A sphere of radius $R$   is floating half submerged in a liquid of density  $\rho$ . If the sphere is slightly pushed down and released, the frequency of vertical oscillations is

Since, the sphere floats half immersed in the liquid of density  $\rho$ ,  so we have

$\left(\begin{array}{l}\text{Mass of}\\ \text{thesphere}\end{array}\right)=\left(\begin{array}{l}\text{Mass of liquid}\\ \text{displaced}\end{array}\right)=\frac{2}{3}\pi R^3\rho$

When pushed down by a small distance y, excess upthrust (restoring force) generated is

$t-\left(\pi R^{2}y\right)\rho g$

Therefore, force constant of SHM is

$k=\frac{F}{y}=\left(\pi R^2\right)\rho g$

Hence, frequency of oscillations is

$f-\frac{1}{2\pi }\sqrt{\frac{k}{m}}-\frac{1}{2\pi }\sqrt{\frac{\pi R^2\rho g}{\frac{2}{3}\pi R^3\rho }}-\frac{1}{2\pi }\sqrt{\frac{3g}{2R}}$