A rod of lengths L is supported by two ideal strings of length $l$ such that the system hangs in a vertical plane.
Case 1: Rod is kept horizontal, displaced slightly perpendicular to the plane and allowed by oscillate.
Case 2: Rod is given a small twist about central axis and then allowed to oscillate.
Let $T_{1}, T_{2}$ be the periods of oscillations in the two respective cases.

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$T_{2} = 2 π \sqrt{\frac{l}{6 g}}$

$T_{1} = 2 π \sqrt{\frac{l}{2 g}}$

$T_{2} = 2 π \sqrt{\frac{l}{3 g}}$

$T_{1} = 2 π \sqrt{\frac{l}{g}}$

answer is A, D.

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Detailed Solution

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Let $θ$ be the angle that the rod rotates about its central axis and be the angle made by string with the vertical
$- (2 T ϕ) \frac{L}{2} = \frac{m L^{2}}{12} \frac{d^{2} θ}{d t^{2}} \Rightarrow \frac{d^{2} θ}{d t^{2}} = - (\frac{6 g}{L}) ϕ \Rightarrow T_{2} = 2 π \sqrt{\frac{l}{3 g}}$ Or $\frac{d^{2} θ}{d t^{2}} = - (\frac{3 g}{l}) θ$