreversible process, find expressions for the change in entropy of (a) the hot reservoir, (b) the cold reservoir, and (c) the Universe, neglecting any change in entropy of the metal rod. • Neutron stars are extremely dense objects formed from the remnants of supernova explosions. Many rotate very rapidly. Suppose the mass of a certain spherical neutron star is twice the mass of the Sun and its radius is 10.0 km. Determine the greatest possible angular speed it can have so that the matter at the surface of the star on its equator is just held in orbit by the gravitational force. • A loop of wire in the shape of a rectangle of width ww and length LL and a long, straight wire carrying a current II lie on a tabletop as shown in Figure P31.11. (a) Determine the magnetic flux through the loop due to the current II (b) Suppose the current is changing with time according to I=a+bt,I=a+bt, where aa and bb are constants. Determine the emf that is induced in the loop if b=10.0A/s,h=1.00cm,b=10.0A/s,h=1.00cm, w=10.0cm,w=10.0cm, and L=1.00m.L=1.00m. (c) What is the direction of the induced current in the rectangle? • How much diffraction spreading does a light beam undergo? One quantitative answer is the full width at half maximum of the central maximum of the single-slit Fraun-hofer diffraction pattern. You can evaluate this angle of spreading in this problem. (a) In Equation 38.2,38.2, define ϕ=ϕ= πasinθ/λπasin⁡θ/λ and show that at the point where I=0.5ImaxI=0.5Imax we must have ϕ=2–√sinϕ.ϕ=2sin⁡ϕ. (b) Let y1=sinϕy1=sin⁡ϕ and y2=ϕ/2–√y2=ϕ/2 . Plot y1y1 and y2y2 on the same set of axes over a range from ϕ=ϕ= 1 rad to ϕ=π/2ϕ=π/2 rad. Determine ϕϕ from the point of inter-section of the two curves. (c) Then show that if the fraction λ/aλ/a is not large, the angular full width at half maximum of the central diffraction maximum is θ=0.885λ/aθ=0.885λ/a . (d) What If? Another method to solve the transcendental equation ϕ=2–√sinϕϕ=2sin⁡ϕ in part (a) is to guess a first value of ϕ,ϕ, use a computer or calculator to see how nearly it fits, and continue to update your estimate until the equation balances. How many steps (iterations) does this process take? • A helium-neon laser (λ=632.8nm)(λ=632.8nm) is used to calibrate a diffraction grating. If the first-order maximum occurs at 20.5∘,20.5∘, what is the spacing between adjacent grooves in the grating? • A circular radar antenna on a Coast Guard ship has a diameter of 2.10 m and radiates at a frequency of 15.0 GHz. Two small boats are located 9.00 km away from the ship. How close together could the boats be and still be detected as two objects? • Why is the following situation impossible? At a blind corner in an outdoor shopping mall, a convex mirror is mounted so pedestrians can see around the corner before arriving there and bumping into someone traveling in the perpendicular direction. The installers of the mirror failed to take into account the position of the Sun, and the mirror focuses the Sun’s rays on a nearby bush and sets it on fire. • Two small spheres of mass mm are suspended from strings of length ℓℓ that are connected at a common point. One sphere has charge QQ and the other charge 2Q.2Q. The strings make angles θ1θ1 and θ2θ2 with the vertical. (a) Explain how θ1θ1 and θ2θ2 are related. (b) Assume θ1θ1 and θ2θ2 are small. Show that the distance rr between the spheres is approximately r≈(4keQ2ℓmg)1/3r≈(4keQ2ℓmg)1/3 • Coherent light of wavelength 501.5 nm is sent through two parallel slits in an opaque material. Each slit is 0.700μmμm wide. Their centers are 2.80μmμm apart. The light then falls on a semi cylindrical screen, with its axis at the mid line between the slits. We would like to describe the appearance of the pattern of light visible on the screen. (a) Find the direction for each two-slit interference maximum on the screen as an angle away from the bisector of the line joining the slits. (b) How many angles are there that represent two-slit interference maxima? (c) Find the direction for each single-slit interference minimum on the screen as an angle away from the bisector of the line joining the slits. (d) How many angles are there that represent single-slit interference minima? (e) How many of the angles in part (d) are identical to those in part (a)? (f) How many bright fringes are visible on the screen? (g) If the intensity of the central fringe is Imax,Imax, what is the intensity of the last fringe visible on the screen? • Four parallel metal plates P1,P2,P3,P1,P2,P3, and P4,P4, each of area 7.50cm2,7.50cm2, are separated successively by a distance d=d= 1.19 mmmm as shown in Figure P26.55.P26.55. Plate P1P1 is connected to the negative terminal of a battery, and P2P2 is connected to the positive terminal. The battery maintains a potential difference of 12.0 VV . (a) If P3P3 connected to the negative terminal, what is the capacitance of the three-plate system P1P2P3?P1P2P3? (b) What is the charge on P2?(c)P2?(c) If P4P4 is now connected to the positive terminal, what is the capacitance of the four-plate system P1P2P3P4?P1P2P3P4? (d) What is the charge on P4?P4? • An object of mass m=m= 5.00 kg, attached to a spring scale, rests on a frictionless, horizontal surface as shown in Figure P6.20. The spring scale, attached to the front end of a boxcar, reads zero when the car is at rest. (a) Determine the acceleration of the car if the spring scale has a constant reading of 18.0 N when the car is in motion. (b) What constant reading will the spring scale show if the car moves with constant velocity? Describe the forces on the object as observed (c) by someone in the car and (d) by someone at rest outside the car. • An underwater scuba diver sees the Sun at an apparent angle of 45.0∘0∘ above the horizontal. What is the actual elevation angle of the Sun above the horizontal? • Use the definition of the vector product and the definitions of the unit vectors ˆi,ˆj,i^,j^, and ˆkk^ to prove Equations 11.7.11.7. You may assume the xx axis points to the right, the yy axis up, and the zz axis horizontally toward you (not away from you). This choice is said to make the coordinate system a right-handed system. • Energy is to be transmitted over a pair of copper wires in a transmission line at a rate PP with only a fractional loss ff over a distance ℓℓ at potential difference ΔVrmsΔVrms between the wires. Assuming the current density is uniform in the conductors, what is the diameter required for each of the two wires? • The work function for zinc is 4.31 eV. (a) Find the cutoff wavelength for zinc. (b) What is the lowest frequency of light incident on zinc that releases photoelectrons from its surface? (C) If photons of energy 5.50 eV are incident on zinc, what is the maximum kinetic energy of the ejected photoelectrons? • Native people throughout North and South America used a bola to hunt for birds and animals. A bola can consist of three stones, each with mass m,m, at the ends of three light cords, each with length ℓ.ℓ. The other ends of the cords are tied together to form a Y. The hunter holds one stone and swings the other two above his head (Figure P 11.57 a). Both these stones move together in a horizontal circle of radius 2ℓℓ with speed v0v0 . At a moment when the horizontal component of their velocity is directed toward the quarry, the hunter releases the stone in his hand. As the bola flies through the air, the cords quickly take a stable arrangement with constant 120-degree angles between them (Fig. P 11.57 b). In the vertical direction, the bola is in free fall. Gravitational forces exerted by the Earth make the junction of the cords move with the downward acceleration g→g→ . You may ignore the vertical motion as you proceed to describe the horizontal motion of the bola. In terms of m,ℓ,m,ℓ, and v0,v0, calculate (a) the magnitude of the momentum of the bola at the moment of release and, after release, (b) the horizontal speed of the center of mass of the bola and (c) the angular momentum of the bola about its center of mass. (d) Find the angular speed of the bola about its center of mass after it has settled into its Y shape. Calculate the kinetic energy of the bola (e) at the instant of release and (f) in its stable Y shape. (g) Explain how the conservation laws apply to the bola as its configuration changes. Robert Beichner suggested the idea for this problem. • The distance between an object and its upright image is$d .$If the magnification is$M,$what is the focal length of the lens being used to form the image? • An unstable particle with mass m=3.34×10−27kgm=3.34×10−27kg is initially at rest. The particle decays into two fragments that fly off along the xx axis with velocity components u1=u1= 0.987cc and u2=−0.868cu2=−0.868c . From this information, we wish to determine the masses of fragments 1 and 2.2. (a) Is the initial system of the unstable particle, which becomes the system of the two fragments, isolated or nonisolated? (b) Based on your answer to part (a), what two analysis models are appropriate for this situation? (c) Find the values of γγ for the two fragments after the decay. (d) Using one of the analysis models in part (b), find a relationship between the masses m1m1 and m2m2 of the fragments. (e) Using the second analysis model in part (b), find a second relationship between the masses m1m1 and m2.m2. (f) Solve the relationships in parts (d) and (e) simultaneously for the masses m1m1 and m2m2 . • The expression F=arv+br2v2F=arv+br2v2 gives the magnitude of the resistive force (in newtons) exerted on a sphere of radius rr (in meters) by a stream of air moving at speed vv (in meters per second), where aa and bb are constants with appropriate SI units. Their numerical values are a=3.10×10−4a=3.10×10−4 and b=0.870.b=0.870. Using this expression, find the terminal speed for water droplets falling under their own weight in air, taking the following values for the drop radii: (a) 10.0μmμm , (b) 100μm,100μm, (c) 1.00 mmmm . For parts (a) and (c), you can obtain accurate answers without solving a quadratic equation by considering which of the two contributions to the air resistance is dominant and ignoring the lesser contribution. • A linearly polarized microwave of wavelength 1.50 cm is directed along the positive x axis. The electric field vector has a maximum value of 175 V/m and vibrates in the xy