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 $T$.

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

Answer 1

Answer: (A), (C), (D)

The correct options are
A $T\propto \sqrt{m}$
C $T\propto \frac{1}{\sqrt{A}}$
D $T\propto \frac{1}{\sqrt{\rho }}$

Time period of SHM of small vertical oscillations in a liquid is given
by $T=2\pi \sqrt{\frac{l}{g}}$, where $l$ is the length of
cube/cylinder/block dipped in the liquid.
So according to law of floatation,
weight of the block = weight of the liquid displaced
$mg=Al\rho g\Rightarrow l=\frac{m}{A\rho }\Rightarrow T=2\pi \sqrt{\frac{m}{A\rho g}}$
So options A,C and D are true.