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

Question:

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 :

  1. $\frac{2}{\sqrt{3}}$

  2. $\frac{\sqrt{2}}{3}$

  3. $\frac{2}{3}$

  4. $\frac{3}{\sqrt{2}}$


Correct Option: 1

Solution:

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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