Special Relativity, Spring '10

Homework 5 (covering Mermin Chapter 6)

1. Summarize the three rules for moving clocks and meter sticks which, as of the end of chapter 6, have been established. (One has to do with synchronized clocks, one has to do with how fast moving clocks tick, and one has to do with how much moving sticks shrink.)

2. Alice (as usual) lives on a train whose length she measures to be L. She sets up two synchronized clocks, at the back and front of the train. She is at the back of the train, and when the clock there reads "t=0" she sends a pulse of light forward toward the front of the train. There is a light-pulse-detection device wired up to the clock there which records the reading of the clock at the moment the pulse arrives. Clearly, since the light moves at speed c over a distance L, and since the clocks were synchronized, the device ends up recording "t = L/c". Now the question is: what precisely is the story by which Bob (who as usual lives on the tracks) accounts for this result? (As usual, the train moves at speed v w.r.t. the tracks.) Your answer should be in the form of a story composed of complete English sentences. The first sentence might begin something like this: "Silly Alice thought she synchronized her two clocks, but in fact, at the moment the light pulse was emitted from the rear of the train, the clock in the front of the train reads ..." (You'll have to do some math/physics/algebra work on the side to figure out exactly how to make the story work, of course.)

3. Here is a cute alternative way of deriving the "moving clocks run slow" rule that doesn't rely on the synchronized clocks rule. Imagine a crude sort of clock that consists of two mirrors (attached a distance L apart to a stick or something) between which light bounces back and forth. Suppose there is a little device on one end (next to one of the mirrors) that increments a counter each time the light bounces off that mirror. Thus the counter keeps time, in units of 2L/c. But now suppose the clock is moving at speed c -- perpindicular to the axis of the stick or whatever connects the two mirrors. (This is relevant because moving sticks only shrink along their direction of motion -- so with the clock moving this way, the spatial separation between the mirrors remains L, and we don't have to worry about shrinkage.) Draw a little picture of the path that the blip of light takes through space as it bounces back and forth between the (now moving) mirrors, and then use math -- and of course the postulate that light always moves at c -- to determine the amount of time that elapses between successive increments of the (moving) counter.
