JEE Main 2025 Jan 23 Shift 1, Physics Q2: SHM in Spring-Mass System

JEE Main2025Jan 23, Shift 1Physics

A light hollow cube of side length 10 cm and mass 10 g, is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $y π \times 1 0^{- 2}$ s, where the value of $y$ is

(Acceleration due to gravity, $g = 10$ m/s $^{2}$ , density of water $= 1 0^{3}$ kg/m $^{3}$ )

A
$2$
B
$6$
C
$4$
D
$1$

Solution

When the cube is depressed by a small distance $x$ , the extra buoyant force is $F = ρ g L^{2} x$ (additional submerged volume $L^{2} x$ ), directed upward. This is a Hooke-type restoring force with effective stiffness $k_{eff} = ρ g L^{2}$ .

$T = 2 π \frac{m}{ρ g L ^{2}}$ .

Plugging in $m = 0.01$ kg, $ρ = 1 0^{3}$ kg/m $^{3}$ , $g = 10$ m/s $^{2}$ , $L = 0.1$ m:

$\frac{m}{ρ g L ^{2}} = \frac{0.01}{1 0 ^{3} \cdot 10 \cdot 1 0 ^{- 2}} = 1 0^{- 4}$ s $^{2}$ .

$T = 2 π \cdot 1 0^{- 2}$ s $= 2 π \times 1 0^{- 2}$ s, so $y = 2$ .

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JEE Main 2025 Jan 23 Shift 1, Physics Q2: SHM in Spring-Mass System

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