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A uniform rod of mass m and length $l$ is suspended through a light wire of length $l$ and torsional constant k as shown in figure. The time periods of the system for small oscillations in the vertical plane about the suspension point and angular oscillations in the horizontal plane about the centre of the rod are

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$2 π \sqrt{\frac{13 l}{12 g}}; 2 π \sqrt{\frac{m l^{2}}{12 K}}$

$2 π \sqrt{\frac{l}{g}}; 2 π \sqrt{\frac{m l^{2}}{3 K}}$

$2 π \sqrt{\frac{13 l}{12 g}}; 2 π \sqrt{\frac{m l^{2}}{3 K}}$

$2 π \sqrt{\frac{l}{g}}; 2 π \sqrt{\frac{m l^{2}}{12 K}}$

answer is A.

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Detailed Solution

$T_{1} = 2 π \sqrt{\frac{I}{m g l}}; I = \frac{m l^{2}}{12} + m l^{2}; T_{2} = 2 π \sqrt{\frac{I}{K}}; I = \frac{m l^{2}}{12}$

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