A ring is hung on a nail.


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 :

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

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

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

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

Correct Option: 1


(1) Let $I_{1}$ and $I_{2}$ be the moment of inertia in first and second case respectively.

$I_{1}=2 M R^{2}$

$I_{2}=M R^{2}+\frac{M R^{2}}{2}=\frac{3}{2} M R^{2}$

Time period, $T=2 \pi \sqrt{\frac{I}{m g d}}$

$T \propto I$

$\therefore \frac{T_{1}}{T_{2}}=\sqrt{\frac{I_{1}}{I_{2}}}=\sqrt{\frac{2 M R^{2}}{\frac{3}{2} M R^{2}}}=\frac{2}{\sqrt{3}}$


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