plane. Assuming the magnetic field component of the wave can be written in the form B=Bmaxsin(kx−ωt)B=Bmaxsin⁡(kx−ωt) give values for (a) Bmax,Bmax, (b) k,k, and (c)(c) \omega. (d) Determine in which plane the magnetic field vector vibrates. (e) Calculate the average value of the Poynting vector for this wave. (f) If this wave were directed at normal incidence onto a perfectly reflecting sheet, what radiation pressure would it exert? (g) What acceleration would be imparted to a 500-g sheet (perfectly reflecting and at normal incidence) with dimensions of 1.00m×0.750m1.00m×0.750m ? • A lobsterman’s buoy is a solid wooden cylinder of radius r and mass M. It is weighted at one end so that it floats upright in calm seawater, having density r. A passing shark tugs on the slack rope mooring the buoy to a lobster trap, pulling the buoy down a distance x from its equilibrium position and releasing it. (a) Show that the buoy will execute simple harmonic motion if the resistive effects of the water are ignored. (b) Determine the period of the oscillations. • A particle of mass 2.00×10−28kg is confined to a one- dimensional box of length 1.00×10−10m. For n=1, what are (a) the particle’s wavelength, (b) its momentum, and (c) its ground-state energy? • Show that the proton-decay p→e++γp→e++γ cannot occur because it violates the conservation of baryon number. (b) What If? Imagine that this reaction does occur and the proton is initially at rest. Determine the energies and magnitudes of the momentum of the positron and photon after the reaction. (c) Determine the speed of the positron after the reaction. • An object of mass MM is held in place by an applied force FF and a pulley system as shown in Figure P5.61.P5.61. The pulleys are massless and frictionless. (a) Draw diagrams showing the forces on each pulley. Find (b) the tension in each section of rope, T1,T2,T1,T2, T3,T4,T3,T4, and T5T5 and (c)(c) the magnitude of →F.F→. • Why is the following situation impossible? A team of researchers discovers a new gas, which has a value of γ=CP/CVγ=CP/CV of 1.75.1.75. • One technique for measuring the apex angle of a prism is shown in Figure P35.50. Two parallel rays of light are directed onto the apex of the prism so that the rays reflect the prism. The angular separation γγ of the two reflected rays can be measured. Show that ϕ=12γϕ=12γ . • Many machines employ cams for various purposes, such as opening and closing valves. In Figure P10.32, the cam is a circular disk of radius RR with a hole of diameter RR cut through it. As shown in the figure, the hole does not pass through the center of the disk. The cam with the hole cut out has mass MM . The cam is mounted on a uniform, solid, cylindrical shaft of diameter RR and also of mass MM . What is the kinetic energy of the cam-shaft combination when it is rotating with angular speed ωω about the shaft’s axis? • A hollow, metallic, spherical shell has exterior radius 0.750 m, carries no net charge, and is supported on an insulating stand. The electric field everywhere just outside its surface is 890 N/C radially toward the center of the sphere. Explain what you can conclude about (a) the amount of charge on the exterior surface of the sphere and the distribution of this charge, (b) the amount of charge on the interior surface of the sphere and its distribution, and (c) the amount of charge inside the shell and its distribution. • Two identical particles, each of mass 1000 kgkg , are coasting in free space along the same path, one in front of the other by 20.0 mm . At the instant their separation distance has this value, each particle has precisely the same velocity of 800 im/sim/s . What are their precise velocities when they are 2.00 mm apart? • Determine the work done on a gas that expands from ii to ff as indicated in Figure P20.24P20.24 . (b) What If? How much work is done on the gas if it is compressed from ff to ii along the same path? • As the driver steps on the gas pedal, a car of mass 1 160 kg accelerates from rest. During the first few seconds of motion, the car’s acceleration increases with time according to the expression a=1.16t−0.210t2+0.240t3a=1.16t−0.210t2+0.240t3 where tt is in seconds and aa is in m/s2m/s2 . (a) What is the change in kinetic energy of the car during the interval from t=0t=0 to t=2.50s2t=2.50s2 (b) What is the minimum average power out- put of the engine over this time interval? (c) Why is the value in part (b) described as the minimum value? • In Niels Bohr’s 1913 model of the hydrogen atom, an electron circles the proton at a distance of 5.29×10−11m5.29×10−11m with a speed of 2.19×106m/s2.19×106m/s . Compute the magnitude of the magnetic field this motion produces at the location of the proton. • For the arrangement described in Problem 42, calculate the electric potential at point B,B, which lies on the perpendicular bisector of the rod a distance bb above the xx axis. • A spherical balloon of volume 4.00×103cm34.00×103cm3 contains helium at a pressure of 1.20×105Pa1.20×105Pa . How many moles of helium are in the balloon if the average kinetic energy of the helium atoms is 3.60×10−22J3.60×10−22J ? • Why is the following situation impossible? After learning about de Broglie’s hypothesis that material particles of momentum pp move as waves with wavelength λ=h/p,λ=h/p, an 80−kg80−kg student has grown concerned about being diffracted when passing through a doorway of width w=75cm.w=75cm. Assume significant diffraction occurs when the width of the diffraction aperture is less than ten times the wavelength of the wave being diffracted. Together with his classmates, the student performs precision experiments and finds that he does indeed experience measurable diffraction. • A 7.00 -L vessel contains 3.50 moles of gas at a pressure of 1.60×106Pa1.60×106Pa . Find (a)(a) the temperature of the gas and (b)(b) the average kinetic energy of the gas molecules in the vessel. (c) What additional information would you need if you were asked to find the average speed of the gas molecules? • In addition to cable and satellite broadcasts, television stations still use VHFVHF and UHF bands for digitally broadcasting their signals. Twelve VHF television channels (channels 2 through 13)) lie in the range of frequencies between 54.0 MHzMHz and 216 MHzMHz . Each channel is assigned a width of 6.00MHz,6.00MHz, with the two ranges 72.0−76.0MHz72.0−76.0MHz and 88.0−88.0− 174 MHz reserved for non-TV purposes. (Channel 2, for example, lies between 54.0 and 60.0 MHz.) Calculate the broadcast wavelength range for (a) channel 4,4, (b) channel 6,6, and (c) channel 8. • A projectile of mass m moves to the right with a speed vivi (Fig. P 11.51 a). The projectile strikes and sticks to the end of a stationary rod of mass M,M, length d,d, pivoted about a friction less axle perpendicular to the page through OO (Fig. P 11.51 b). We wish to find the fractional change of kinetic energy in the system due to the collision. (a) What is the appropriate analysis model to describe the projectile and the rod? (b) What is the angular momentum of the system before the collision about an axis through OO? (c) What is the moment of inertia of the system about an axis through OO after the projectile sticks to the rod? (d) If the angular speed of the system after the collision is ww, what is the angular momentum of the system after the collision? (e) Find the angular speed ww after the collision in terms of the given quantities. (f) What is the kinetic energy of the system before the collision? (g) What is the kinetic energy of the system after the collision? (h) Determine the fractional change of kinetic energy due to the collision. • Figure CQ37.2CQ37.2 shows an unbroken soap film in a circular frame. The film thickness increases from top to bottom, slowly at first and then rapidly. As a simpler model, consider a soap film (n=1.33)(n=1.33) contained within a rectangular wire frame. The frame is held vertically so that the film drains downward and forms a wedge with flat faces. The thickness of the film at the top is essentially zero. The film is viewed in reflected white light with near-normal incidence, and the first violet (λ=420nm)(λ=420nm) interference band is observed 3.00 cmcm from the top edge of the film. (a) Locate the first red (λ=680nm)(λ=680nm) interference band. (b) Determine the film thickness at the positions of the violet and red bands. (c) What is the wedge angle of the film? • According to classical physics, a charge ee moving with an acceleration aa radiates energy at a rate dEdt=−16πϵ0e2a2c3dEdt=−16πϵ0e2a2c3 (a) Show that an electron in a classical hydrogen atom (see 42.5 ) spirals into the nucleus at a rate drdt=−e412π2ϵ20m2ec3(1r2)drdt=−e412π2ϵ20m2ec3(1r2) (b) Find the time interval over which the electron reaches r=0,r=0, starting from r0=2.00×10−10mr0=2.00×10−10m • A uniform electric field of magnitude 640 N/CN/C exists between two parallel plates that are 4.00 cmcm apart. A proton is released from rest at the positive plate at the same instant an electron is released from rest at the negative plate. (a) Determine the distance from the positive plate at which the two pass each other. Ignore the electrical attraction between the proton and electron. (b) What If? Repeat part (a) for a sodium ion (Na+)(Na+) and a chloride ion (Cl−)(Cl−) . • A spool of wire of mass MM and radius RR is unwound under a constant force F→F→ (Fig. PIO. 76)) . Assuming the spool is a uniform, solid cylinder that doesn’t slip, show that (a) the acceleration of the center of mass is 4 F→/3MF→/3M and (b)(b) the force of friction is to the right and equal in magnitude to F/3.F/3. (c) If the cylinder starts from rest and rolls without slipping, what is the speed of its center of mass after it has rolled through a distance dd ? • In places such as hospital operating rooms or factories for electronic circuit boards, electric sparks must be avoided. A person standing on a grounded floor and touching nothing else can typically have a body capacitance of 150 pF, in parallel with a foot capacitance of 80.0 pF produced by the dielectric soles of his or her shoes. The person acquires static electric charge from interactions with his or her surroundings. The static charge flows to ground through the equivalent resistance of the two shoe soles in parallel with each other. A pair of rubber-soled street shoes can present an equivalent resistance of 5.00 3 103 MV. A pair of shoes with special static-dissipative soles can have an equivalent resistance of 1.00 MV. Consider the person’s body and shoes as forming an RC circuit with the ground. (a) How long does it take the rubber-soled shoes to reduce a person’s potential from 3.00×103V3.00×103V to 100 V?V? (b) How long does it take the static-dissipative shoes to do the same thing? • A converging lens has a focal length of$10.0 \mathrm{cm} .