(36.13) for approaching emitter

Special Relativity

c 3.2 10⁹ J

3.0 10⁸ m/s 1.1 10¹⁷ kgm/s

1244 CHAPTER 36 The Theory of Special Relativity

S U M M A RY

PRINCIPLE OF RELATIVITY All the laws of physics are the same in all inertial reference frames.

(36.5)

¢t ¢t

21 V²

^c²

PRINCIPLE OF UNIVERSALITY OF SPEED OF LIGHT The speed of light is the same in all inertial reference frames.

TIME DILATION registered by clock in its own reference frame.)

(¢t _1/

V 1–V ²/c²

5.0 4.0 3.0 2.0 1.0

0.2c

0 0.4c 0.6c 0.8c 1.0c

for receding emitter

(36.13) for approaching emitter

f B

1 V

^c^f

f B

1 V

^c^f

RELATIVISTIC DOPPLER SHIFT

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Summary 1245

LENGTH CONTRACTION is length of body in its own reference frame.)

(L by observers on Earth are short.

v at speeds close to c.

Kinetic energy at speeds close to c.

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1246 CHAPTER 36 The Theory of Special Relativity

4George Gamow, Mr. Tompkins in Wonderland.

10. According to the arguments of Section 36.3, a light signal trav-eling along a track placed perpendicular to the direction of motion of the spaceship (see Fig. 36.11) takes a longer time to complete a round trip when measured by the clocks on the Earth than when measured by the clocks on the spaceship.

Would the same be true for a light signal traveling along a track placed parallel to the direction of motion? Explain qualitatively.

11. A cannonball is perfectly round in its own reference frame.

Describe the shape of this cannonball in a reference frame rel-ative to which it has a speed of 0.95c. Is the volume of the cannonball the same in both reference frames?

12. A rod at rest in the ground makes an angle of with the x axis in the reference frame of the Earth. Will the angle be larger or smaller in the reference frame of a spaceship moving along the x axis?

13. In the charming tale “City Speed Limit” by George Gamow,⁴ the protagonist, Mr. Tompkins, finds himself riding a bicycle in a city where the speed of light is very low, roughly 30 km/h.

What weird effects must Mr. Tompkins have noticed under these circumstances?

14. A long spaceship is accelerating away from the Earth. In the reference frame of the Earth, are the instantaneous speeds of the nose and of the tail of the spaceship the same?

15. Suppose that a very fast runner holding a long horizontal pole runs through a barn open at both ends. The length of the pole (in its rest frame) is 6 m, and the length of the barn (in its rest frame) is 5 m. In the reference frame of the barn, the pole will suffer length contraction and, at one instant of time, all of the pole will be inside the barn. However, in the reference frame of the runner, the barn will suffer length contraction and all of the pole will never be inside the barn at one instant of time. Is this a contradiction?

16. Why can a spaceship not travel as fast as or faster than the speed of light?

17. If the beam from a revolving searchlight is intercepted by a distant cloud, the bright spot will move across the surface of the cloud very quickly, with a speed that can easily exceed the speed of light. Does this conflict with our conclusion of Section 36.6, that the speed of light is unattainable?

Q U E S T I O N S F O R D I S C U S S I O N

1. An astronaut is inside a closed space capsule coasting through interstellar space. Is there any way the astronaut can measure the speed of the capsule without looking outside?

2. Why did Michelson and Morley use two light beams, rather than a single light beam, in their experiment?

3. When Einstein was a boy he wondered about the following question: A runner holds a mirror at arm’s length in front of his face. Can he see himself in the mirror if he runs at (almost) the speed of light? Answer this question both according to the ether theory and according to the theory of Special Relativity.

4. Consider the piece of paper on which one page of this book is printed. Which of the following properties of the piece of paper are absolute, that is, which are independent of whether the paper is at rest or in motion relative to you? (a) The thick-ness of the paper, (b) the mass of the paper, (c) the volume of the paper, (d) the number of atoms in the paper, (e) the chemi-cal composition of the paper, (f ) the speed of light reflected by the paper, and (g) the color of the colored print on the paper.

5. Two streetlamps, one in Boston and the other in New York City, are turned on at exactly 6:00 P.M. Eastern Standard Time. Find a reference frame in which the streetlamp in New York was turned on late.

