Question

A disc of radius $R$ and mass $m$ is mounted on a vertical axis through its centre and perpendicular to its plane. A rat of same mass $m$ runs around the rim of the disc with speed $v$ anticlockwise relative to stationary observer. The disc rotates clockwise with angular speed $\omega$. The rat finds a piece of bread on the rim and stops. Find the angular velocity of the disc after the rat stops.

Answer 1

Angular momentum of system when rat was running $=\frac{M{R}^2}{2}\omega -M{R}^2\frac{v}{R}=M{R}^2(\frac{\omega }{2}-\frac{v}{R})$

Angular momentum when disc stops$=(\frac{M{R}^2}{2}+M{R}^2){\omega }^{{}^{\prime }}=\frac{3}{2}M{R}^2{\omega }^{{}^{\prime }}$

Since, there is no external torque $\Rightarrow \frac{3}{2}M{R}^2{\omega }^{{}^{\prime }}=M{R}^2(\frac{\omega }{2}-\frac{v}{R})$

$\Rightarrow {\omega }^{{}^{\prime }}=\frac{2}{3}(\frac{\omega }{2}-\frac{v}{R})$