Question

A cockroach of mass $m$ is moving on the rim of a disc with velocity $V$ in the anticlockwise direction. The moment of inertia of the disc about its own axis is $I$ and it is rotating in the clockwise direction with angular speed $\omega$. If the cockroach stops moving then the angular speed of the disc will be

• A. $\frac{I\omega }{I+m{R}^2}$
• B. $\frac{I\omega +mVR}{I+m{R}^2}$
• C. $\frac{I\omega -mVR}{I+m{R}^2}$
• D. $\frac{I\omega -mVR}{I}$

Answer 1

The correct option is C $\frac{I\omega -mVR}{I+m{R}^2}$

Angular momentum of the cockroach about the axis passing through the center of the disc: $L_c=mVR$
Angular momentum of the disc $L_d=I\omega$
$L_{total}=L_c+L_d=I\omega -mVR$ since both angular momentum are opposite to each other.
When the cockroach stops moving, due to conservation of angular momentum,
$(I+m{R}^2){\omega }^{\prime }=I\omega -mVR$
$\Rightarrow {\omega }^{\prime }=\frac{I\omega -mVR}{I+m{R}^2}$