$Locate the object if a real image is located at a distance from the lens of$(\mathrm{a}) 20.0 \mathrm{cm}$and$(\mathrm{b}) 50.0 \mathrm{cm} .$What If? Redo the calculations if the images are virtual and located at a distance from the lens of (c) 20.0$\mathrm{cm}$and (d)$50.0 \mathrm{cm} .$• A bismuth target is struck by electrons, and x-rays are emitted. Estimate (a) the M- to L-shell transitional energy for bismuth and (b) the wavelength of the x-ray emitted when an electron falls from the M shell to the L shell. • A microwave oven is powered by a magnetron, an electronic device that generates electromagnetic waves of frequency 2.45 GHz. The microwaves enter the oven and are reflected by the walls. The standing-wave pattern produced in the oven can cook food unevenly, with hot spots in the food at antinodes and cool spots at nodes, so a turntable is often used to rotate the food and distribute the energy. If a microwave oven intended for use with a turntable is instead used with a cooking dish in a fixed position, the antinodes can appear as burn marks on foods such as carrot strips or cheese. The separation distance between the burns is measured to be 6cm±5%. From these data, calculate the speed of the microwaves. • A thin rod of length hh and mass MM is held vertically with its lower end resting on a frictionless, horizontal surface. The rod is then released to fall freely. (a) Determine the speed of its center of mass just before it hits the horizontal surface. (b) What If? Now suppose the rod has a fixed pivot at its lower end. Determine the speed of the rod’s center of mass just before it hits the surface. • Identical thin rods of length 2aa carry equal charges +Q uniformly distributed along their lengths. The rods lie along the xx axis with their centers separated by a distance b>2a(Fig.P23.75)b>2a(Fig.P23.75) . Show that the magnitude of the force exerted by the left rod on the right one is F=(keQ24a2)ln(b2b2−4a2)F=(keQ24a2)ln⁡(b2b2−4a2) • A material having an index of refraction nn is surrounded by vacuum and is in the shape of a quarter circle of radius R(Fig.P35.63).R(Fig.P35.63). A light ray parallel to the base of the material is incident from the left at a distance LL above the base and emerges from the material at the angle θ.θ. Determine an expression for θθ in terms of n,R,n,R, and L.L. • Consider a common mirage formed by superheated air immediately above a roadway. A truck driver whose eyes are 2.00 mm above the road, where n=1.000293,n=1.000293, looks forward. She perceives the illusion of a patch of water ahead on the road. The road appears wet only beyond a point on the road at which her line of sight makes an angle of 1.20∘20∘ below the horizontal. Find the index of refraction of the air immediately above the road surface. • A particle in a one-dimensional box of length L is in its first excited state, corresponding to n=2. Determine the probability of finding the particle between x=0 and x= L/4. • A 200 -g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. The total energy of the system is 2.00 JJ . Find (a) the force constant of the spring and (b) the amplitude of the motion. • A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of vi=18.0m/svi=18.0m/s . The cliff is h=50.0mh=50.0m above a body of water as shown in Figure P4.23. ( a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity of the stone? (c) What is the appropriate analysis model-for the vertical motion of the stone? (d) What is the appropriate analysis model for the horizontal motion of the stone? (e) Write symbolic equations for the x and y components of the velocity of the stone as a function of time. (f) Write symbolic equations for the position of the stone as a function of time. (g) How long after being released does the stone strike the water below the cliff? (h) With what speed and angle of impact does the stone land? • Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is one kilocalorie, defined as 1 kcal 5 4 186 J. Metabolizing 1 g of fat can release 9.00 kcal. A student decides to try to lose weight by exercising. He plans to run up and down the stairs in a football stadium as fast as he can and as many times as necessary. To evaluate the program, suppose he runs up a flight of 80 steps, each 0.150 m high, in 65.0 s. For simplicity, ignore the energy he uses in coming down (which is small). Assume a typical efficiency for human muscles is 20.0%. This statement means that when your body converts 100 J from metabolizing fat, 20 J goes into doing mechanical work (here, climbing stairs). The remainder goes into extra internal energy. Assume the student’s mass is 75.0 kg. (a) How many times must the student run the flight of stairs to lose 1.00 kg of fat? (b) What is his average power output, in watts and in horsepower, as he runs up the stairs? (c) Is this activity in itself a practical way to lose weight? • A worker pushing a 35.0-kg wooden crate at a constant speed for 12.0 m along a wood floor does 350 J of work by applying a constant horizontal force of magnitude F on the crate. (a) Determine the value of F . (b) If the worker now applies a force greater than F , describe the subsequent motion of the crate. (c) Describe what would happen to the crate if the applied force is less than F . • The speed of the Earth in its orbit is 29.8 km/skm/s . If that is the magnitude of the velocity →vv→ of the ether wind in Figure P39.3,P39.3, find the angle ϕϕ between the velocity of light →cc→ in vacuum and the resultant velocity of light if there were an ether. • Calculate the magnitude of the magnetic field at a point 25.0 cmcm from a long, thin conductor carrying a current of 2.00A.2.00A. • A 140−mH140−mH inductor and a 4.90−Ω4.90−Ω resistor are connected with a switch to a 6.00−V6.00−V battery as shown in Figure P32.27P32.27 (a) After the switch is first thrown to aa (connecting the battery), what time interval elapses before the current reaches 220 mAmA ? (b) What is the current in the inductor 10.0 ss after the switch is closed? (c) Now the switch is quickly thrown from aa to b.b. What time interval elapses before the current in the inductor falls to 160 mAmA ? • A car of mass 2 000 kg moving with a speed of 20.0 m/s collides and locks together with a 1 500-kg car at rest at a stop sign. Show that momentum is conserved in a reference frame moving at 10.0 m/s in the direction of the moving car. • An electron of momentum pp is at a distance rr from a stationary proton. The electron has kinetic energy K=p2/2me.K=p2/2me. The atom has potential energy U=−kee2/rU=−kee2/r and total energy E=K+U.E=K+U. If the electron is bound to the proton to form a hydrogen atom, its average position is at the proton but the uncertainty in its position is approximately equal to the radius rr of its orbit. The elec- tron’s average vector momentum is zero, but its average squared momentum is approximately equal to the squared uncertainty in its momentum as given by the uncertainty principle. Treating the atom as a one-dimensional system, (a) estimate the uncertainty in the electron’s momentum in terms of rr . Estimate the electron’s (b) kinetic energy and (c) total energy in terms of rr . The actual value of rr is the one that minimizes the total energy, resulting in a stable atom. Find (d) that value of rr and (e) the resulting total energy. (f) State how your answers compare with the predictions of the Bobr theory. • Eight molecules have speeds of 3.00km/s,4.00km/s3.00km/s,4.00km/s 80km/s,2.50km/s,3.60km/s,1.90km/s,3.80km/s5.80km/s,2.50km/s,3.60km/s,1.90km/s,3.80km/s and 6.60 km/skm/s . Find (a) the average speed of the molecules and (b)(b) the rms speed of the molecules. • Figure P33.80 a shows a parallel RLC circuit. The instantaneous voltages (and rms voltages) across each of the three circuit elements are the same, and each is in phase with the current in the resistor. The currents in C and L lead or lag the current in the resistor as shown in the current phasor diagram, Figure P33.80b . (a) Shown in the rms current delivered by the source is Irms=ΔVrms[1R2+(ωC−1ωL)2]1/2 (b) Show that the phase angle ϕ between ΔVrms and Irms is given by tanϕ=R(1XC−1XL) • Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed vivi . After the collision, the orange disk moves along a direction that makes an angle θθ with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk. • Find the equivalent capacitance between points a and b for the group of capacitors connected as shown in Figure P26.28P26.28 . Take C1=5.00μF,C2=C1=5.00μF,C2= 10.0μF,10.0μF, and C3=2.00μF.C3=2.00μF. (b) What charge is stored on C3C3 if the potential difference between points aa and bb is 60,0V60,0V ? • Fill in the missing particle. Assume reaction ( a) occurs via the strong interaction and reactions (b) and (c) involve the weak interaction. Assume also the total strangeness changes by one unit if strangeness is not conserved. (a) K++p→?+pK++p→?+p (b) Ω−→?+π−Ω−→?+π− (c) K+→?+μ++νμK+→?