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All Textbook Solutions for Modern Physics for Scientists and Engineers
Michelson used the motion of the Earth around the sun to try to determine the effects of the ether. Can you think of a more convenient experiment with a higher speed that Michelson might have used in the 1880s? What about today?If you wanted to set out today to find the effects of the ether, what experimental apparatus would you want to use? Would a laser he included? Why?
3Q4Q5Q6Q7Q8QDevise a system for you and three colleagues, at rest with you, to synchronize your clocks if your clocks are too large to move and are separated by hundreds of miles.In the experiment to verify time dilation by flying the cesium clocks around the Earth, what is the order of the speed of the four clocks in a system fixed at the center of the Earth, but not rotating?Can you think of an experiment to verify length contraction directly? Explain.Would it be easier to perform the muon decay experiment in the space station orbiting above Earth and then compare with the number of muons on Earth? Explain.On a spacetime diagram, can events above t = 0 but not in the shaded area in Figure 2.25 affect the future? Explain.14QWhat would be a suitable name for events connected by s2 = 0?16Q17QExplain how in the twin paradox, we might arrange to compare clocks at the beginning and end of Marys journey and not have to worry about acceleration effects.In each of the following pairs, which is the more massive: a relaxed or compressed spring, a charged or uncharged capacitor, or a piston-cylinder when closed or open?20Q21QA salesman driving a very fast car was arrested for driving through a traffic light while it was red, according to a policeman parked near the traffic light. The salesman said that the light was actually green to him, because it was Doppler shifted. Is he likely to be found innocent? Explain.A salesman driving a very fast car was arrested for driving through a traffic light while it was red, according to a policeman parked near the traffic light. The salesman said that the light was actually green to him, because it was Doppler shifted. Is he likely to be found innocent? Explain.Show that the form of Newtons second law is invariant under the Galilean transformation.2P3PA swimmer wants to swim straight across a river with current flowing at a speed of v1 = 0.350 m/s. If the swimmer swims in still water with speed v2 = 1.25 m/s, at what angle should the swimmer point upstream from the shore, and at what speed will the swimmer swim across the river?5P6P7P8PProve that the constancy of the speed of light (postulate 2) is inconsistent with the Galilean transformation.10P11P12PTwo events occur in an inertial system K as follows: Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2 = 3a/ 2c, y2 = 0, z2 = 0 In what frame K will these events appear to occur at the same time? Describe the motion of system K.Is there a frame K in which the two events described in Problem 13 occur at the same place? Explain. 13. Two events occur in an inertial system K as follows: Event 1: x1 = a, t1 = 2a/c, y1 = 0, z1 = 0 Event 2: x2 = 2a, t2 = 3a/ 2c, y2 = 0, z2 = 0 In what frame K will these events appear to occur at the same time? Describe the motion of system K.15PAn event occurs in system K at x = 2 m, y = 3.5 m, z = 3.5 m, and t = 0. System K and K have their axes coincident at t = t = 0, and system K travels along the x axis of system K with a speed 0.8c. What are the coordinates of the event in system K?17P18PA rocket ship carrying passengers blasts off to go from New York to Los Angeles, a distance of about 5000 km. (a) How fast must the rocket ship go to have its own length shortened by 1%? (b) Ignore effects of general relativity and determine how much time the rocket ships clock and the ground-based clocks differ when the rocket ship arrives in Los Angeles.20PParticle physicists use particle track detectors to determine the lifetime of short-lived particles. A muon has a mean lifetime of 2.2 s and makes a track 9.5 cm long before decaying into an electron and two neutrinos. What was the speed of the muon?The Apollo astronauts returned from the moon under the Earths gravitational force and reached speeds of almost 25,000 mi/h with respect to Earth. Assuming (incorrectly) they had this speed for the entire trip from the moon to Earth, what was the time difference for the trip between their clocks and clocks on Earth?A clock in a spaceship is observed to run at a speed of only 3/5 that of a similar clock at rest on Earth. How fast is the spaceship moving?A spaceship of length 40 m at rest is observed to be 20 m long when in motion. How fast is it moving?25PA mechanism on Earth used to shoot down geosynchronous satellites that house laser-based weapons is finally perfected and propels golf balls at 0.94c. (Geosynchronous satellites are placed 3.58 104 km above the surface of the Earth.) (a) What is the distance from the Earth to the satellite, as measured by a detector placed inside the golf ball? (b) How much time will it take the golf ball to make the journey to the satellite in the Earths frame? How much time will it take in the golf balls frame?27PImagine that in another universe the speed of light is only 100 m/s. (a) A person traveling along an interstate highway at 120 km/h ages at what fraction of the rate of a person at rest? (b) This traveler passes by a meterstick at rest on the highway. How long does the meterstick appear?29P30P31PA proton and an antiproton are moving toward each other in a head-on collision. If each has a speed of 0.8c with respect to the collision point, how fast are they moving with respect to each other?Imagine the speed of light in another universe to be only 100 m/s. Two cars are traveling along an interstate highway in opposite directions. Person 1 is traveling 110 km/h, and person 2 is traveling 140 km/h. How fast does person 1 measure person 2 to be traveling? How fast does person 2 measure person 1 to be traveling?34PThree galaxies are aligned along an axis in the order A, B, C. An observer in galaxy B is in the middle and observes that galaxies A and C are moving in opposite directions away from him, both with speeds 0.60c. What is the speed of galaxies B and C as observed by someone in galaxy A?36P37PConsider a reference system placed at the U.S. Naval Observatory in Washington, D.C. Two planes take off from Washington Dulles Airport, one going eastward and one going westward, both carrying a cesium atomic clock. The distance around the Earth at 39 latitude (Washington, D.C.) is 31,000 km, and Washington rotates about the Earths axis at a speed of 360 m/s. Calculate the predicted differences between the clock left at the observatory and the two clocks in the airplanes (each traveling at 300 m/s) when the airplanes return to Washington. Include the rotation of the Earth but no general relativistic effects. Compare with the predictions given in the text.39P40PUse the Lorentz transformation to prove that s2 = s2.42PProve that for a spacelike interval, two events cannot occur at the same place in space.Given two events, (x1, t1) and (x2, t2), use a spacetime diagram to find the speed of a frame of reference in which the two events occur simultaneously. What values may Δs2 have in this case?
