The Relativity of Time — Question 7

Question 7

A circular clock dilates despite its changing velocity direction

An ideal clock circles a ring of laboratory radius R at constant speed $v=0.60c$. Assume the clock hypothesis: acceleration has no additional direct effect on its rate. Use $d\tau=dt\sqrt{1-v^2/c^2}$ for an ideal clock, with $c=3.00\times10^8\,\mathrm{m/s}$ when needed.

1. Find the proper time per revolution in terms of R and c.

2. Compare with a stationary clock at the center over one laboratory revolution.

Question 7 - Solution

A circular clock dilates despite its changing velocity direction

(a) Method and calculation

$$T=2\pi R/v=\frac{10\pi R}{3c}.$$$$\tau=T\sqrt{1-0.60^2}=\boxed{\frac{8\pi R}{3c}}.$$

(b) Interpretation and further calculation

$$\tau/T=\boxed{0.80}.$$$$\boxed{T-\tau=\frac{2\pi R}{3c}}.$$

Physical check

The integral depends on the speed magnitude in this inertial frame. Changing direction does not remove the accumulated time dilation.