Question

A ring is hung on a nail. It can oscillate, without slipping or sliding $\left(i\right)$ in its plane with a time period $T_1$ and $\left(ii\right)$ back and forth in a direction perpendicular to its plane, with a period $T_2$. The ratio $\frac{T_1}{T_2}$ will be

• A. $\frac{2}{\sqrt{3}}$
• B. $\frac{2}{3}$
• C. $\frac{3}{\sqrt{2}}$
• D. $\frac{\sqrt{2}}{3}$

Answer 1

The correct option is A $\frac{2}{\sqrt{3}}$

Let $I_1$ and $I_2$ be the moment of inertia in first and second case respectively.

$I_1=2M{R}^2$
$I_2=M{R}^2+\frac{M{R}^2}{2}=\frac{3}{2}M{R}^2$


Time period,
$T=2\pi \sqrt{\frac{I}{mgd}}$
$T\propto I$
$\therefore \frac{T_1}{T_2}=\sqrt{\frac{I_1}{I_2}}=\sqrt{\frac{2M{R}^2}{\frac{3}{2}M{R}^2}}=\frac{2}{\sqrt{3}}$