A spherical cavity of radius $\frac{R}{2}$  is removed from a solid sphere of radius R as shown in figure. The sphere is placed on a rough horizontal surface as shown. The sphere is given a gentle push. Friction is large enough to prevent slippage. Find the time period of small oscillation

Location of centre of mass of the cavitied sphere is given by
 $$\begin{gathered} \frac{4}{3}\pi (R^3-\frac{R^3}{8})\rho x=\frac{4}{3}\pi \frac{R^3}{8}\rho \frac{R}{2} \\ \Rightarrow \frac{7}{8}x=\frac{R}{16}[\rho =density] \\ \Rightarrow x=\frac{R}{14} \end{gathered}$$
 
 Moment of inertia of the cavitied sphere about an axis (${\perp }^r$ to plane of the figure) through point of contact (P) is calculated as follows
Let M = mass of cavitied sphere
 Mass of sphere of radius $\frac{R}{2}$ is $m=\rho \frac{4}{3}\pi \frac{R^3}{8}=\frac{M}{7}$
Mass of sphere without cavity 
 Required moment of inertia
I = (moment of inertia of complete shphere without cavity about an axis through P)
(moment of inertia of the cavity about the same axis)
 $\frac{7}{5}\frac{8M}{7}{R}^2-\left[\frac{47}{20}\frac{M}{7}{R}^2\right]=\frac{177}{140}M{R}^2$
A purely rolling sphere can be considered to be is pure rotation about the point of contact. Consider the sphere at a slightly displaced position , as shown.

Restoring torque in this position is