6. According to the theory of Special Relativity, the time order of events can be reversed under certain conditions. Does this mean that a sparrow might fall from the sky before it leaves the nest?

7. Because of the rotational motion of the Earth about its axis, a point on the equator moves with a speed of 460 m/s relative to a point on the North Pole. Does this mean that a clock placed on the equator runs more slowly than a similar clock placed on the pole?

8. According to Jacob Bronowski, author of The Ascent of Man, the explanation of time dilation is as follows: If you are moving away from a clock tower at a speed nearly equal to the speed of light, you keep pace with the light that the face of the clock sent out at, say, 11 o’clock. Hence, if you look toward the clock tower, you always see its hands at 11 o’clock. Is this explanation correct? If not, what is wrong with it?

9. Suppose you wanted to travel into the future and see what the twenty-fifth century is like. In principle, how could you do this? Could you ever return to the twenty-first century?

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Problems 1247

†For help, see Online Concept Tutorial 41 at www.wwnorton.com/physics

††For help, see Online Concept Tutorial 42 at www.wwnorton.com/physics

P R O B L E M S

†3 6 . 1 T h e S p e e d o f L i g h t ; t h e E t h e r

1. Consider the case where the Sun moves at a high speed v through the hypothetical ether. What are the minimum and maximum ether-wind speeds on Earth when the Sun moves through the ether at (a) 30 km/s and (b) 60 km/s? Assume the orbital speed of the Earth is 30 km/s.

*2. A Michelson–Morley interferometer determines the shift of two waves traveling in perpendicular directions (see also Section 35.2 and Fig. 35.9).

(a) Assume that one wave travels a distance L along the ether wind with speed to a mirror, and back with speed

as in Figs. 36.3a–b. Show that the round-trip time can be written

(b) Assume the other wave travels the same distance perpen-dicular to the ether wind with speed (see Fig.

36.3c). Show that its round-trip time is

(d) What fraction of a full period is this shift for light with Use the values and

*3. Ordinarily, the two arms of a Michelson–Morley interferome-ter cannot be set exactly equal, and instead have two values, and Insert these respective values into the results of Problem 2a and b and obtain a new expression for (see Problem 2c). Note that this result alone cannot be used to determine the ether-wind speed, since the difference between

and is not accurately known. In an actual experiment, the entire apparatus is rotated (thus interchanging and

Obtain an expression for the net shift by subtracting the differences in arrival times for the two orientations.

†,††3 6 . 2 E i n s t e i n ’s P r i n c i p l e o f R e l a t i v i t y

4. A spaceship traveling at speed c relative to the Earth ejects a spacepod traveling in the forward direction at speed c relative to the spaceship. The spacepod emits a light signal toward the Earth at speed c relative to the spacepod. What is the speed of the light signal relative to the spaceship? What is the speed of the light signal relative to the Earth? Which observer (on the spaceship or on the spacepod) determines that the light strikes the Earth earlier?

5. If a moving clock is to have a time-dilation factor of 10, what must be its speed?

6. Neutrons have an average lifetime of 15 minutes when at rest in the laboratory. What is the average lifetime of neutrons of a speed of 25% of the speed of light? 50%? 90%?

7. Consider an unstable particle, such as a pion, which has a life-time of only when at rest in the laboratory.

What speed must you give such a particle to make its lifetime twice as long as when at rest in the laboratory?

8. The speed of the Sun around the center of our Galaxy is 200 km/s. Clocks in the Solar System will therefore run slow as compared with clocks at rest in the Galaxy. By what factor are the Solar System clocks slow?

9. The orbital speed of the Earth around the Sun is 30 km/s. In one year, how many seconds do the clocks on the Earth lose with respect to the clocks of an inertial reference frame at rest relative to the Sun? [Hint: If Vc is small, the approximation

is valid.]

10. In 1961, the cosmonaut G. S. Titov circled the Earth for 25 h at a speed of 7.8 km/s. According to Eq. (36.5), what was the time-dilation factor of his body clock relative to the clocks on Earth? By how many seconds did his body clock fall behind during the entire trip? (Hint: Use the approximation given in Problem 9.)

11. At a speed V, the time-dilation factor has some value. Suppose that at speed 2V, the time-dilation factor has twice the previ-ous value. What is the speed V ?

