A rod of mass $M$ and length $L$ is hinged at its centre of mass so that it can rotate in a vertical plane. Two springs each of stiffness $k$ are connected at its ends, as shown in the figure. The time period of SHM is

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$2 π \sqrt{\frac{M}{6 k}}$

$2 π \sqrt{\frac{M}{3 k}}$

$2 π \sqrt{\frac{ML}{k}}$

$π \sqrt{\frac{M}{6 k}}$

answer is A.

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Detailed Solution

The restoring torque (for small $θ$ )

$τ_{r e s t} = - [\frac{k L θ}{2} \times \frac{L}{2}] \times 2 = \frac{k L^{2}}{2} (- θ) ∴ α = \frac{τ_{r e s t}}{I} = \frac{k L^{2} / 2}{M L^{2} / 12} (- θ) = \frac{6 k}{M} (- θ) ∴ T = 2 π \sqrt{\frac{M}{6 k}}$

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