A solid sphere of radius R is floating in a liquid of density $\rho$ with half of its volume submerged. If the sphere is slightly pushed and released, it starts performing simple harmonic motion. The frequency of these oscillations is

Half of the volume of sphere is submerged. For equilibrium of sphere, Weight = upthrust

$\therefore V{\rho }_{s}g=\frac{V}{2}\left({\mathrm{r}}_L\right)\left(g\right);{\rho }_s=\frac{{\rho }_L}{2}$

When slightly pushed downwards by x, weight will remain as it is while upthrust will increase. The increased upthrust will become the net restoring force (upwards).

F          = –(extra upthrust)

             = –(extra volume immersed)(r_L)(g)

$\text{ or }ma=-\left(\pi R^2\right)x{\rho }_L\mathrm{g}(a=\text{ acceleration })$

$\therefore \frac{4}{3}\pi R^3\left(\frac{{\rho }_L}{2}\right)a=-\left(\pi R^2\right)x{\rho }_{L}g(a=\text{ acceleration })$

$\therefore a=-\left(\frac{3g}{2R}\right)x\text{ as the given motion is simple harmonic }$

Frequency of oscillation, $f=\frac{1}{2\pi }\sqrt{\left|\frac{a}{x}\right|}=\frac{1}{2\pi }\sqrt{\frac{3g}{2R}}$