When a floating object is displaced by a small distance x, the restoring force is provided by the additional buoyant force. The buoyant force is equal to the weight of the displaced liquid: F = -Aρgx, where A is the cross-sectional area of the cork and ρ is the density of the liquid.
Using Newton's Second Law, ma = -Aρgx. Since m = Vρ_c = (Ah)ρ_c (where h is the height of the cork and ρ_c is the density of the cork), the equation becomes a = -(Aρg / Vρ_c)x. This represents Simple Harmonic Motion with angular frequency ω = sqrt(Aρg / m). The time period T is given by: $$T = 2\pi \sqrt{\frac{m}{A\rho g}}$$
From the formula, we observe that T is inversely proportional to the square root of the liquid density: T ∝ 1 / sqrt(ρ). Therefore, T_1 / T_2 = sqrt(ρ_2 / ρ_1).
Given T_1 = T and T_2 = 2T, we have: $$\frac{T}{2T} = \sqrt{\frac{\rho_2}{\rho_1}}$$ $$\frac{1}{2} = \sqrt{\frac{\rho_2}{\rho_1}}$$ $$\frac{\rho_2}{\rho_1} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$$
Final Answer: 1/4
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