Question

A hemisphere of mass $3$ m and radius $R$ 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. If particle is displaced with a negligible velocity, then find the angular velocity of the particle relative to the centre of the hemisphere at an 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. $\frac{2c}{R\cos \theta }$

Answer 1

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

where the hemisphere has moved right by the distance x particle slide down through angle theta there are no external forces some positional horizontally

$m(Rsin\theta -x)=Mx$

$mRsin\theta =(M+m)x$

differentiate

$mR\omega cos\theta =(M+m)v$

$\omega =\frac{5v}{Rcos\theta }$