A cylindrical cork of uniform density floats in a liquid of density ρ₁. When depressed slightly and released it oscillates harmonically with time period T. If the same cork floats in another liquid of density ρ₂, the oscillation has period 2T. The value of (ρ₂)/(ρ₁) is:

Solution & Explanation

A floating cylinder of cross-section A, length L, density ρ_s undergoes SHM when pushed down: restoring force per unit extra depression x is ρ_liquid·A·g·x, with mass m = ρ_s·A·L. ω² = (ρ_liquid·A·g)/(ρ_s·A·L) = ρ_liquid·g/(ρ_s·L), so T = 2π√(ρ_s·L/(ρ_liquid·g)). For the SAME cork, ρ_s and L are fixed, so T ∝ 1/√(ρ_liquid). Take the ratio of the two liquids: T₁/T₂ = √(ρ₂/ρ₁). Given T₂ = 2T₁, i.e. T₁/T₂ = 1/2: 1/2 = √(ρ₂/ρ₁). Square both sides: ρ₂/ρ₁ = 1/4. So ρ₂/ρ₁ = 1/4, option D. (A denser liquid → stiffer restoring force → shorter period, so the longer period 2T must come from the lighter liquid ρ₂ < ρ₁, consistent with 1/4.)