I D a. :ntta1 I C a m I. Homework Problems 83. Figure R4.7. M is f v I = V GM/R, where G is the universal gravitational

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3 Homework Problems 83 Event E Event F Brian :ntta1 I C a m I (a) Tram (b) Train Brian I D a ve... itl 11 i 11 i i i Figure R4.7 (a) Event E and (b) event Fin the situation described in problem R4M.3. R4M.3 Alice is driving a race car around an essentially circular track at a constant speed of 60 m/s. Brian, who is sitting at a fixed position at the edge of the track, measures the time that Alice takes to complete a lap by starting his watch when Alice passes by his position (call this event E) and stopping it when Alice passes his position again (call this event f). This situation is also observed by Cara and Dave, who are passengers in a train that passes very close to Brian. Cara happens to be passing Brian just as Alice p sses Brian the first time, and Dave happens to pass Brian just as Alice passes Brian the second time (see fi gu re R4.7). Assume the clocks used by Alice, Brian, and Cara are close enough together that we can consider them all to be "present" at event E; similarly, that those used by Alice, Brian and Dave are "present" at event f. Assume the ground frame is an inertial reference frame. (a) Who measures the shortest time between these events? Who measures the longest? Explain. (b) If Brian measures 100 s between the events, how much less time does Alice measure between the events? (c) If Cara's and Dave's train moves at a speed of 30 m/s, how much larger or smaller is the time that they measure compared to Brian's time? Explain carefully. (d) Chris and Dylan are moving in the ground frame. Shouldn't they therefore measure less time between the events than Bob? Explain why the "moving clocks run slow idea" is very misleading here. R4M.4 The half-life of a muon at rest is 1.52 µs. One can store muons for a much longer ti e (as measured in the laboratory) by accelerating them to a speed very close to that of light and then keeping them circulating at that speed in an evacuated ring. Assume you want to design a ring that can keep muons moving so fast that they have a laboratory half-life of 0.25 s (about an eye-blink). (a) How fast must the muons be moving? (Hint: Define u = D.T/ D.t, write an equation that links u to Iv I, then solve for I v I and use the binomial approximation. You may need to do the final calculation of Iv I by hand.) (b) If the ring is 7.01 min diameter, how long will it take a muon to go once around the ring in the lab frame? R4M.5 Suppose some astronauts travel in a near-earth orl::lit at an altitude of 200 km for 225 orbits. (a) In unit N, we saw that the speed of an object in a circular orbit of radius R around an object with mass M is f v I = V GM/R, where G is the universal gravitational constant. Ar gu e that in SR units, G = G 15 /J/ c 3 = X s/kg. (b) About how much less time passed between the departure and arrival of the spaceship according to the astronauts' clocks than passed on the ground? Assume for the sake of simplicity that the surface of the earth defines an inertial reference frame. R4M.6 The satellites used in the Global Positioning System go around the earth in circular orbits whose radius is 26,600 km and period is 12 h exactly. Assume for the sake of simplicity that the earth is not rotating, so that a clock on its surface is in an inertial frame. (a) In unit N, we saw that the speed of an object in a circular orbit of radius R around an object with mass M is Iv I=./GM/R, where G is the universal gravitational constant. Argue that in SR units, G = G 15 /J/ c 3 = X s/kg. (b) Let event A be a certain GPS satellite passing a given position in space and event B be it passing that point again after one complete orbit. At each event, this satellite sends a radio signal to a clock directly below it on the (nonrotating) earth, which receives the signals at events C and D, respectively. What is the difference between the time an atomic clock on board the satellite registers between events A and B and the time a clock on the earth's surface registers between events C and D? Express your result symbolically in terms of G, M, and R (don't crunch numbers yet), though you can assume that GM/R << 1. (Hint: Argue that the signal's travel time is the same in both cases.) (c) Now calculate numerically how much less time a clock on the GPS satellite measures for a complete orbit than the clock on the ground does. (See problem R4A.3 for a discussion of how the earth's gravity affects GPS satellite clock rates.) R4M.7 Integrating equation R4.6 is a lot less tricky if Iv I is always small enough that we can use the binomial approximation. Suppose a spaceship starts from rest from Space Station Alpha floating in deep space and accelerates at a