+μ++νμ • Consider a flat, circular current loop of radius RR carrying a current II . Choose the xx axis to be along the axis of the loop, with the origin at the loop’s center. Plot a graph of the ratio of the magnitude of the magnetic field at coordinate xx to that at the origin for x=0x=0 to x=5Rx=5R . It may be helpful to use a programmable calculator or a computer to solve this problem. • A 5.00-kg particle starts from the origin at time zero. Its velocity as a function of time is given by →v=6t2ˆi+2tˆjv→=6t2i^+2tj^ where →vv→ is in meters per second and tt is in seconds. (a) Find its position as a function of time. (b) Describe its motion qualitatively. Find (c) its acceleration as a function of time, (d) the net force exerted on the particle as a function of time, (e) the net torque about the origin exerted on the particle as a function of time, (f) the angular momentum of the particle as a function of time, (g) the kinetic energy of the particle as a function of time, and (h) the power injected into the system of the particle as a function of time. • As a person moves about in a dry environment, electric charge accumulates on the person’s body. Once it is at high voltage, either positive or negative, the body can discharge via sparks and shocks. Consider a human body isolated from ground, with the typical capacitance 150 pFpF . (a) What charge on the body will produce a potential of 10.0 kVkV ? (b) Sensitive electronic devices can be destroyed by electrostatic discharge from a person. A particular device can be destroyed by a discharge releasing an energy of 250μJμJ . To what voltage on the body does this situation correspond? • A long solenoid, with its axis along the x axis, consists of 200 turns per meter of wire that carries a steady current of 15.0 A. A coil is formed by wrapping 30 turns of thin wire around a circular frame that has a radius of 8.00 cm . The coil is placed inside the solenoid and mounted on an axis that is a diameter of the coil and coincides with the y axis. The coil is then rotated with an angular speed t=0. Determine the emf generated in the coil as a function of time. • Submarine A travels horizontally at 11.0 m/s through ocean water. It emits a sonar signal of frequency f=f= 5.27×103Hz5.27×103Hz in the forward direction. Submarine BB is in front of submarine AA and traveling at 3.00 m/sm/s relative to the water in the same direction as submarine A. A crewman in submarine B uses his equipment to detect the sound waves (‘pings”) from submarine A. We wish to determine what is heard by the crewman in submarine B. (a) An observer on which submarine detects a frequency f′f′ as described by Equation 17.19? (b) In Equation 17.19, should the sign of vSvS be positive or negative? (c) In Equation 17.19,17.19, should the sign of vOvO be positive or negative? (d) In Equation 17.19,17.19, what speed of sound should be used? (e) Find the frequency of the sound detected by the crewman on submarine BB . • A Carnot engine has a power output of 150 kWkW . The engine operates between two reservoirs at 20.0∘0∘C and 500∘C500∘C . (a) How much energy enters the engine by heat per hour? (b) How much energy is exhausted by heat per hour? • A model airplane of mass 0.750 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 60.0-m-long control wire as shown in Figure P6.63a. The forces exerted on the airplane are shown in Figure P6.63b: the tension in the control wire, the gravitational force, and aerodynamic lift that acts at θ=20.0∘θ=20.0∘ inward from the vertical. Compute the tension in the wire, assuming it makes a constant angle of θ=20.0∘θ=20.0∘ with the horizontal. • A science student is riding on a flatcar of a train traveling along a straight, horizontal track at a constant speed of 10.0 m/sm/s . The student throws a ball into the air along a path that he judges to make an initial angle of 60.0∘0∘ with the horizontal and to be in line with the track. The student’s professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise? • A rifle bullet with a mass of 12.0 g traveling toward the right at 260 m/s strikes a large bag of sand and penetrates it to a depth of 23.0cm. Determine the magnitude and direction of the friction force (assumed constant) that acts on the bullet. • The atoms of an NaCl molecule are separated by a distance r=0.280nm.r=0.280nm. Calculate (a) the reduced mass of an NaCl molecule, (b) the moment of inertia of an NaCl molecule, and (c) the wavelength of radiation emitted when an NaCl molecule undergoes a transition from the J=2J=2 state to the J=1J=1 state. • A sound wave in air has a pressure amplitude equal to 4.00×10−34.00×10−3 Pa. Calculate the displacement amplitude of the wave at a frequency of 10.0 kHzkHz . • A young man owns a canister vacuum cleaner marked 2535W2535W [at] 120 VnVn and a Volkswagen Beetle, which he wishes to clean. He parks the car in his apartment parking lot and uses an inexpensive extension cord 15.0 m long to plug in the vacuum cleaner. You may assume the cleaner has constant resistance. (a) If the resistance of each of the two conductors in the extension cord is 0.900 V, what is the actual power delivered to the cleaner? (b) If instead the power is to be at least 525 W, what must be the diameter of each of two identical copper conductors in the cord he buys? (c) Repeat part (b) assuming the power is to be at\ least 532 W. • A certain quaternary star system consists of three stars, each of mass m, moving in the same circular orbit of radius r about a central star of mass M. The stars orbit in the same sense and are positioned one-third of a revolution apart from one another. Show that the period of each of the three stars is given by T=2πr3G(M+m/3–√)−−−−−−−−−−−−−√T=2πr3G(M+m/3) • In the Bohr model of the hydrogen atom (which will be covered in detail in Chapter 42),42), an electron in the lowest energy state moves at a speed of 2.19×106m/s2.19×106m/s in a circular path of radius 5.29×10−11m5.29×10−11m . What is the effective current associated with this orbiting electron? • A spring cannon is located at the edge of a table that is 1.20 mm above the floor. A steel ball is launched from the cannon with speed vivi at 35.0∘0∘ above the horizontal. (a) Find the horizontal position of the ball as a function of vivi at the instant it lands on the floor. We write this function as x(vi)x(vi) Evaluate xx for (b )vi=0.100m/sand forvi=100m/s(b )vi=0.100m/sand forvi=100m/s (d) Assume vivi is close to but not equal to zero. Show that one term in the answer to part (a) dominates so that the function x(vi)x(vi) reduces to a simpler form. (e) If vi is very large, what is the approximate form of x(vi)?(( ) Describe the overall shape of the graph of the function x(vi). • The mass of the deuterium molecule (D2)(D2) is twice that of the hydrogen molecule (H2).(H2). If the vibrational frequency of H2H2 is 1.30×1014Hz1.30×1014Hz , what is the vibrational frequency of D2?D2? Assume the “spring constant” of attracting forces is the same for the two molecules. • A nonconducting sphere has mass 80.0 g and radius 20.0 cm. A flat, compact coil of wire with five turns is wrapped tightly around it, with each turn concentric with the sphere. The sphere is placed on an inclined plane that slopes downward to the left (Fig. P 29.65), making an angle θθ with the horizontal so that the coil is parallel to the inclined plane. A uniform magnetic field of 0.350 T vertically upward exists in the region of the sphere. (a) What current in the coil will enable the sphere to rest in equilibrium on the inclined plane? (b) Show that the result does not depend on the value of θθ. • Two spheres having masses M and 2M and radii R and 3R, respectively, are simultaneously released from rest when the distance between their centers is 12R. Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What two isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for v→1,v→1, the velocity of the sphere of mass MM at any time after release in terms of v→2,v→2, the velocity of 2M.2M. (c) Write an equation from the other model and solve it for speed v1v1 in terms of speed v2v2 when the spheres collide. (d) Combine the two equations to find the two speeds v1v1 and v2v2 when the spheres collide. • An electric generating station is designed to have an electric output power of 1.40 MW using a turbine with two-thirds the efficiency of a Carnot engine. The exhaust cnergy is transferred by heat into a cooling tower at 110∘C110∘C . (a) Find the rate at which the station exhausts energy by heat as a function of the fuel combustion temperature ThTh . (b) If the firebox is modified to run hotter by using more advanced combustion technology, how does the amount of cnergy exhaust change? (c) Find the exhaust power for Th=800∘Th=800∘C. (d) Find the value of ThTh for which the exhaust power would be only half as large as in part (c).(c). (e) Find the value of ThTh for which the exhaust power would be one-fourth as large as in part (c). • Consider an array of parallel wires with uniform spacing of 1.30 cm between centers. In air at 20.0°C, ultrasound with a frequency of 37.2 kHz from a distant source is incident perpendicular to the array. (a) Find the number of directions on the other side of the array in which there is a maximum of intensity. (b) Find the angle for each of these directions relative to the direction of the incident beam. • The magnetic field 40.0 cmcm away from a long, straight wire carrying current 2.00 AA is 1.00μTμT . (a) At what distance is it 0.100μTμT ? (b) What If? At one instant, the two conductors in a long household extension cord carry equal 2.00−A2.00−A currents in opposite directions. The two wires are 3.00 mmmm apart. Find the magnetic field 40.0 cmcm away from the middle of the straight cord, in the plane of the two wires. (c) At what distance is it one-tenth as large? (d) The center wire in a coaxial cable carries current 2.00 AA in one direction, and the sheath around it carries current 2.00 AA in the opposite direction. What magnetic field does the cable create at points outside the cable? • A projectile is launched from the point (x=0,y=0)(x=0,y=0), with velocity (12.0i+49.0j)m/s,(12.0i+49.0j)m/s, at t=0.t=0. (a) Make a table listing the projectile’s distance |→r||r→| from the origin at the end of each second thereafter, for 0≤t≤10s0≤t≤10s . Tabulating the xx and yy coordinates and the components of velocity vxvx and vyvy will also be useful. (b) Notice that the projectile’s distance from its starting point increases with time, goes through a maximum, and starts to decrease. Prove that the distance is a maximum when the position vector is perpendicular to the velocity. Suggestion: Argue that if →vv⃗ is not perpendicular to →r,r→, then |→r||r→| must be increasing or decreasing. (c) Determine the magnitude of the maximum displacement. (d) Explain your method for solving part (c). • Figure P 38.69a is a three-dimensional sketch of a birefringent crystal. The dotted lines illustrate how a thin, parallel-faced slab of material could be cut from the larger specimen with the crystal’s optic axis parallel to the faces of the plate. A section cut from the crystal in this manner is known as a retardation plate. When a beam of light is incident on the plate perpendicular to the direction of the optic axis as shown in Figure P 38.69b, the O ray and the E ray travel along a single straight line, but with different speeds. The figure shows the wave fronts for the two rays. (a) Let the thickness of the plate be d.d. Show that the phase difference between the O ray and the E ray after traveling the thickness of the plate is θ=2πdλ|nO−nE|θ=2πdλ|nO−nE| where λλ is the wavelength in air. (b) In a particular case, the incident light has a wavelength of 550 nm. Find the minimum value of dd for a quartz plate for which θ=π/2θ=π/2 . Such a plate is called a quarter-wave plate. Use values of nOnO and nEnE from Table 38.1 . • A projectile is fired from the top of a cliff of height hh above the ocean below. The projectile is fired at an angle θθ above the horizontal and with an initial speed vivi . (a) Find a symbolic expression in terms of the variables vi,g,vi,g, and θθ for the time at which the projectile reaches its maximum height. (b) Using the result of part (a), find an expression for the maximum height hmaxhmax above the ocean attained by the projectile in terms of h,vi,g,h,vi,g, and θ.