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Question

A uniform rod of length $L$ $L$ and mass $M$ $M$ is pivoted at its center.It's 2 ends are attached to 2 springs of equal spring constants $k$ $k$ .The springs are fixed to rigid supports as shown and the rod is free to oscillate in the horizontal plane.It's gently pushed through a small angle $θ$ $theta$ in one direction and released.What is the frequen cy of oscillation?

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Answer 1 · 2018-04-15T09:44:10

$ω = \sqrt{\frac{1}{12} \frac{M}{k}}$ $omega = sqrt(1/12 M/k)$

Assuming a small $θ$ $theta$ such that $sin θ \approx θ$ $sintheta approx theta$

$I ddot theta =- L ^2 k theta$

here

$I = 1/12 M L^2$ then

$1/12 M L^2 ddot theta = L^2 k theta$

or

$1/12 M/k ddot theta + theta = 0 rArr omega = sqrt(1/12 M/k)$