12. An astronaut traveling at V 0.80c taps her foot 3.0 times per second. What is the frequency of taps determined by an observer on the Earth?

13. An atomic clock aboard a spaceship runs slow compared with an Earth-based atomic clock at a rate of 1.0 second per day.

What is the speed of the spaceship?

14. A spaceship equipped with a chronometer is sent on a round-trip to Alpha Centauri, 4.4 light-years away. The spaceship travels at 0.10 c, and returns immediately.

(a) According to clocks on the Earth, how long does this trip take?

(b) According to the chronometer on the spaceship, how long does this trip take?

15. Consider the Doppler-shift formula for a receding source. By what factor does the frequency decrease for For

For V 0.90c?

V 0.70c?

V 0.50c?

21 (V^c)²¹¹²^(V^c)²

2.6 10⁸ s

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16. The frequencies of light received from distant galaxies and quasars are shifted due to the Doppler effect. Frequencies 5.0 times smaller than expected for a stationary source have been detected from receding quasars. What is Vc for such a quasar?

*17. In a test of the relativistic time-dilation effect, physicists com-pared the rates of vibration of nuclei of iron moving at differ-ent speeds. One sample of iron nuclei was placed on the rim of a high-speed rotor; another sample of similar nuclei was placed at the center. The radius of the rotor was 10 cm, and it rotated at 35 000 rev/min. Under these conditions, what was the speed of the rim of the rotor relative to the center?

According to Eq. (36.5), what was the time-dilation factor of the sample at the rim compared with the sample at the center?

(Hint: Use the approximation given in Problem 9.)

*18. If cosmonauts from the Earth wanted to travel to the Andromeda galaxy in a time of no more than 10 years as reck-oned by clocks aboard their spaceship, at what (constant) speed would they have to travel? How much time would have elapsed on Earth after 10 years of time on the spaceship? The distance to the Andromeda galaxy is light-years.

*19. Because of the rotation of the Earth, a point on the equator has a speed of 460 m/s relative to a point at the North Pole.

According to the time-dilation effect of Special Relativity, by what factor do the rates of two clocks differ if one is located on the equator and the other at the North Pole? After 1.00 year has elapsed, by how many seconds will the clocks differ?

Which clock will be ahead? (Although the special-relativistic time dilation slows one clock at the equator, there is an addi-tional gravitaaddi-tional time dilation that slows the other clock.

These two time-dilation effects balance, and the two clocks actually run at the same rate.)

**20. The star Alpha Centauri is 4.4 light-years away from us.

Suppose that we send a spaceship on an expedition to this star.

Relative to the Earth, the spaceship accelerates at a constant rate of 0.10g until it reaches the midpoint, 2.2 light-years from Earth. The spaceship then decelerates at a constant rate of 0.10g until it reaches Alpha Centauri. The spaceship performs the return trip in the same manner.

(a) What is the time required for the complete trip according to the clocks on the Earth? Ignore the time that the spaceship spends at its destination.

(b) What is the time required for the complete trip according to the clocks on the spaceship? Assume that the instanta-neous time-dilation factor is still even though the speed V is a function of time.

3 6 . 4 L e n g t h C o n t r a c t i o n

21. A meterstick is moving by an observer in a direction parallel to its length. The speed of the meterstick is 0.50c. What is its measured length in the reference frame of the observer?

22. According to the manufacturer’s specifications, a spaceship has a length of 200 m. At what speed (relative to the Earth) will

21 V²^c²

2.2 10⁶

this spaceship have a length of 100 m in the reference frame of the Earth?

23. A cannonball flies through our laboratory at a speed of 0.30c.

Measurement of the transverse diameter of the cannonball gives a result of 0.20 m. What can you predict for the measurement of the length, or the longitudinal diameter, of the cannonball?

24. What is the percent length contraction of an automobile traveling at 96 km/h? (Hint: Use the approximation given in Problem 9.) 25. A hangar for housing spaceships is 100 m long. How fast must

a 200-m-long spaceship be traveling to (briefly) fit in the hangar?

26. A right triangle of sheet metal with two angles lies in the x–y plane, with one of its sides along the x axis (see Fig.