4 Answers to Exercises 85 offs. The simplest curved worldline between the starbases is a circle in space with radius ½ D. If you follow such a worldline, will you beat a spaceship that simply travels directly at a constant velocity between the starbases? Advanced R4A.1 If you know about Taylor series, you can prove the binomial approximation quite generally. Any continuous and differentiable function f (x) can be expressed in terms of a Taylor series expansion as follows: [df] x 2 [d 2 f] x 3 [d 3 J] f(x) = f (O) + x (R4.25) dx x=d 2. dx x=o 3. dx x=d Apply this to the functionf(x) = (1 + x)" and show that if you drop terms in this power series involving x 2 or higher, you end up with the binomial approximation. Also show how you would write the approximation if you were to keep terms involving x 2 but drop higher-order terms. R4A.2 Consider an inertial frame at rest with respect to the earth. We observe an alien spaceship to move along the x axis of this frame in such a way that x(t) = ti [sin(wt + ¼1r) - b] (R4.26) where both x and t are measured in the inertial reference frame, w = 1f /2 rad/h, and b = sin( 7f / 4). Assume also that the earth is located at the origin (x = 0) in this frame. (a) Argue that the ship passes the earth at t = 0 and again at t = 1.0 h. (Hint: The value of wt is 1r/2 at this time.) (b) Draw a quantitatively accurate spacetime diagram of the spaceship's worldline, labeling the events where and when it passes the earth as events A and B. (c) Show that the ship's x-velocity is Vx = cos(wt + 1r/4) as measured in the inertial frame attached to the earth. (Hint: You don't need to use any relativity!) (d) Find the proper time measured by clocks on the alien ship between the events where it passes earth the first and second times. (Hint: l - cos 2 x = sin 2 x.) R4A.3 Consider the Global Positioning System satellites described in problem R4M.6. Again, for simplicity's sake, suppose the earth is not rotating. Let t:..t 0 be the time between two events that bracket one complete satellite orbit as measured by a clock at rest with respect to the earth but so far away that the effect of the earth's gravity on its rate is negligible. (We'll call this "the clock at infinity.") (a) The satellite's speed is Iv I =./GM/ R, where G is the universal gravitational constant (2.475 x s/kg in SR units), M is the earth's mass, and R = 26,600 km is the orbit's radius. Assuming that GM/R << 1, use the binomial approximation to find an expression (in terms of G, M, and R) for the discrepancy ot,. = t:..t 0 - t:..t between what the clock at infinity and the satellite's clock measure for a full orbit due to the satellite's motion. (b) General relativity states that a clock at rest a distance R from the center of a planet of mass M runs more slowly than the clock at infinity by the factor,/1-2gm/r (in SR units) due to the planet's gravitational field. Find a symbolic expression for the gravity-induced discrepancy ot g between what the clock at infinity and the stationary clock at R measure for a full orbit. (c) Similarly find the gravity-induced discrepancy ot, between what the clock at infinity and a clock on the earth's surface at radius R, measure for a full orbit. (d) Find a symbolic expression for the total discrepancy ot between what the satellite's clock and the clock on the earth's surface measure for a full orbit, taking into account all of these effects. (e) Evaluate ot numerically and interpret its sign. ANSWERS TO EXERCISES R4X.l See the table below. All results are rounded to nine decimal places. lvl J1- lvl ½lvl We see that the approximation is accurate to four decimal places even when Iv I is as large as 0.1. R4X.2 The ratio of a muon's acceleration I a I in the laboratory frame to I g I is I a I IV ( 3.0 X 108 m ) 2 lgl = lglr = (9.8m//)(7.0lm) ls o::: 1.3 X (R4.27) R4X.3 Andrea is not moving at a constant speed, so the use of the constant-speed formula for proper time is not appropriate, even when one uses the average speed. On the other hand, she spends most of her time traveling at a constant speed, so this will be a reasonable approximation. To see whether this is likely to yield an answer too high or too low, let us consider a specific and fairly extreme case. Suppose Andrea travels at a speed of 0.86 for four-fifths of the time and is at rest for the remaining time, as measured in the sun-based frame: this yields the desired average of ¼(0.86) = In this case, though, we can compute the proper time more accurately by breaking each half of the path into two segments, one where she is moving at a constant speed of 0.86 and one where she is at rest. Andrea's total elapsed proper time in this particular case is (1 -_ ) 112 ¼ (25 y) + (1 ) 112 ½ (25 y) = 15 y (R4.28) instead of 18 y. If this example is any indication, using the average speed yields an estimate that is too high.

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