Question

A hemisphere of mass 3$m$ and radius $R$ is free to slide with its base on asmooth horizontal table. A particle of mass $m$ is placed on the top of thehemisphere. If particle is displaced with a negligible velocity, then find the angular velocity of the particle relative to the centre of the hemisphere atan angular displacement $\theta$ , when velocity of hemisphere is $v:$

• A. $\frac{4v}{R\cos \theta }$
• B. $\frac{3v}{R\cos \theta }$
• C. $\frac{5v}{R\cos \theta }$
• D. $\frac{2v}{R\cos \theta }$

Answer 1

Answer: (B)

$\frac{3v}{R\cos \theta }$

$\begin{array}{c}\text{Let the velocity of the Particle at the instant}\\ \text{be}{v}^{\prime }\text{and angular velocity}w\\ \text{Then By Linear momentum conseration-}\\ \begin{array}{c}3mv=m{v}^{\prime }\cos \theta \\ \Rightarrow v^{\prime }=\frac{3v}{\cos \theta }\end{array}\end{array}$

$\begin{array}{c}\Rightarrow \text{Angular velocity}w=\frac{v^{\prime }}{R}\\ \Rightarrow w=\frac{3v}{R\cos \theta }\\ \text{Hence, option (B) is correct.}\end{array}$