Question

A particle of mass $m$ is placed on top of a smooth hemispherical wedge of mass $4m$ and radius $R$ at rest. The hemispherical wedge is kept on a frictionless horizontal surface. The particle is given a gentle push. The angular velocity of the particle relative to the wedge, if wedge has velocity $v$ when particle has falls by an angle $\theta$ with respect to the wedge is

• A. $\frac{5v}{R\cos \theta }$
• B. $\frac{5v}{2R\cos \theta }$
• C. $\frac{4v}{R\cos \theta }$
• D. $\frac{5v}{R\sin \theta }$

Answer 1

The correct option is A $\frac{5v}{R\cos \theta }$

Let $v_1$ be velocity of $m$ wrt $4m,$
Horizontal component of velocity of $m$ wrt $4m$ is $v_1\cos \theta$ towards right
Horizontal component of velocity of $m$ is $(v_1\cos \theta -v)$ towards right

Applying conservation of momentum of the system along horizontal direction
$-4mv+m(v_1\cos \theta -v)=0$
$v_1=\frac{5v}{\cos \theta }$
Angular velocity at this instant is $\omega =\frac{v_1}{R}=\frac{5v}{R\cos \theta }$
$\therefore \left(a\right)$