Question

A hemisphere of radius $R$ and of mass $4m$ is free to slide with its base on a smooth horizontal table. A particle of mass $m$ is placed on the top of the hemisphere. Find the angular velocity of the particle relative to hemisphere at an angular displacement $\theta$ when velocity of hemisphere has become $v$.

Answer 1

Let the hemisphere be moved by a distance of x through angle $\theta$. CM remains same as there are no forces acting horizontally

$\begin{array}{c}m\left(R\sin \theta -x\right)=Mx\\ mR\sin \theta =\left(M=m\right)x\\ Differentiating,\\ mR\omega \cos \theta =\left(M+m\right)v\\ wherev=xand\omega =\theta .\\ wkt,M=4m\\ \omega =\frac{5v}{R\cos \theta }\end{array}$