The Rotational Variables — Question 5

Question 5

Rigid rotation shares angles but not arc lengths

Two marks A and B lie at radii $0.20$ and $0.50\,\mathrm m$ on the same rigid disk. The disk rotates by $1.2\,\mathrm{rad}$ without reversing.

1. Find the arc length traced by each mark and compare their angular displacements.

2. Can the center mark be assigned the same changing polar angle from its position alone?

Question 5 - Solution

Rigid rotation shares angles but not arc lengths

(a) Derivation and calculation

Rigid rotation imposes a common orientation change: $$\boxed{\Delta\theta_A=\Delta\theta_B=1.2\ \mathrm{rad}}.$$ Using radian measure in $s=r\Delta\theta$, $$s_A=(0.20)(1.2)=\boxed{0.24\ \mathrm m},\qquad s_B=(0.50)(1.2)=\boxed{0.60\ \mathrm m}.$$

(b) Result and interpretation

At the center, $$r=0\quad\Longrightarrow\quad s=0,\qquad (x,y)=(0,0).$$ A position-based polar angle is undefined at this point. The changing orientation is instead assigned to a body-fixed line, which shares the disk’s $\Delta\theta$.

Physical check

The arc ratio $s_B/s_A=2.5$ equals the radius ratio, although both angular displacements are equal.