JEE Main 2025 Apr 4 Shift 2, Physics Q22: Angular Momentum, Its Conservation and Angular Impulse

JEE Main2025Apr 4, Shift 2Physics

A solid sphere with uniform density and radius $R$ is rotating initially with constant angular velocity $ω_{1}$ about its diameter. After some time during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become $R /2$ is $x ω_{1}$ . The value of $x$ is ______.

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Solution

The sphere keeps its shape and density, so when the radius shrinks from $R$ to $R /2$ the mass scales by the volume ratio:

$M^{'} = M \cdot (\frac{R /2}{R})^{3} = \frac{M}{8}$ .

No external torque acts about the rotation axis, so angular momentum about that axis is conserved:

$I_{1} ω_{1} = I_{2} ω_{2} \Rightarrow \frac{2}{5} M R^{2} ω_{1} = \frac{2}{5} \cdot \frac{M}{8} \cdot (\frac{R}{2})^{2} ω_{2}$ .

$M R^{2} ω_{1} = \frac{M R ^{2}}{32} ω_{2} \Rightarrow ω_{2} = 32 ω_{1}$ .

$∴ x = 32$ .

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JEE Main 2025 Apr 4 Shift 2, Physics Q22: Angular Momentum, Its Conservation and Angular Impulse

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