45PConsider a fixed and a moving system with their clocks synchronized and their origins aligned at t = t′ = 0. (a) Draw on a spacetime diagram in the fixed system a line expressing all the events occurring at t′ = 0. (b) Draw on this diagram a line expressing all the events occurring at x′ = 0. (c) Draw all the world-lines for light that pass through t = t′ = 0. (d) Are the x′ and ct′ axes perpendicular? Explain.
47PAn astronaut is said to have tried to get out of a traffic violation for running a red light ( = 650 nm) by telling the judge that the light appeared green ( = 540 nm) to her as she passed by in her high-powered transport. If this is true, how fast was the astronaut going?49PDo the complete derivation for Equation (2.33) when the source and receiver are receding with relative velocity v.A spacecraft traveling out of the solar system at a speed of 0.95c sends back information at a rate of 1400 kHz. At what rate do we receive the information?52P53P54PNewtons second law is given by F=dp/dt. If the force is always perpendicular to the velocity, show that F=ma, where a is the acceleration.Use the result of the previous problem to show that the radius of a particles circular path having charge q traveling with speed v in a magnetic field perpendicular to the particles path is r= p/qB. What happens to the radius as the speed increases as in a cyclotron?57P58PA particle having a speed of 0.92c has a momentum of 1016 kg m/s. What is its mass?A particle initially has a speed of 0.5c. At what speed does its momentum increase by (a) 1%, (b) 10%, (c) 100%?61P62P63P64P65P66P67P68P69P70PWhat is the speed of an electron when its kinetic energy is (a) 10% of its rest energy, (b) equal to the rest energy, and (c) 10 times the rest energy?72P73P74P75PCalculate the energy needed to accelerate a spaceship of mass 10,000 kg to a speed of 0.3c for intergalactic space exploration. Compare this with a projected annual energy usage on Earth of 1021 J.
77P78P79P80PThe Large Hadron Collider at Europes CERN facility is designed to produce 7.0 TeV (that is, 7.0 1012 eV) protons. Calculate the speed, momentum, and total energy of the protons.What is the kinetic energy of (a) an electron having a momentum of 40 GeV/c? (b) a proton having a momentum of 40 GeV/c?
A muon has a mass of 106 MeV/c2. Calculate the speed, momentum, and total energy of a 200-MeV muon.
84PThe reaction 2H + 3H → n + 4He is one of the reactions useful for producing energy through nuclear fusion, (a) Assume the deuterium (2H) and tritium (3H) nuclei are at rest and use the atomic mass units of the masses in Appendix 8 to calculate the mass-energy imbalance in this reaction. This amount of energy is given up when this nuclear reaction occurs. (b) What percentage of the initial rest energy is given up?
Instead of one positive charge outside a conducting wire, as was discussed in Section 2.14 and shown in Figure 2.34, consider a second conducting wire parallel to the first one. Both wires have positive and negative charges, and the wires are electrically neutral. Assume that in both wires the positive charges travel to the right and negative charges to the left. (a) Consider an inertial frame moving with the negative charges of wire 1. Show that the second wire is attracted to the first wire in this frame. (b) Now consider an inertial frame moving with the positive charges of the second wire. Show that the first wire is attracted to the second. (c) Use this argument to show that electrical and magnetic forces are relative.
87PShow that the following form of Newton’s second law satisfies the Lorentz transformation. Assume the force is parallel to the velocity.
89PFor the twins Frank and Mary described in Section 2.8, consider Mary’s one-way trip at a speed of 0.8c to the star system 8 lightyears from Earth. Compute the spacetime interval s in the fixed frame and s′ in the moving frame, and compare the results.
