Gauss' law for Oscillations

As shown below left, a solid sphere of radius $R$ has density $\rho$ in the top half and $2\rho$ in the bottom half. It is kept in a uniform fluid of density $3\rho$ with a hinge at the bottom such that it can freely perform oscillations about any horizontal axis through the hinge.

If all the points of the sphere are to oscillate in their respective planes parallel to the plane $x = -\sqrt3 z$ (shown below right), the period of small oscillations is $2\pi \sqrt { \frac { \alpha R }{ \beta g } },$ where $\alpha, \beta$ are coprime positive integers. Find $\alpha - \beta$ .