θ. • A particle with charge QQ is located at the center of a cube of edge L.L. In addition, six other identical charged particles qq are positioned symmetrically around QQ as shown in Figure P 24.19. For each of these particles, qq is a negative number. Determine the electric flux through one face of the cube. • A solid cube of side 2aa and mass MM is sliding on a friction less surface with uniform velocity v→v→ as shown in Figure P 11.63 a. It hits a small obstacle at the end of the table that causes the cube to tilt as shown in Figure P 11.63 b. Find the minimum value of the magnitude of v→v→ such that the cube tips over and falls off the table. Note: The cube undergoes an inelastic collision at the edge. • The rotational spectrum of the HCl molecule contains lines with wavelengths of 0.060 4, 0.069 0, 0.080 4, 0.096 4, and 0.120 4 mm. What is the moment of inertia of the molecule? • A 650 -kg elevator starts from rest. It moves upward for 3.00 s with constant acceleration until it reaches its cruising speed of 1.75 m/sm/s . (a) What is the average power of the elevator motor during this time interval? (b) How does this power compare with the motor power when the elevator moves at its cruising speed? • Given that the period of the Moon’s orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is 3.84×108m,3.84×108m, use Equation 13.8 to calculate the mass of the Earth. (b) Why is the value you calculate a bit too large? • A playground is on the flat roof of a city school, 6.00 mm above the street below (Fig. P4.19). The vertical wall of the building is h=7.00mh=7.00m high, forming a 1 -m-high railing around the playground. A ball has fallen to the street below, and a passerby returns it by launching it an angle of θ=53.0∘θ=53.0∘ above the horizontal at a point d=24.0md=24.0m from the base of the building wall. The ball takes 2.20 ss to reach a point vertically above the wall. (a) Find the speed at which the ball was launched. (b) Find the vertical distance by which the ball clears the wall. (c) Find the horizontal distance from the wall to the point on the roof where the ball lands. • Measurements are made of the intensity distribution within the central bright fringe in a Young’s interference pattern (see Fig. 37.6).37.6). At a particular value of y,y, it is found that I/Imax=0.810I/Imax=0.810 when 600 -nm light is used. What wavelength of light should be used to reduce the relative intensity at the same location to 64.0%% of the maximum intensity? • Jane, whose mass is 50.0 kgkg , needs to swing across a river (having width DD ) filled with person-eating crocodiles to save Tarzan from danger. She must swing into a wind exerting constant horizontal force F,F, on a vine having length LL and initially making an angle θθ with the vertical (Fig. P8.77). Take D=50.0m,F=110N,L=40.0m,D=50.0m,F=110N,L=40.0m, and θ=50.0∘.θ=50.0∘. (a) With what minimum speed must Jane begin her swing to just make it to the other side? (b) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing? Assume Tarzan has a mass of 80.0 kgkg . • A transmission line that has a resistance per unit length of 4.50×10−4Ω/m4.50×10−4Ω/m is to be used to transmit 5.00 MWMW across 400mi(6.44×105m).400mi(6.44×105m). The output voltage of the source is 4.50 kVkV . (a) What is the line loss if a transformer is used to step up the voltage to 500 kV?kV? (b) What fraction of the input power is lost to the line under these circumstances? (c) What If? What difficulties would be encountered in attempting to transmit the 5.00 MWMW at the source voltage of 4.50 kVkV ? • An air-filled spherical capacitor is constructed with inner-and outer-shell radii of 7.00 cmcm and 14.0cm,14.0cm, respectively. (a) Calculate the capacitance of the device. (b) What potential difference between the spheres results in a 4.00−μC4.00−μC charge on the capacitor? • Complete the calculation in Example 32.3 by proving that ∫∞0e−2Rt/Ldt=L2R∫0∞e−2Rt/Ldt=L2R • Lightning can be studied with a Van de Graaff generator, which consists of a spherical dome on which charge is continuously deposited by a moving belt. Charge can be added until the electric field at the surface of the dome becomes equal to the dielectric strength of air. Any more charge leaks off in sparks as shown in Figure P25.49. Assume the dome has a diameter of 30.0 cmcm and is surrounded by dry air with a “breakdown” electric field of 3.00×106V/m.3.00×106V/m. (a) What is the maximum potential of the dome? (b) What is the maximum charge on the dome? • An electron with kinetic energy E=5.00 eV is incident on a barrier of width L=0.200nm and height U=10.0eV (Fig. P41.30). What is the probability that the electron (a) tunnels through the barrier? (b) Is reflected? • A block of mass m=2.00kgm=2.00kg is attached to a spring of force constant k=500N/mk=500N/m as shown in Figure P8.15.P8.15. The block is pulled to a position xi=xi= 00 cmcm to the right of equilibrium and released from rest. Find the speed the block has as it passes through equilibrium if (a)(a) the horizontal surface is frictionless and (b) the coef- ficient of friction between block and surface is μk=0.350μk=0.350 . • A beam of light both reflects and refracts at the surface between air and glass as shown in Figure P35.29. If the refractive index of the glass is ng,ng, find the angle of incidence θ1θ1 in the air that would result in the reflected ray and the refracted ray being perpendicular to each other. • The electric field everywhere on the surface of a thin, spherical shell of radius 0.750 m is of magnitude 890 N/C and points radially toward the center of the sphere. (a) What is the net charge within the sphere’s surface? (b) What is the distribution of the charge inside the spherical shell? • A proton moves with a velocity of →v=(2ˆi−4ˆj+ˆk)m/sv→=(2i^−4j^+k^)m/s in a region in which the magnetic field is →B=(→i+2ˆj−ˆk)TB→=(i→+2j^−k^)T . What is the magnitude of the magnetic force this particle experiences? • A proton is projected into a magnetic field that is directed along the positive xx axis. Find the direction of the magnetic force exerted on the proton for each of the following directions of the proton’s velocity: (a) the positive yy direction, (b) the negative yy direction, (c) the positive xx direction. • A clown balances a small spherical grape at the top of his bald head, which also has the shape of a sphere. After drawing sufficient applause, the grape starts from rest and rolls down without slipping. It will leave contact with the clown’s scalp when the radial line joining it to the center of curvature makes what angle with the vertical? • The creation and study of new and very massive elementary particles is an important part of contemporary physics. To create a particle of mass MM requires an energy Mc2Mc2 . With enough energy, an exotic particle can be created by allowing a fast-moving proton to collide with a similar target particle. Consider a perfectly inelastic collision between two protons: an incident proton with mass mp,mp, kinetic energy K,K, and momentum magnitude pp joins with an originally stationary target proton to form a single product particle of mass MM . Not all the kinetic energy of the incoming proton is available to create the product particle because conservation of momentum requires that the system as a whole still must have some kinetic energy after the collision. Therefore, only a fraction of the energy of the incident particle is available to create a new particle. (a) Show that the energy available to create a product particle is given by Mc2=2mρc2√1+K2mρc2Mc2=2mρc21+K2mρc2−−−−−−−−−√ This result shows that when the kinetic energy KK of the incident proton is large compared with its rest energy mpc2mpc2 , then MM approaches (2mpK)1/2/c(2mpK)1/2/c . Therefore, if the energy of the incoming proton is increased by a factor of 9,9, the mass you can create increases only by a factor of 3,3, not by a factor of 9 as would be expected. (b) This problem can be alleviated by using colliding beams as is the case in most modern accelerators. Here the total momentum of a pair of interacting particles can be zero. The center of mass can be at rest after the collision, so, in principle, all the initial kinetic energy can be used for particle creation. Show that Mc2=2mc2(1+Kmc2)Mc2=2mc2(1+Kmc2) where KK is the kinetic energy of each of the two identical colliding particles. Here, if K>>mc2,K>>mc2, we have MM directly proportional to KK as we would desire. • The current in a 90.0 -mH inductor changes with time as I=1.00t2−6.00t,I=1.00t2−6.00t, where II is in amperes and tt is in seconds. Find the magnitude of the induced emf at (a) t=1.00st=1.00s and (b)t=4.00s.(b)t=4.00s. (c) At what time is the emf zero? • Batteries are rated in terms of ampere-hours (A⋅h).