JEE Main 2026 — Jan 24, Shift 2 — Physics solution

JEE Main2026Jan 24, Shift 2Physics

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_{1}$ . A small co-centric cylinder of length $L /2$ and radius $R /3$ carved from this cylinder has moment of inertia about its axis equals to $I_{2}$ . The ratio $I_{1} / I_{2}$ is ____.

Your answer

Solution

Let the density of the cylinder be $ρ$ . Original mass $M = ρ π R^{2} L$ , so $I_{1} = \frac{1}{2} M R^{2}$ .

Mass of the carved small cylinder:

$m = ρ π (\frac{R}{3})^{2} (\frac{L}{2}) = \frac{ρ π R ^{2} L}{18} = \frac{M}{18}$ .

Moment of inertia of small cylinder about its own axis:

$I_{2} = \frac{1}{2} m (\frac{R}{3})^{2} = \frac{1}{2} \cdot \frac{M}{18} \cdot \frac{R ^{2}}{9} = \frac{M R ^{2}}{324}$ .

$\frac{I _{1}}{I _{2}} = \frac{M R ^{2} /2}{M R ^{2} /324} = \frac{324}{2} = 162$ .

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