A system of two identical rods ($L-$ shaped) of mass $m$ and length $l$ are resting on a peg $P$ as shown in the figure. If the system is displaced in its plane by a small angle $\theta ,$ find the period of oscillations :
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$\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{I}}{\mathrm{mg} \mathrm{L}}}$
Here $I=\frac{m l^{2}}{3}+\frac{m l^{2}}{3}=\frac{2 m g l^{2}}{3}$
From figure: $\sin 45^{\circ}=\frac{L}{l / 2}$
$\therefore \mathrm{L}=\frac{l}{2 \sqrt{2}}$
$\therefore \quad \mathrm{T}=2 \pi \sqrt{\frac{2 \mathrm{m} l^{2}}{3 \times \frac{l}{2 \sqrt{2}} \mathrm{mg}}}=2 \pi \sqrt{\frac{2 \sqrt{2} l}{3 \mathrm{g}}}$
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