Notes on Relativity and Cosmology - Physics Department, UCSB

140 CHAPTER 5. ACCELERATING REFERENCE FRAMES . . .
the clocks in rockets B and C. How do the clocks A, B, and C compare?
(Just say which is running fastest, slowest, etc. I don’t need a
quantitative answer.)
(c) What happens if an observer in rocket B compares the clocks?
(d) What happens if an observer in rocket C compares the clocks?
(e) Suppose that B and C make measurements of their separation from
A along their lines of simultaneity. How ‘fast’ do they find A to be
moving away from them? If they measure their separation from both
the front and back of A, what happens to their measurements of the
length of A as time passes?
5-9. Consider a wagon wheel of radius r with rigid spokes that starts at rest in
some inertial frame but then begins to spin rapidly so that the outside of
the wheel is moving at .8c relative to the inertial frame.
Since length contraction occurs only in the direction of motion, the radius
r of the wheel remains unchanged (as measured in the inertial frame).
(a) Suppose that you are in the original inertial frame. What value will
you measure for the circumference of the wheel? (Assume here that
the wheel does not break under the stress.)
(b) Suppose that you measure the circumference by tacking down measuring
rods around the outside of the spinning wheel (so that the
rods spin with the wheel). What value of the circumference will this
measurement produce?
(c) If the spokes of the wheel are connected by pieces of thread, what
will happen to the thread while the wheel is getting up to speed?
(d) Suppose that identical clocks are placed at the end of each spoke
and that the clocks are all synchronized in the original inertial frame
before the wheel is spun up. As viewed from the inertial frame, do the
clocks remain synchronized with each other after the wheel is spun
up? Do they remain synchronized with a clock that remains inertial?
(e) Consider again the clocks in (D). If you are standing at the end of
one of the spokes and make measurements of the clocks next to you,
what do you find? Are they synchronized with yours? Do they run
at the same rate?