The Rotational Variables — Question 9

Question 9

A short arc measures an angle only after units are settled

A pointer tip at radius $0.30\,\mathrm m$ traces $0.12\,\mathrm m$ of circular arc. A student inserts the resulting angle as degrees into a calculation that requires radians.

1. Find the angle in radians and degrees.

2. If the arc is measured within $[0.118,0.122]\,\mathrm m$ and the radius within $[0.295,0.305]\,\mathrm m$, find sharp angle bounds in radians.

Question 9 - Solution

A short arc measures an angle only after units are settled

(a) Derivation and calculation

The arc formula requires radian measure: $$\theta=\frac sr=\frac{0.12}{0.30}=\boxed{0.400\ \mathrm{rad}}.$$ $$\theta_{\mathrm{deg}}=0.400\frac{180}{\pi}=\boxed{22.92^\circ}.$$ Inserting $22.92$ into $s=r\theta$ without conversion would overestimate the arc by $180/\pi$.

(b) Result and interpretation

For positive independent intervals, $\partial(s/r)/\partial s>0$ and $\partial(s/r)/\partial r<0$. Therefore $$\theta_{\min}=\frac{0.118}{0.305}=0.386885,\qquad \theta_{\max}=\frac{0.122}{0.295}=0.413559.$$ $$\boxed{0.3869\le\theta\le0.4136\ \mathrm{rad}}.$$ Each extreme is attainable at the corresponding opposite endpoint pair.

Physical check

The nominal value $0.400$ rad lies within the sharp bounds.