A uniform pole of length $2 L$ is laid on a smooth horizontal table as shown in figure. The mass of pole is M and it is connected to a frictionless axis at O. A spring with force constant $k$ is connected to the other end. The pole is displaced by a small angle from equilibrium position and released such that it performs small oscillations. Its frequency of oscillations is $\sqrt{\frac{z k}{M}}$ . Find the value of z?

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answer is 3.

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Detailed Solution

The restoring torque for small $θ$

$τ_{rest} = - [k (2 L θ) \times 2 L]$

And $α = \frac{τ_{r e s t}}{I}$

$= \frac{4 k L^{2}}{\frac{M (2 L)^{2}}{3}} (- θ) = \frac{3 k}{M} (- θ)$

On comparing with $α = - ω^{2} θ$ , we get

$ω = \sqrt{\frac{3 k}{M}}$

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