A uniform thin hemispherical shall is kept at rest and in equilibrium on an inclined plane of angle of inclination $θ = 30^{o}$ as shown in figure. If the surface of the inclined plane is sufficiently rough to prevent sliding then the angle $α$ made by the plane of hemisphere with inclined plane is:

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TestHub · Accepted Answer

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

$f = m g \sin θ$ ...(i)

$N - m g \cos θ$ ...(ii)

$N \times \frac{R}{2} \sin α = [R \frac{R}{2} \cos α] f$ ...(iii)

$∴ \tan θ = \frac{\sin α}{2 - \cos α} \Rightarrow α = 60^{o}$

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