The Rotational Variables — Question 6

Question 6

Sampling can disguise a rapidly turning rotor

A strobe flashes every $0.10\,\mathrm s$. A single mark appears $30^\circ$ farther counterclockwise at each flash, modulo one turn. Assume constant angular velocity.

1. Find every angular velocity consistent with the observations.

2. If independent information gives $|\omega|<20\,\mathrm{rad/s}$, identify the unique value.

Question 6 - Solution

Sampling can disguise a rapidly turning rotor

(a) Derivation and calculation

The sampled increment includes an unknown whole number of turns: $$\Delta\theta_k=\frac{\pi}{6}+2\pi k,\qquad k\in\mathbb Z.$$ $$\boxed{\omega_k=\frac{\Delta\theta_k}{0.10} =\frac{5\pi}{3}+20\pi k\ \mathrm{rad/s}}.$$

(b) Result and interpretation

Apply the independent speed bound: $$-20<\frac{5\pi}{3}+20\pi k<20 \quad\Longrightarrow\quad -0.402<k<0.235.$$ $$\boxed{k=0,\qquad\omega=5.236\ \mathrm{rad/s}}.$$ All other integer branches are rejected by the bound.

Physical check

Adjacent possible rates differ by $2\pi/\Delta t=20\pi$ rad/s, more than the full allowed speed interval.