In arrangement shown if the rod is given a small displacement in vertical direction, then find the time period of small oscillations. The mass of rod is $m = 27 kg$ , the length is ' 1 m ' and the spring constant is equal to $k = 4 π^{2} N / m$ (Initially the rod is horizontal and in equilibrium).

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Initial tension from moments about P: T 0 ​ ℓ = 2 m g ℓ ​ ⇒ T 0 ​ = 2 m g ​ , Δ x = 2 k m g ​ For a small angular displacement θ, the net restoring torque is τ = k ( Δ x + ℓ θ ) ℓ − m g 2 ℓ ​ cos θ ≈ k ℓ 2 θ I α = − k ℓ 2 θ , I = 3 m ℓ 2 ​ ⇒ α = − m 3 k ​ θ ω = m 3 k ​ ​ ⇒ T = 2 π 3 k m ​ ​ = 2 π 3 × 4 π 2 27 ​ ​ = 3 s

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