The mass of a sphere placed on a smooth horizontal surface is distributed non-uniformly. A horizontal force equal to weight of sphere is applied at a distance 'a' from centre of the sphere for a short time. The sphere is rotated by an angle $\theta$ but it behaves as if in stable equilibrium. Find the distance between centre and CG of the sphere:

The sphere is in stable equilibrium. It will come back to original position after rotation. Rotating Moment = Fa. For stable equilibrium, 

Restoring moment = Rotating moment
Since $F=W,Wa=Fa$
Let O be the centre and C and C' be the positions of CG before and after rotation.

$OC=O{C}^{\mathrm{\prime }}\text{ and }{C}^{\mathrm{\prime }}B=a........(from equation(i\left)\right)$

$\text{ In }\mathrm{\Delta }OB{C}^{\mathrm{\prime }}\text{ , OC'= }{C}^{\mathrm{\prime }}B\mathrm{cosec}\theta =a\mathrm{cosec}\theta$