Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length ℓ and mass m . The rod is pivoted at its centre ' O ' and

Question

Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length ℓ and mass m . The rod is pivoted at its centre ' O ' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is:

TestHub · Accepted Answer

For a small angular displacement θ , each rod end shifts by x = 2 ℓ ​ θ , so each spring exerts k x . Restoring torque τ = 2 ( k x ) 2 ℓ ​ = 2 k ℓ 2 ​ θ ; moment of inertia about O is I = 12 m ℓ 2 ​ . ω = m ℓ 2 /12 k ℓ 2 /2 ​ ​ = m 6 k ​ ​ , so f = 2 π 1 ​ m 6 k ​ ​ - option (A).

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