Consider the object shown in the figure. It consist of a solid hemisphere of mass M and radius R. There is a solid rod welded at its centre. The object is placed on a flat surface so that the rod is vertical. Mass of the rod per unit length is $\frac{\mathrm{M}}{2\mathrm{R}}$.What is the maximum length of the rod that can be welded so that the object can perform oscillations about the position shown in diagram? Note : Centre of mass of a solid hemisphere is at a distance of $\frac{3\mathrm{R}}{8}$from its base  

The object will perform oscillations if it is in stable equilibrium. 

It will be in stable equilibrium if its COM lies below centre of the sphere, C. Let the length of the rod be L. For COM of the system to lie below C
$$\begin{gathered} \text{M}\frac{3\mathrm{R}}{8}>\left(\frac{\mathrm{M}}{2\mathrm{R}}\mathrm{L}\right)\frac{\mathrm{L}}{2} \\ \Rightarrow \frac{3{\mathrm{R}}^2}{2}>{\mathrm{L}}^2\Rightarrow \mathrm{L}<\sqrt{\frac{3}{2}}\cdot \mathrm{R} \end{gathered}$$
Note :For better understanding, consider three spheres : (A) a uniform sphere (B) a non uniform sphere whose COM is above the geometrical centre (C) a non uniform sphere whose COM is below its geometrical centre.
 

In (A), the COM is always above contact point, hence force mg causes no torque about the point of contact. The sphere is in neutral equilibrium. In (B), if the sphere rotates a little, the COM moves sideways and produces a torque that tends to rotate the sphere farther away. The sphere is in unstable equilibrium. In (C), if the sphere moves a little, the COM moves sideways and this produces a restoring torque. Hence this is position of stable equilibrium.