Question

A uniform thin hemispherical shell is kept at rest and in equilibrium on an inclined plane of angle of inclination $\theta =30^{\circ }$ as shown in the figure. If the surface of the inclined plane is sufficiently rough to prevent sliding then the angle $\alpha$ (in degree) made by the plane of hemisphere with inclined plane is

Answer 1

Answer: 60

$O$ is the centre of mass of the hollow hemisphere and is at $\frac{R}{2}$ from $C$ .


$f=mgsin\theta$
$N=mgcos\theta$
$N\times \frac{R}{2}sin\alpha =\left[R-\frac{R}{2}cos\alpha \right]f$
$\therefore tan\theta =\frac{sin\alpha }{2-cos\alpha }\Rightarrow \alpha =60^o$