Question

A rectangular block of mass m and area of crosssection $A$ floats in a liquid of density $\rho$. If it is given a small vertical displacement from equilibrium it undergoes oscillation with a time period $T$. Then:-

• A. $T\propto \sqrt{\rho }$
• B. $T\propto \frac{1}{\sqrt{A}}$
• C. $T=\frac{1}{\rho }$
• D. $T\propto \frac{1}{\sqrt{m}}$

Answer 1

The correct option is B $T\propto \frac{1}{\sqrt{A}}$

Restoring force,F = mg=$Ax\rho g$ (density=mass/volume)

$F=kx$

Equating both

$k=A\rho g$

And we know that $T=2\pi \sqrt{\frac{m}{k}}$

$\Rightarrow$ T = 2$\pi \sqrt{\frac{m}{A\rho g}}\Rightarrow$ Ans $\left(2\right)$