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 $\mathrm{T}_{2}$. the ratio $\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}$ will be :
Correct Option: 1
Moment of inertia in case (i) is $\mathrm{I}_{1}$
Moment of inertia in case (ii) is $\mathrm{I}_{2}$
$\mathrm{I}_{1}=2 \mathrm{MR}^{2}$
$\mathrm{I}_{2}=\frac{3}{2} \mathrm{MR}^{2}$
$\mathrm{T}_{1}=2 \pi \sqrt{\frac{\mathrm{I}_{1}}{\mathrm{Mgd}}} \quad ; \mathrm{T}_{2}=2 \pi \sqrt{\frac{\mathrm{I}_{2}}{\mathrm{Mgd}}}$
$\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\sqrt{\frac{\mathrm{I}_{1}}{\mathrm{I}_{2}}}=\sqrt{\frac{2 \mathrm{MR}^{2}}{\frac{3}{2} \mathrm{MR}^{2}}}=\frac{2}{\sqrt{3}}$
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