A ring is hung on a nail. It can oscillate, without slipping or… | JEE Main 2020 (5 Sep, Shift 2)

Q151JEE Main 2020 · 5 SEP · SHIFT 2Difficult

Moment of Inertia & Radius of Gyration

A ring is hung on a nail. It can oscillate, without slipping or sliding

(i) in its plane with a time period T1 and,

(ii) back and forth in a direction perpendicular to its plane,

with a period T2. The ratio $\frac{T_1}{T_2}$ will be:

Show answer and solution

Answer: (b)

Physical pendulum: $T = 2\pi\sqrt{\frac{I}{mgR}}$ , so $T \propto \sqrt{I}$ (same $R$ ).
(i) In plane, axis $\perp$ plane at rim: $I_1 = MR^2 + MR^2 = 2MR^2$
(ii) $\perp$ to plane, axis tangent in plane: $I_2 = \frac{MR^2}{2} + MR^2 = \frac{3}{2}MR^2$
$\frac{T_1}{T_2} = \sqrt{\frac{2}{3/2}} = \frac{2}{\sqrt{3}}$ , so the answer is (b).

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