Question

A rectangular block of mass $m$ and area of cross-section $A$ 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. $T\propto {\rho }^0$
• C. $T\propto \frac{1}{\rho }$
• D. $T\propto \frac{1}{\sqrt{\rho }}$

Answer 1

The correct option is D $T\propto \frac{1}{\sqrt{\rho }}$

For a block of mass $m$ and cross-sectional area $A$ floating in a liquid of density $\rho$, if it is given a very small vertical push it will start to perform simple harmonic motion due to restoring force provided by buoyancy.

The time period of oscillation is given by,
$\Rightarrow T=2\pi \sqrt{\frac{m}{\rho Ag}}$
where $m=\sigma \times V$, is the mass of block and $\sigma$ be the density of block.

Hence time period of oscillation of block is inversely proportional to square root of density of liquid.
$T\propto \frac{1}{\sqrt{\rho }}$