(A⋅h). For example, a battery that can produce a current of 2.00 A for 3.00 hh is rated at 6.00A⋅6.00A⋅h. (a) What is the total energy, in kilowatt-hours, stored in a 12.0−V12.0−V battery rated at 55.0 A⋅hA⋅h ? (b) At$0.110$0.110 per kilowatt-hour, what is the value of the electricity produced by this battery? • Why is the following situation impossible? The perpendicular distance of a lightbulb from a large plane mirror is twice the perpendicular distance of a person from the mirror. Light from the lightbulb reaches the person by two paths: (1) it travels to the mirror and reflects from the mirror to the person, and (2) it travels directly to the person without reflecting off the mirror. The total distance traveled by the light in the first case is 3.10 times the distance traveled by the light in the second case. • Show that the frequency ff and wavelength λλ of a freely moving quantum particle with mass are related by the expression (fc)2=1λ2+1λ2C(fc)2=1λ2+1λC2 where λC=h/mcλC=h/mc is the Compton wavelength of the particle. (b) Is it ever possible for a particle having nonzero mass to have the same wavelength and frequency as a photon? Explain. • Vectors A→A→ and B→B→ have equal magnitudes of 5.00.5.00. The sum of A→A→ and B→B→ is the vector 6.00j^j^ . Determine the angle between A→A→ and B→.B→. • Light of wavelength 632.8 nm illuminates a single slit, and a diffraction pattern is formed on a screen 1.00 m from the slit. (a) Using the data in the following table, plot relative intensity versus position. Choose an appropriate value for the slit width aa and, on the same graph used for the experimental data, plot the theoretical expression for the relative intensity IImax=sin2ϕϕ2IImax=sin2⁡ϕϕ2 where ϕ=(πasinθ)/λ.ϕ=(πasin⁡θ)/λ. (b) What value of aa gives the best fit of theory and experiment? • On a dry winter day, you scuff your leather-soled shoes across a carpet and get a shock when you extend the tip of one finger toward a metal doorknob. In a dark room, you see a spark perhaps 5 mm long. Make order-of-magnitude estimates of (a) your electric potential and (b) the charge on your body before you touch the doorknob. Explain your reasoning. • An insulating solid sphere of radius aa has a uniform volume charge density and carries a total positive charge Q.Q. A spherical gaussian surface of radius r,r, which shares a common center with the insulating sphere, is inflated starting from r=0.r=0. (a) Find an expression for the electric flux passing through the surface of the gaussian sphere as a function of rr for r<ar<a . (b) Find an expression for the electric flux for r>ar>a (c) Plot the flux versus r.r. • A placekicker must kick a football from a point 36.0 m (about 40 yards) from the goal. Half the crowd hopes the ball will clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 20.0 m/s at an angle of 53.0∘ to the horizontal. (a) By how much does the ball clear or fall short of clearing the crossbar? (b) Does the ball approach the crossbar while still rising or while falling? • A Σ0Σ0 particle at rest decays according to Σ0→Λ0+γ.Σ0→Λ0+γ. Find the gamma-ray energy. • The polar coordinates of a certain point are (r=4.30cm,(r=4.30cm, θ=214∘).θ=214∘). (a) Find its Cartesian coordinates xx and y.y. Find the polar coordinates of the points with Cartesian coordinates (b) (−x,y),(−x,y), (c) (−2x,−2y),(−2x,−2y), and (d) (3x,−3y)(3x,−3y) . • When, after a reaction or disturbance of any kind, a nucleus is left in an excited state, it can return to its normal (ground) state by emission of a gamma-ray photon (or several photons). This process is illustrated by Equation 44.25. The emitting nucleus must recoil to conserve both energy and momentum. (a) Show that the recoil energy of the nucleus is Er=(ΔE)22Mc2 where ΔE is the difference in energy between the excited and ground states of a nucleus of mass M. (b) Calculate the recoil energy of the 57 Fe nucleus when it decays by gamma emission from the 14.4 -keV excited state. For this calculation, take the mass to be 57 u. Suggestion: Assume hf<< Mc2. • As a 1.00 -mol sample of a monatomic ideal gas expands adiabatically, the work done on it is −2.50×103J−2.50×103J . The initial temperature and pressure of the gas are 500 KK and 60 atmatm . Calculate (a) the final temperature and (b) the final pressure. • To repair a power supply for a stereo amplifier, an electronics technician needs a 100−μF100−μF capacitor capable of withstanding a potential difference of 90 VV between the plates. The immediately available supply is a box of five 100−μF100−μF capacitors, each having a maximum voltage capability of 50 VV . (a) What combination of these capacitors has the proper electrical characteristics? Will the technician use all the capacitors in the box? Explain your answers. (b) In the combination of capacitors obtained in part (a), what will be the maximum voltage across each of the capacitors used? • The object in Figure$\mathrm{P} 36.67$is midway between the lens and the mirror, which are separated by a distance$d=25.0 \mathrm{cm} .$The magnitude of the mirror’s radius of curvature is$20.0 \mathrm{cm},$and the lens has a focal length of$-16.7 \mathrm{cm} .$(a) Considering only the light that leaves the object and travels first toward the mirror, locate the final image formed by this system. (b) Is this image real or virtual? (c) Is it upright or inverted? (d) What is the overall magnification? • An emf of 96.0 mVmV is induced in the windings of a coil when the current in a nearby coil is increasing at the rate of 1.20 A/sA/s . What is the mutual inductance of the two coils? • Light of wavelength 540 nm passes through a slit of width 0.200 mm. (a) The width of the central maximum on a screen is 8.10 mm. How far is the screen from the slit? (b) Determine the width of the first bright fringe to the side of the central maximum. • In a piece of rock from the Moon, the 87Rb content is assayed to be 1.82×1010 atoms per gram of material and the 87Sr content is found to be 1.07 ×109 atoms per gram. The relevant decay relating these nuclides is 87Rb→87Sr+ e−+¯ν. The half-life of the decay is 4.75×1010yr . (a) Calculate the age of the rock. (b) What If? Could the material in the rock actually be much older? What assumption is implicit in using the radioactive dating method? • An electron in an infinitely deep potential well has a ground-state energy of 0.300 eV. (a) Show that the photon emitted in a transition from the n=3 state to the n=1 state has a wavelength of 517nm, which makes it green visible light. (b) Find the wavelength and the spectral region for each of the other five transitions that take place among the four lowest energy levels. • An astronaut is traveling in a space vehicle moving at 0.500 c relative to the Earth. The astronaut measures her pulse rate at 75.0 beats per minute. Signals generated by the astronaut’s pulse are radioed to the Earth when the vehicle is moving in a direction perpendicular to the line that connects the vehicle with an observer on the Earth. (a) What pulse rate does the Earth-based observer measure? (b) What If? What would be the pulse rate if the speed of the space vehicle were increased to 0.990cc ? • An electric current in a conductor varies with time according to the expression I(t)=100sin(120πt),I(t)=100sin(120πt), where II is in amperes and tt is in seconds. What is the total charge passing a given point in the conductor from t=0t=0 to t=1240st=1240s ? • Assume the magnitude of the electric field on each face of the cube of edge L=1.00mL=1.00m in Figure P 24.30 is uniform and the directions of the fields on each face as indicated. Find (a) the net electric flux through the cube and (b) the net charge inside the cube. (c) Could the net charge be a single point charge? • A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection? • Consider a quantum particle moving in a one- dimensional box for which the walls are at x=−L/2 and x=L/2. (a) Write the wave functions and probability densities for n=1,n=2, and n=3. (b) Sketch the wave functions and probability densities. • A vessel contains 1.00×1041.00×104 oxygen molecules at 500 KK . (a) Make an accurate graph of the Maxwell speed distribution function versus speed with points at speed intervals of 100 m/sm/s . (b) Determine the most probable speed from this graph. (c) Calculate the average and rms speeds for the molecules and label these points on your graph. (d) From the graph, estimate the fraction of molecules with speeds in the range 300 m/sm/s to 600 m/sm/s . • You want to find out how many atoms of the isotope 65 CuCu are in a small sample of material. You bombard the sample with neutrons to ensure that on the order of 1%% of these copper nuclei absorb a neutron. After activation, you turn off the neutron flux and then use a highly efficient detector to monitor the gamma radiation that comes out of the sample. Assume half of the 66 CuCu nuclei emit a 1.04 -MeV gamma ray in their decay. (The other half of the activated nuclei decay directly to the ground state of 66Ni.)66Ni.) If after 10 minmin (two half-lives) you have detected 1.00×104MeV1.00×104MeV gamma ray in their decay. (The other half of the activated nuclei decay directly to the ground state of 66 NiNi . If after 10 min (two half-lives) you have detected 1.00×104MeV1.00×104MeV of photon energy at 1.04MeV,(a)1.04MeV,(a) approximately how many 65 CuCu atoms are in the sample? (b) Assume the sample contains natural copper. Refer to the isotopic abundances listed in Table 44.2 and estimate the total mass of copper in the sample. • A uniform magnetic field of magnitude 0.150 TT is directed along the positive xx axis. A positron moving at a speed of 5.00×106m/s5.00×106m/s enters the field along a direction that makes an angle of θ=85.0∘θ=85.0∘ with the xx axis (Fig. P 29.73). The motion of the particle is expected to be a helix as described in Section 29.2 . Calculate (a) the pitch pp and (b)(b) the radius rr of the trajectory as defined in Figure P 29.73. • Scanning through Figure 42.19 in order of increasing atomic number, notice that the electrons usually fill the subshells in such a way that those subshells with the lowest values of n+ℓn+ℓ are filled first. If two subshells have the same value of n+ℓ,n+ℓ, the one with the lower value of nn is generally filled first. Using these two rules, write the order in which the subshells are filled through n+ℓ=7n+ℓ=7 . • A series RLCRLC circuit has resonance angular frequency 2.00×103rad/s2.00×103rad/s . When it is operating at some input frequency, XL=12.0ΩXL=12.0Ω and XC=8.00DΩ.XC=8.00DΩ. (a) Is this input frequency higher than, lower than, or the same as the resonance