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A uniform rod of length L and mass M is pivoted at the centre

Question :- A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constants k. The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $θ$ in one direction and released. The frequency of oscillation is

Solution :- A uniform rod of length L and mass M is pivoted at the centre….

Restoring torque,

$\displaystyle \tau =-2\times \left( {{F}_{spring}}\frac{L}{2} \right)$

Here F_spring = kdx

For small angular displacement θ, $\displaystyle dx=\frac{L}{2}\theta$

Restoring torque,

$\displaystyle \tau =-2\times \left( kdx\frac{L}{2} \right)=-2\times \left( k\frac{L}{2}\theta \frac{L}{2} \right)=-k\frac{{{L}^{2}}}{2}\theta$ …..(1)