First slide

Applications of SHM

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 $T_{2}$ . The ratio $\frac{T_{1}}{T_{2}}$ will be

Moderate

A

$\frac{2}{3}$
B

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

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

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

Solution

As time period of physical pendulum $T = 2 π \sqrt{\frac{I}{m g l_{c}}}$
Thus $T_{1} = 2 π \sqrt{\frac{m R^{2} + m R^{2}}{m g R}} = 2 π \sqrt{\frac{2 R}{g}}$ ,
$\begin{array}{l} T_{2} = 2 π \sqrt{\frac{\frac{m R^{2}}{2} + m R^{2}}{m g R}} = 2 π \sqrt{\frac{3 R}{2 g}} \\ \Rightarrow \frac{T_{1}}{T_{2}} = \frac{2}{\sqrt{3}} \end{array}$

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