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

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


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