A system of two identical rods (L-shaped) of m ass m and length I are resting on a peg P as shown in the figure. If the system is displaced in its plane by a small angle, find the period of oscillations.

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$2 π \sqrt{\frac{\sqrt{2} l}{3 g}}$

$2 π \sqrt{\frac{2 l}{3 g}}$

$2 π \sqrt{\frac{2 \sqrt{2} l}{3 g}}$

$3 π \sqrt{\frac{l}{3 g}}$

answer is C.

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Detailed Solution

$\begin{array}{l} T = 2 π \sqrt{\frac{I}{m g L}} \\ Here I = \frac{m l^{2}}{3} + \frac{m l^{2}}{3} = \frac{2 m g l^{2}}{3} \\ From figure, \sin 45^{\circ} = \frac{L}{l / 2} \\ ∴ L = \frac{l}{2 \sqrt{2}} \\ ∴ T = 2 π \sqrt{\frac{2 m l^{2}}{3 \times \frac{l}{2 \sqrt{2}} m g}} = 2 π \sqrt{\frac{2 \sqrt{2} l}{3 g}} \end{array}$

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