A ring is hung on a nail. It can oscillate, without slipping or sliding $(i)$ in its plane with a time period $T_{1}$ and, $(ii)$ back and forth in a direction perpendicular to its plane, with a period $T _{2}$. the ratio $\frac{ T _{1}}{ T _{2}}$ will be
JEE MAIN 2020, Diffcult
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Moment of inertia in case $(i)$ is $I _{1}$
Moment of inertia in case $(ii)$ is $I_{2}$
$I_{1}=2 M R^{2}$
$I _{2}=\frac{3}{2} MR ^{2}$
$T _{1}=2 \pi \sqrt{\frac{ I _{1}}{ Mgd }} ; T _{2}=2 \pi \sqrt{\frac{ I _{2}}{ Mgd }}$
$\frac{ T _{1}}{ T _{2}}=\sqrt{\frac{ I _{1}}{ I _{2}}}=\sqrt{\frac{2 MR ^{2}}{\frac{3}{2} MR ^{2}}}=\frac{2}{\sqrt{3}}$
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