A floating cylindrical cork of fixed mass and cross-sectional area performs small vertical SHM in a liquid of density ρ₁ with period T. In another liquid of density ρ₂ it oscillates with period 2T. Find ρ₂/ρ₁.
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Question

A floating cylindrical cork of fixed mass and cross-sectional area performs small vertical SHM in a liquid of density ρ₁ with period T. In another liquid of density ρ₂ it oscillates with period 2T. Find ρ₂/ρ₁.

Options

G1 Physics · Accepted Answer

Step 1: If the cork is displaced downward by a small distance x, the extra displaced liquid volume is Ax. Step 2: Extra buoyant restoring force is F = −ρgAx, so the effective spring constant is k_eff = ρgA. Step 3: The period of vertical SHM is T = 2π√[m/(ρgA)], hence T ∝ 1/√ρ for the same cork. Step 4: T₂/T₁ = √(ρ₁/ρ₂) = 2. Step 5: Square both sides: ρ₁/ρ₂ = 4. Final answer: ρ₂/ρ₁ = 1/4 (Option D).