Question

A spherical ball of radius R is floating in a liquid with half of its volume submerged in the liquid. Now the ball is displaced vertically by small distance inside the liquid. Find the frequency of oscillation of ball.

• A. $f=\frac{1}{3\pi }\sqrt{\frac{3g}{2R}}$.
• B. $f=\frac{1}{2\pi }\sqrt{\frac{3g}{2R}}$.
• C. $f=\frac{1}{4\pi }\sqrt{\frac{3g}{2R}}$.
• D. $f=\frac{1}{5\pi }\sqrt{\frac{3g}{2R}}$.

Answer 1

The correct option is B $f=\frac{1}{2\pi }\sqrt{\frac{3g}{2R}}$.

Restoring force $=\pi R^{2}x\rho g$ (for a small x)

$-m\frac{d^{2}x}{d{t}^2}=\pi R^{2}x\rho g$

$\frac{d^{2}x}{d{t}^2}=-\frac{3}{2}\frac{g}{R}x$ [as $\frac{4}{3}\frac{\pi R^3}{2}\rho g=mg$]

$\therefore$ Motion is SHM

$\Rightarrow {\omega }^2=\frac{3}{2}\frac{g}{R}$

$\Rightarrow f=\frac{1}{2\pi }\sqrt{\frac{3g}{2R}}$