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 constant k . The rod is gently pushed through a small angle θ in one direction a

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 constant k . The rod is gently pushed through a small angle θ in one direction and released. The frequency of oscillation is (the springs are fixed to the walls)

TestHub · Accepted Answer

Consider the situation below

From the above diagram

$θ = \frac{x}{l / 2}$

$\Rightarrow x = \frac{L}{2} θ$

In the above diagram the rod is displaced through an angle $θ$ as in above diagram

Restoring torque $= (- 2 kx) \times \frac{L}{2} = - kxL$
angular acceleration,
$\Rightarrow a = \frac{- k \times \frac{L}{2} θ \times L}{Ι} = (\frac{- 6 k}{M}) θ$

$= - ω^{2} θ$

$ω = \sqrt{\frac{6 K}{M}}$

$f = \frac{1}{2 π} \sqrt{\frac{6 K}{M}}$

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