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A cylindrical cork of uniform density floats in a liquid of density $\rho_1$. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density $\rho_2$, then the similar oscillation has time period 2T. The value of $\rho_2/\rho_1$ is:
Detailed Solution
For a small depression x, the restoring force is $F=-\rho gAx$. Comparing with $F=-kx$ gives $k=\rho Ag$. $T=2\pi\sqrt{\frac{m}{\rho Ag}} \Rightarrow T\propto\frac{1}{\sqrt{\rho}}$. $\frac{T}{2T}=\sqrt{\frac{\rho_2}{\rho_1}} \Rightarrow \frac{\rho_2}{\rho_1}=\frac{1}{4}$.