Frank and Mary are twins. Mary jumps on a spaceship and goes to the star system Alpha Centauri (4.30 light-years away) and returns. She travels at a speed of 0.8c with respect to Earth and emits a radio signal every week. Frank also sends out a radio signal to Mary once a week, (a) How many signals does Mary receive from Frank before she turns around? (b) At what time does the frequency of signals Frank receives suddenly change? How many signals has he received at this time? (c) How many signals do Frank and Mary receive for the entire trip? (d) How much time does the trip take according to Frank and to Mary? (e) How much time does each twin say the other twin will measure for the trip? Do the answers agree with those for (d)?
A police radar gun operates at a frequency of 10.5 GHz. The officer, sitting in a patrol car at rest by the highway, directs the radar beam toward a speeding car traveling 80 mph directly away from the patrol car. What is the frequency shift of the reflected beam, relative to the original radar beam?
93P94PA proton moves with a speed of 0.90c. Find the speed of an electron that has (a) the same momentum as the proton, and (b) the same kinetic energy.
A high-speed K0 meson is traveling at a speed of 0.90c when it decays into a π+ and a π− meson. What are the greatest and least speeds that the mesons may have?
97PThe International Space Federation constructs a new spaceship that can travel at a speed of 0.995c. Mary, the astronaut, boards the spaceship to travel to Barnard’s star, which is the second nearest star to our solar system after Alpha Centauri and is 5.98 lightyears away. After reaching Barnard’s star, the spaceship travels slowly around the star system for three years doing research before returning back to Earth. (a) How much time does her journey take? (b) How much older is her twin Frank than Mary when she returns?
99P100PA spaceship is coming directly toward you while you are in the International Space Station. You are told that the spaceship is shining sodium light (with an intense yellow doublet of wavelengths 588.9950 and 589.5924 nm). You have an apparatus that can resolve two closely spaced wavelengths if the difference is Δλ < 0.55 nm. If you find that you can just resolve the doublet, how fast is the spaceship traveling with respect to you?
Quasars are among the most distant objects in the universe and are moving away from us at very high speeds, as discussed in Chapter 16. Astrophysicists use the redshift parameter z to determine the redshift of such rapidly moving objects. The parameter z is determined by observing a wavelength λ′ of a known spectral line of wavelength λsource on Earth; z = Δλ/λsource = (λ′ − λsource) /λsource. Find the speed of two quasars having z values of 1.9 and 4.9.
One possible decay mode of the neutral kaon is K0 → π0 + π0. The rest energies of the K0 and π0 are 498 MeV and 135 MeV, respectively. The kaon is initially at rest when it decays. (a) How much energy is released in the decay? (b) What are the momentum and relative directions of the two neutral pions (π0)?
104P105PSmall differences in the wavelengths in the sun’s spectrum are detected when measurements are taken from different parts of the sun’s disk. Specifically, measurements of the 656-nm line in hydrogen taken from opposite sides on the sun’s equator—one side approaching Earth and the other receding—differ from each other by 0.0090 nm. Use this information to find the rotational period of the sun’s equator. Express your answer in days. (The sun’s equatorial radius is 6.96 × 108 m.)
1Q2Q3Q4Q5Q6Q7Q8Q9QIn the experiment of Example 3.2, how could you explain an experimental value of q = 0.81019 C?11Q12Q13Q14Q15Q16Q17Q18Q19Q20Q21Q22Q23Q24Q25Q26Q1P2PAcross what potential difference does an electron have to be accelerated to reach the speed v = 1.8 107 m/s? Work the problem both nonrelativistically and relativistically and compare the results.4P5P6P7P8P9P10P11P12P13P14P15P16PCalculate max for blackbody radiation for (a) liquid helium (4.2 K), (b) room temperature (293 K), (c) a steel furnace (2500 K), and (d) a blue star (9000 K).18P19P20PWhite dwarf stars have been observed with a surface temperature as hot as 200,000C. What is the wave-length of the maximum intensity produced by this star?22P23P24P25P26P27P32P33P34P35P36P37P38P39P40P41P42P43P44P45P46P47P48P49P50P52P53P54P55P56P57P58P59P60P61P62P63P64P65P66P67P68PThe Fermi Gamma-ray Space Telescope, launched in 2008, can detect gamma rays with energies ranging from 10 keV to 300 MeV. For each of those energy extremes, find the resulting kinetic energy and speed of an electron created by the gamma ray as part of an electron-positron pair. Assume that the electron has half of the gamma ray’s energy.
70P1Q2Q3Q4Q5Q6Q7Q8Q9Q10Q11Q12Q13Q2P3P4P5P6P7PWhat fraction of 5-MeV α particles will be scattered through angles greater than 8° from a gold foil (Z = 79, density = 19.3 g/cm3) of thickness 10−8 m?
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