Uniform Circular Motion — Question 8

Question 8

Repeated photographs can hide whole revolutions

A point is constrained to a circle of radius 1 m. Photographs taken at every integer second always show it at $(1,0)$ m. Between photographs, its angular speed is constant, but its value and sense are unknown.

1. Give a stationary history and two nonstationary uniform histories compatible with all photographs.

2. Find the possible angular speeds and linear speeds for a complete family of such histories.

3. Can acceleration magnitude be inferred uniquely? What additional observation could distinguish two candidate histories?

Question 8 - Solution

Repeated photographs can hide whole revolutions

(a) Account for unobserved full turns.

$$\theta(t)=\omega t:\quad\omega=0,\ +2\pi,\ -2\pi\quad\mathrm{s^{-1}}\text{ all fit the samples}$$ Rest at $(1,0)$ is compatible. So are counterclockwise motion at $2\pi$ s$^{-1}$ and clockwise motion at $-2\pi$ s$^{-1}$. Each returns to the same position after every 1 s sampling interval.

(b) Use phase agreement modulo a full revolution.

$$\omega(1\,\mathrm s)=2\pi n\ \Rightarrow\ \boxed{\omega=2\pi n\,\mathrm{s^{-1}},\quad v=2\pi|n|\,\mathrm{m/s}},\quad n\in\mathbb Z$$ The family is $\omega=2\pi n$ s$^{-1}$ for any integer n. Speed is $2\pi|n|$ m/s. The sign chooses the sense; n=0 gives rest. The nonzero choices represent arbitrarily many complete turns between samples.

(c) Do not infer a derivative from aliased positions.

$$a=\omega^2R=4\pi^2n^2\,\mathrm{m/s^2},\quad\theta(.5\,\mathrm s)=n\pi$$ Acceleration magnitude is $\omega^2R=4\pi^2n^2$ m/s$^2$, not uniquely determined. A direct velocity measurement would distinguish these candidates. An extra image at a suitable noninteger time can distinguish chosen candidates. A half-second image distinguishes rest from one turn per second, but not rest from two turns per second.