The Rotational Variables — Question 2

Question 2

Reversal separates angular displacement from angular travel

A disk turns from unwrapped angle $0$ to $5\,\mathrm{rad}$ in $2\,\mathrm s$, then reverses and reaches $2\,\mathrm{rad}$ after another $1\,\mathrm s$.

1. Find net angular displacement, total angular travel and both corresponding time averages.

2. Can either average specify the instantaneous angular velocity at reversal?

Question 2 - Solution

Reversal separates angular displacement from angular travel

(a) Derivation and calculation

Signed displacement adds algebraically; angular travel adds magnitudes: $$\Delta\theta=(5-0)+(2-5)=\boxed{2\ \mathrm{rad}},\qquad \mathcal S=|5|+|-3|=\boxed{8\ \mathrm{rad}}.$$ $$\bar\omega=\frac{2}{2+1}=\boxed{\frac23\ \mathrm{rad/s}},\qquad \overline{|\omega|}=\frac{8}{3}=\boxed{\frac83\ \mathrm{rad/s}}.$$

(b) Result and interpretation

These are interval averages. If the reversal is smooth at $t_r$, continuity and the sign change imply $$\omega(t_r)=\left.\frac{d\theta}{dt}\right|_{t_r}=\boxed{0}.$$ The endpoint data alone do not impose smoothness or determine the instantaneous rate.

Physical check

Angular travel exceeds the magnitude of net displacement because part of the motion is retraced.