frequency? Explain how you can tell. (b) Explain whether it is possible to determine the values of both LL and C.C. (c) If it is possible, find LL and C.C. If it is not possible, give a compact expression for the condition that LL and CC must satisfy. • Aluminum and copper wires of equal length are found to have the same resistance. What is the ratio of their radii? • A person walks into a room that has two flat mirrors on opposite walls. The mirrors produce multiple images of the person. Consider only the images formed in the mirror on the left. When the person is 2.00$\mathrm{m}$from the mirror on the left wall and 4.00$\mathrm{m}$from the mirror on the right wall, find the distance from the person to the first three images seen in the mirror on the left wall. • Find the mass mm of the counterweight needed to balance a truck with mass M=1500kgM=1500kg on an incline of θ=45∘θ=45∘ (Fig. Pl2.9). Assume both pulleys are frictionless and massless. • A uniformly charged, straight filament 7.00 m in length has a total positive charge of 2.00μC.2.00μC. An uncharged cardboard cylinder 2.00 cm in length and 10.0 cm in radius surrounds the filament at its center, with the filament as the axis of the cylinder. Using reasonable approximations, find (a) the electric field at the surface of the cylinder and (b) the total electric flux through the cylinder. • A rod of mass 0.720 kg and radius 6.00 cm rests on two parallel rails (Fig. P29.37)P29.37) that are d=12.0cmd=12.0cm apart and L=45.0cmL=45.0cm long. The rod carries a current of I=48.0AI=48.0A in the direction shown and rolls along the rails without slipping. A uniform magnetic field of magnitude 0.240 T is directed perpendicular to the rod and the rails. If it starts from rest, what is the speed of the rod as it leaves the rails? • A shower stall has dimensions 86.0cm×86.0cm×210cm.86.0cm×86.0cm×210cm. Assume the stall acts as a pipe closed at both ends, with nodes at opposite sides. Assume singing voices range from 130 Hz to 2 000 Hz and let the speed of sound in the hot air be 355 m/s. For someone singing in this shower, which frequencies would sound the richest (because of resonance)? • A parallel-plate capacitor with plates of area LW and plate separation tt has the region between its plates filled with wedges of two dielectric materials as shown in Figure P26.72.P26.72. Assume tt is much less than both LL and W.W. (a) Determine its capacitance. (b) Should the capacitance be the same if the labels κ1κ1 and κ2κ2 are interchanged? Demonstrate that your expression does or does not have this property. (c) Show that if κ1κ1 and κ2κ2 approach equality to a common value κ,κ, your result becomes the same as the capacitance of a capacitor containing a single dielectric: C=κϵ0LW/tC=κϵ0LW/t • Two particles with masses m1 and m2 are joined by a light spring of force constant k. They vibrate along a straight line with their center of mass fixed. (a) Show that the total energy 12m1u21+12m2u22+12kx2 can be written as 12μu2+12kx2, where u=|u1|+|u2| is the relative speed of the particles and μ=m1m2/(m1+m2) is the reduced mass of the system. This result demonstrates that the pair of freely vibrating particles can be precisely modeled as a single particle vibrating on one end of a spring that has its other end fixed. (b) Differentiate the equation$$\frac{1}{2} \mu u^{2}+\frac{1}{2} k x^{2}=$ $constant with respect to$x$. Proceed to show that the system executes simple harmonic motion. (c) Find its frequency. • At 11:00 a.m. on September 7, 2001, more than one million British schoolchildren jumped up and down for one minute to simulate an earthquake. (a) Find the energy stored in the children’s bodies that was converted into internal energy in the ground and their bodies and propagated into the ground by seismic waves during the experiment. Assume 1 050 000 children of average mass 36.0 kg jumped 12 times each, raising their centers of mass by 25.0 cm each time and briefly resting between one jump and the next. (b) Of the energy that propagated into the ground, most produced high-frequency “microtremor” vibrations that were rapidly damped and did not travel far. Assume 0.01% of the total energy was carried away by long-range seismic waves. The magnitude of an earthquake on the Richter scale is given by M=logE−4.81.5M=logE−4.81.5 where EE is the seismic wave energy in joules. According to this model, what was the magnitude of the demonstration quake? • In the situation described in Problem 57 and Figure P5.57, the masses of the rope, spring balance, and pulley are negligible. Nick’s feet are not touching the ground. (a) Assume Nick is momentarily at rest when he stops pulling down on the rope and passes the end of the rope to another child, of weight 440 NN , who is standing on the ground next to him. The rope does not break. Describe the ensuing motion. (b) Instead, assume Nick is momentarily at rest when he ties the end of the rope to a strong hook projecting from the tree trunk. Explain why this action can make the rope break. • Ecotourists use their global positioning system indicator to determine their location inside a botanical garden as latitude 0.002 43 degree south of the equator, longitude 75.642 38 degrees west. They wish to visit a tree at latitude 0.001 62 degree north, longitude 75.644 26 degrees west. (a) Determine the straight-line distance and the direction in which they can walk to reach the tree as follows. First model the Earth as a sphere of radius 6.37 Mm to determine the westward and northward displacement components required, in meters. Then model the Earth as a flat surface to complete the calculation. (b) Explain why it is possible to use these two geometrical models together to solve the problem. • Two objects move with initial velocity$-8.00 \mathrm{m} / \mathrm{s}$, final velocity 16.0$\mathrm{m} / \mathrm{s}$, and constant accelerations. (a) The first object has displacement 20.0$\mathrm{m}$. Find its acceleration. (b) The second object travels a total distance of 22.0$\mathrm{m}\$ . Find its acceleration.
• The electric potential inside a charged spherical conductor of radius RR is given by V=keQ/R,V=keQ/R, and the potential outside is given by V=keQ/rV=keQ/r . Using Er=−dV/dr,Er=−dV/dr, derive the electric field (a) inside and (b) outside this charge distribution.
• A 0.60 -kg block attached to a spring with force constant 130 N/mN/m is free to move on a frictionless, horizontal surface as in Active Figure 15.1.15.1. The block is released from rest when the spring is stretched 0.13 mm . At the instant the block is released, find (a) the force on the block and (b) its acceleration.
• The distance between two telephone poles is 50.0m.50.0m. When a 1.00−kg1.00−kg bird lands on the telephone wire midway between the poles, the wire sags 0.200m.0.200m. (a) Draw a free-body diagram of the bird. (b) How much tension does the bird produce in the wire? Ignore the weight of the wire.
• The longest wavelength of radiation absorbed by a certain semiconductor is 0.512μm.0.512μm. Calculate the energy gap for this semiconductor.
• Three forces acting on an object are given by →F1=F→1=
(−2.00ˆi+2.00ˆj)N,→F2=(5.00ˆi−3.00ˆj)N,(−2.00i^+2.00j^)N,F→2=(5.00i^−3.00j^)N, and →F3=
(−45.0ˆi)N. The object experiences an acceleration of magnitude 3.75 m/s2 . (a) What is the direction of the acceleration? (b) What is the mass of the object: (c) If the object is initially at rest, what is its speed after 10.0 s ? (d) What are the velocity components of the object after 10.0 s ?
• A 1.00 -g cork ball with charge 2.00μCμC is suspended vertically on a 0.500 -m-long light string in the presence of a uniform, downward-directed electric field of magnitude E=1.00×105N/CE=1.00×105N/C . If the ball is displaced slightly from the vertical, it oscillates like a simple pendulum. (a) Determine the period of this oscillation. (b) Should the effect of gravitation be included in the calculation for part (a)? Explain.
• A rigid tank contains 1.50 moles of an ideal gas. Determine the number of moles of gas that must be withdrawn from the tank to lower the pressure of the gas from 25.0 atm to 5.00 atmatm . Assume the volume of the tank and the temperature of the gas remain constant during this operation.
• Consider the situation shown in Figure P34.1.P34.1. An electric field of 300 V/mV/m is confined to a circular area d=10.0cmd=10.0cm in diameter and directed outward perpendicular to the plane of the figure. If the field is increasing at a rate of 20.0 V/m⋅sV/m⋅s , what are (a) the direction and (b) the magnitude of the magnetic field at the point P,r=15.0cmP,r=15.0cm from the center of the circle?
• Show that the wave function y=eb(x−c)y=eb(x−c) is a solution of the linear wave equation (Eq. 16.27),16.27), where bb is a constant.
• The Solar and Heliospheric Observatory (SOHO) spacecraft has a special orbit, located between the Earth and the Sun along the line joining them, and it is always close enough to the Earth to transmit data easily. Both objects exert gravitational forces on the observatory. It moves around the Sun in a near-circular orbit that is smaller than the Earth’s circular orbit. Its period, however, is not less than 1 yr but just equal to 1 yr. Show that its distance from the Earth must be 1.48×109m.1.48×109m. In 1772,1772, Joseph Louis Lagrange determined theoretically the special location allowing this orbit. Suggestions: Use data that are precise to four digits. The mass of the Earth is 5.974×1024kg.5.974×1024kg. You will not be able to easily solve the equation you generate; instead, use a computer to verify that 1.48×109m1.48×109m is the correct value.
• This problem describes one experimental method for determining the moment of inertia of an irregularly shaped object such as the payload for a satellite. Figure P10.52 shows a counterweight of mass mm suspended by a cord wound around a spool of radius r,r, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance h,h, acquiring a speed vv Show that the moment of inertia II of the rotating apparatus (including the turntable) is mr2(2gh/v2−1)mr2(2gh/v2−1)
• Standing at a crosswalk, you hear a frequency of 560 Hz from the siren of an approaching ambulance. After the ambulance passes, the observed frequency of the siren is 480 Hz. Determine the ambulance’s speed from these observations.
