JEE Main 2026 — Jan 24, Shift 1 — Physics solution

JEE Main2026Jan 24, Shift 1Physics

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________.

A
$2 π \frac{M}{ρ A g}$
B
$π \frac{2 M}{ρ A g}$
C
$π \frac{ρ A}{M g}$
D
$2 π \frac{ρ A}{M g}$

Solution

At equilibrium the block is submerged to a depth $h$ such that buoyancy balances weight: $ρ A h g = M g$ .

When the block is depressed by a further distance $x$ , additional liquid of volume $A x$ is displaced, producing an extra upward restoring force $ρ A xg$ .

Equation of motion:

$M \overset{x}{¨} = - ρ A g x \Rightarrow \overset{x}{¨} = - (\frac{ρ A g}{M}) x$ .

Comparing with $\overset{x}{¨} = - ω^{2} x$ :

$ω = \frac{ρ A g}{M} \Rightarrow T = \frac{2 π}{ω} = 2 π \frac{M}{ρ A g}$ .

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