A cylindrical block of mass M and area of cross section A is floating in a liquid of density ρ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is \ \ \ \ .

Physics · Simple Harmonic Motion

JEE Main 2026 — 24 January, Morning Shift — Question 26

A cylindrical block of mass M and area of cross section A is floating in a liquid of density $\rho$ and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is $\_\_\_\_$ .

Option A:
$2 \pi \sqrt{\frac{M}{\rho A g}}$
Correct
Option B:
$\pi \sqrt{\frac{2 \mathrm{M}}{\rho \mathrm{Ag}}}$
Option C:
$\pi \sqrt{\frac{\rho \mathrm{A}}{\mathrm{Mg}}}$
Option D:
$2 \pi \sqrt{\frac{\rho \mathrm{~A}}{\mathrm{Mg}}}$

Answer: A

Step-by-step solution

At equilibrium $\rho \mathrm{Ahg}=\mathrm{Mg}$ After displacing by x , $\mathrm{Ma}=-\rho \mathrm{A}(\mathrm{h}+\mathrm{x}) \mathrm{g}+\mathrm{Mg}$ $\mathrm{Ma}=-\rho \mathrm{Ahg}-\rho \mathrm{Axg}+\mathrm{Mg}$ $\mathrm{Ma}=-\rho \mathrm{Axg}$ $\mathrm{a}=\left(\frac{-\rho \mathrm{Ag}}{\mathrm{M}}\right) \mathrm{x}$ on comparing with, $\mathrm{a}=-\omega^{2} \mathrm{x}$ $\omega=\sqrt{\frac{\rho \mathrm{Ag}}{\mathrm{M}}}$ $\mathrm{T}=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{\mathrm{M}}{\rho \mathrm{Ag}}}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 24 January, Morning Shift
Subject: Physics
Chapter: Simple Harmonic Motion
Topic: Linear SHM and Spring-Pulley-Block Systems