• A particle of mass mm moves in one-dimensional motion through a field for which the potential energy of the particle–field system is
U(x)=Ax3−BxU(x)=Ax3−Bx
where AA and BB are constants. The general shape of this function is shown in Figure P43.61P43.61 . ( a) Find the equilibrium position x0x0 of the particle in terms of m,A,m,A, and BB . (b) Determine the depth U0U0 of this potential well. (c) In moving along the xx axis, what maximum force toward the negative xx direction does the particle experience?
• The intensity of a sound wave at a fixed distance from a speaker vibrating at a frequency ff is II . (a) Determine the intensity that results if the frequency is increased to f′f′ while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to f/2f/2 and the displacement amplitude is doubled.
• The motion of a human body through space can be modeled as the motion of a particle at the body’s center of mass as we will study in Chapter 9. The components of the displacement of an athlete’s center of mass from the beginning to the end of a certain jump are described by the equations
xf=0+(11.2m/s)(cos18.5∘)txf=0+(11.2m/s)(cos18.5∘)t
360m=0.840m+(11.2m/s)(sin18.5∘)t−12(9.80m/s2)t20.360m=0.840m+(11.2m/s)(sin18.5∘)t−12(9.80m/s2)t2
where tt is in seconds and is the time at which the athlete ends the jump. Identify (a) the athlete’s position and (b) his vector velocity at the takeoff point. (c) How far did he jump?
• Why is the following situation impossible? A “clever” technician takes his 20-min coffee break and boils some water for his coffee with an x-ray machine. The machine produces 10.0 rad/s, and the temperature of the water in an insulated cup is initially 50.0°C.
• Find the angular speed of the Earth’s rotation about its axis. (b) How does this rotation affect the shape of the Earth?
• A ball of mass mm is thrown straight up into the air with an initial speed vivi . Find the momentum of the ball (a) at its maximum height and (b) halfway to its maximum height.
• A force →FF→ applied to an object of mass m1m1 produces an acceleration of 3.00 m/s2m/s2 . The same force applied to a second object of mass m2m2 produces an acceleration of 1.00 m/s2m/s2 . (a) What is the value of the ratio m1/m2?(b)m1/m2?(b) If m1m1 and m2m2 are combined into one object, find its acceleration
under the action of the force →FF→ .
• Note: In Problems 9 through 12, calculate numerical answers to three significant figures as usual.
For →A=3ˆi+ˆj−ˆk,→B=−ˆi+2ˆj+5ˆk, and →C=2ˆj−3ˆk find →C⋅(→A−→B).
• In an experiment to measure the speed of light using the apparatus of Armand H. L. Fizeau (see Fig. 35.2), the distance between light source and mirror was 11.45 km and the wheel had 720 notches. The experimentally determined value of cc was 2.998×108m/s2.998×108m/s when the outgoing light passed through one notch and then returned through the next notch. Calculate the minimum angular speed of the wheel for this experiment.
• A uniform, hollow, cylindrical spool has inside radius R/2,R/2, outside radius R,R, and mass M(Fig.P10.81)M(Fig.P10.81) . It is mounted so that it rotates on a fixed, horizontal axle. A counterweight of mass mm is connected to the end of a string wound around the spool. The counterweight falls from rest at t=0t=0 to a position yy at time t.t. Show that the torque due to the friction forces between spool and axle is
τf=R[m(g−2yt2)−M5y4t2]τf=R[m(g−2yt2)−M5y4t2]
• The most recent naked-eye supernova was Supernova Shelton 1987 AA (Fig. P46.53). It was 170000 ly away in the Large Magellanic Cloud, a satellite galaxy of the Milky Way. Approximately 3 h before its optical brightening was noticed, two neutrino detection experiments simultaneously registered the first neutrinos from an identified source other than the Sun. The Irvine-Michigan-Brookhaven experiment in a salt mine in Ohio registered eight neutrinos over a 6−6− s period, and the Kamiokande II experiment in a zinc mine in Japan counted eleven neutrinos in 13 s. (Because the supernova is far south in the sky, these neutrinos entered the detectors from below. They passed through the Earth before they were by chance absorbed by nuclei in the detectors.) The neutrino energies were between approximately 8 MeV and 40 MeV. If neutrinos have no mass, neutrinos of all energies should travel together at the speed of light, and the data are consistent with this possibility. The arrival times could vary simply because neutrinos were created at different moments as the core of the star collapsed into a neutron star. If neutrinos have nonzero mass, lower-energy neutrinos should move comparatively slowly. The data are consistent with a 10 -MeV neutrino requiring at most approximately 10 s more than a photon would require to travel from the supernova to us. Find the upper limit that this observation sets on the mass of a neutrino. (Other evidence sets an even tighter limit.)
• Particles incident from the left in Figure P41.59 are confronted with a step in potential energy. The step has a height U at x=0. The particles have energy E>U . Classically, all the particles would continue moving forward with reduced speed. According to quantum mechanics, however, a fraction of the particles are reflected at the step. (a) Prove that the reflection coefficient R for this case is
R=(k1−k2)2(k1+k2)2
where k1=2π/λ1 and k2=2π/λ2 are the wave numbers for the incident and transmitted particles, respectively. Proceed as follows. Show that the wave function ψ1=Aeik1x+ Be−ik1x satisfies the Schrödinger equation in region 1, for
x<0. Here Aeik1x represents the incident beam and Be−ik1x
represents the reflected particles. Show that ψ2=Ceik2x satisfies the Schrödinger equation in region 2, for x>0. Impose the boundary conditions ψ1=ψ2 and dψ1/dx= dψ2/dx, at x=0, to find the relationship between B and A Then evaluate R=B2/A2 . A particle that has kinetic energy E=7.00eV is incident from a region where the potential energy is zero onto one where U=5.00 eV. Find (b) its probability of being reflected and (c) its probability of being transmitted.
• A light, unstressed spring has length d.d. Two identical particles, each with charge q,q, are connected to the opposite ends of the spring. The particles are held stationary a distance dd apart and then released at the same moment. The system then oscillates on a frictionless, horizontal table. The spring has a bit of internal kinetic friction, so the oscillation is damped. The particles eventually stop vibrating when the distance between them is 3d.3d. Assume the system of the spring and two charged particles is isolated. Find the increase in internal energy that appears in the spring during the oscillations.
• The vector →AA→ has x,y,x,y, and zz components of 8.00,12.0,8.00,12.0, and −4.00−4.00 units, respectively. (a) Write a vector expression for →AA→ in unit-vector notation. (b) Obtain a unit-vector expression for a vector →BB→ one-fourth the length of →AA→ pointing in the same direction as →AA→ . (c) Obtain a unit-vector expression for a vector →CC→ three times the length of →AA→ pointing in the direction opposite the direction of →AA→ .
• A 1.00-kg object slides to the right on a surface having a coefficient of kinetic friction 0.250 (Fig. P8.62a). The object has a speed of vi 5 3.00 m/s when it makes contact with a light spring (Fig. P8.62b) that has a force constant of 50.0 N/m. The object comes to rest after the spring has been compressed a distance d (Fig. P8.62c). The object is then forced toward the left by the spring (Fig. P8.62d) and continues to move in that direction beyond the spring’s unstretched position. Finally, the object comes to rest a distance D to the left of the unstretched spring (Fig. P8.62e).
Find (a) the distance of compression d, (b) the speed v at the unstretched position when the object is moving to the left (Fig. P8.62d), and (c) the distance D where the object comes to rest.
• An ideal gas initially at Pi,Vi,Pi,Vi, and TiTi is taken through a cycle as shown in Figure P20.35. (a) Find the net work done on the gas per cycle. (b) What is the net energy added by heat to the system per cycle?
• Two thermally insulated vessels are connected by a narrow tube fitted with a valve that is initially closed as shown in Figure P20.13. One vessel of volume 16.8 L contains oxygen at a temperature of 300 K and a pressure of 1.75 atm. The other vessel of volume 22.4 L contains oxygen at a temperature of 450 K and a pressure of 2.25 atm. When the valve is opened, the gases in the two vessels mix and the temperature and pressure become uniform throughout. (a) What is the final temperature? (b) What is the final pressure?
• Figure P25.38 shows several equipotential lines, each labeled by its potential in volts. The distance between the lines of the square grid represents 1.00 cm. (a) Is the magnitude of the field larger at AA or at BB ? Explain how you can tell. (b) Explain what you can determine about E→E→ at B.B. (c) Represent what the electric field looks like by drawing at least eight field lines.
• An electron of mass 9.11×10−31kg9.11×10−31kg has an initial speed of 3.00×105m/s3.00×105m/s . It travels in a straight line, and its speed increases to 7.00×105m/s in a distance of 5.00cm. Assuming its acceleration is constant, (a) determine the magnitude of the force exerted on the electron and (b) compare this force with the weight of the electron, which we ignored.
• Gravitation and other forces prevent Hubble’s-law expansion from taking place except in systems larger than clusters of galaxies. What If? Imagine that these forces could be ignored and all distances expanded at a rate described by the Hubble constant of 22×10−3m/s⋅22×10−3m/s⋅ (a) At what rate would the 1.85−m1.85−m height of a basketball player be increasing? (b) At what rate would the distance between the Earth and the Moon be increasing?
• The power output of the Sun is 3.85×1026W3.85×1026W . By how much does the mass of the Sun decrease each second?
• A wire carries a steady current of 2.40 A. A straight section of the wire is 0.750 mm long and lies along the xx axis within a uniform magnetic field, →B=1.60ˆkTB→=1.60k^T. If the current is in the positive xx direction, what is the magnetic force on the section of wire?
• Water falls without splashing at a rate of 0.250 L/sL/s from a height of 2.60 mm into a 0.750 -kg bucket on a scale. If the bucket is originally empty, what does the scale read in newtons