As shown in the figures, a uniform rod O O ′ of length l is hinged at the point O and held in place vertically between two walls using two massless springs of same spring constant. The springs are con

Question

As shown in the figures, a uniform rod O O ′ of length l is hinged at the point O and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end ( O ′ ) of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is f 1 ​ . On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is f 2 ​ . Ignoring gravity and assuming motion only in the plane of the diagram, the value of f 2 ​ f 1 ​ ​ is:

TestHub · Accepted Answer

​ 3 ML 2 ​ θ ¨ + K ⋅ 2 ℓ ​ θ ⋅ 2 ℓ ​ + K ⋅ ℓ ⋅ θ ⋅ ℓ = 0 θ ¨ + 4 15 ​ M k ​ θ = 0 θ ¨ = − ( 4 15 ​ M k ​ ) θ ω 1 ​ = 4 M 15 K ​ ​ ​ 3 1 ​ ML 2 θ ¨ + 2 K 2 L ​ θ ⋅ 2 L ​ = 0 ​ θ ¨ + 2 3 ​ M K ​ θ = 0 θ ¨ = − 2 3 ​ M K ​ θ ω 2 ​ = 2 3 ​ M K ​ ​ ω 2 ​ ω 1 ​ ​ = 4 15 ​ × 3 2 ​ ​ = 2 5 ​ ​ ​

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