36.25). The length of each side is 0.20 m, and the length of the hypotenuse is Suppose that this triangle is observed from an reference frame moving at 0.80c along the x axis. What are the lengths of the sides and of the hypotenuse in this reference frame? What are the angles?

x12 0.20 m., y

45 1248 CHAPTER 36 The Theory of Special Relativity

*27. Two identical spaceships are traveling in the same direction.

An observer on Earth measures the first to have speed 0.80c and observes the second to be 1.50 times as long as the first one. What is the speed of the second spaceship?

*28. Suppose that a meterstick at rest in the reference frame of the Earth lies in the x–y plane and makes an angle of with the x axis. Suppose that one end of the meterstick is at the origin.

At a fixed time t, what are the x and y components of the dis-placement from this end of the meterstick to the other? At a fixed time what are the and components of the dis-placement from one end of the meterstick to the other in a new reference frame moving with velocity in the positive x direction? What is the angle the meterstick makes with the axis of this new reference frame?

*29. Electric charge is uniformly distributed throughout a sphere;

the charge density is If this sphere is put in motion relative to the laboratory at a speed of 0.80c, what will be the charge density? Keep in mind that the total amount of electric charge is unchanged by the motion of the sphere.

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*30. It can be shown that when a point charge moves at uniform velocity of relativistic magnitude, its pattern of electric field lines is contracted by the usual length-contraction factor

in the longitudinal direction and is unchanged in the transverse direction. Figure 36.26 shows the resulting pattern of field lines for a speed Draw a similar picture for a speed of 0.80c.

V 0.60c.

21 V²^c²

34. A spaceship has a length of 300 m, measured in its own refer-ence frame. It is traveling in the positive x direction at a speed of 0.80c relative to the Earth. A strobe light at the nose of the spaceship sends a pulse of light toward the tail of the spaceship.

(a) As measured in the reference frame of the spaceship, how long does this light pulse take to reach the tail?

(b) As measured in the reference frame of the Earth, how long does this light pulse take to reach the tail?

35. A spaceship is moving at a speed of 0.60c toward the Earth. A second spaceship, following the first one, is moving at a speed of 0.90c. What is the speed of the second spaceship as observed in the reference frame of the first?

36. Find the inverse of Eq. (36.36); that is, express in terms of 37. The captain of a spaceship traveling away from Earth in the x

direction at observes that a nova explosion occurs at a point with spacetime coordinates

as measured in the reference frame of the spaceship. He reports this event to the Earth via radio without delay.

(a) What are the spacetime coordinates of the explosion in the reference frame of the Earth? Assume that the master clock of the spaceship coincides with the master clock of the Earth at the instant when the midpoint of the spaceship passes by the Earth, and that the origin of the spaceship coordinates is at the midpoint of the spaceship.

(b) Will the Earth receive the captain’s report before or after astronomers on the Earth see the nova explosion in their telescopes? No calculation is required for this question.

*38. Consider the situation described in Problem 37. Since light takes some time to travel from the nova to the spaceship, the space and time coordinates that the captain reports are not directly measured but, rather, deduced from the time of arrival and the direction of the nova light reaching the spaceship.

(a) At what time time) did the nova light reach the space-ship?

(b) If the captain sends a report to Earth via radio as soon as he sees the nova, at what time (t time) does the Earth receive the report?

*39. At A.M. a boiler explodes in the basement of the Museum of Modern Art in New York City. At

A.M. a similar boiler explodes in the basement of a soup fac-tory in Camden, New Jersey, at a distance of 150 km from the first explosion. Show that, in the reference frame of a space-ship moving at a speed greater than from New York toward Camden, the first explosion occurs after the second.

*40. A radioactive atom in a beam produced by an accelerator has a speed 0.80c relative to the laboratory. The atom decays and ejects an electron of speed 0.50c relative to itself. What is the speed of the electron relative to the laboratory, if ejected in the forward direction? If ejected in the backward direction?

V 0.60c

FIGURE. 36.26 Electric field lines of a charge moving at V 0.60c.

FIGURE 36.27 A drive belt and two flywheels.

*31. A flexible drive belt runs over two flywheels whose axles are mounted on a rigid base (see Fig. 36.27). In the reference frame of the base, the horizontal portions of the belt have a

In document Physics for Engineers and Scientists -- 3rd Ed. Vol. 3 (Page 81-91)

(36.13) for approaching emitter

Special Relativity