A uniform rod OO′ of length l is hinged at O. Two identical massless springs are attached as in Fig. 1: one at l/2 and one at l. In Fig. 2 both springs are attached at l/2. Ignoring gravity and for small angular oscillations, the frequencies are f₁ and f₂. Find f₁/f₂. Image
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Question

A uniform rod OO′ of length l is hinged at O. Two identical massless springs are attached as in Fig. 1: one at l/2 and one at l. In Fig. 2 both springs are attached at l/2. Ignoring gravity and for small angular oscillations, the frequencies are f₁ and f₂. Find f₁/f₂. Image

Options

G1 Physics · Accepted Answer

For a small rotation θ, a spring attached at distance x from the hinge stretches by approximately xθ and produces restoring torque kx²θ. Fig. 1: κ₁ = k(l/2)² + kl² = (5/4)kl². Fig. 2: κ₂ = 2k(l/2)² = (1/2)kl². The rod moment of inertia is the same in both cases, so f₁/f₂ = √(κ₁/κ₂) = √[(5/4)/(1/2)] = √(5/2). Hence option C is correct.