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Question

A thin uniform vertical rod of mass m and length l pivoted at point O is shown in the given figure. The combined stiffness of the springs is equal to k. The mass of the spring is negligible. Find the frequency of small oscillation.
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Solution

As θ is small we consider lθ as linear displacement. So, each spring has change in length is equal to lθ.
The restoring force =2klθ
The moment of inertia of rod about one of its end is I=ml23
Where, l=length of rod
and, m=mass of rod
Moment of restoring force about o is =2klθ.OA=2klθ.lcosθ
Angular acceleration of rod
d2θdt2=Momentofrestoringforceabout0I=2klθ.lcosθ
d2θdt2=2kl2θml23 (as θ is small, cosθ1)
d2θdt2+(6km)θ=0
Angular frequency ω=6km
frequency, f=ω2π=12π6km

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