A cylindrical cork of uniform density floats in a liquid of density...

Let $m$ be the mass of the cork and $A$ be its cross-sectional area.

When the cork is depressed by a small distance $x$ in a liquid of density $\rho_L$, the extra buoyant force provides the restoring force:

$F = -A \rho_L g x$

The acceleration of the cork is:

$a = -\dfrac{A \rho_L g}{m} x$

Comparing this with the standard equation of SHM $a = -\omega^2 x$, we get:

$\omega = \sqrt{\dfrac{A \rho_L g}{m}}$

The time period of oscillation is:

$T = \dfrac{2\pi}{\omega} = 2\pi \sqrt{\dfrac{m}{A \rho_L g}}$

This shows that $T \propto \dfrac{1}{\sqrt{\rho_L}}$.

For the first liquid of density $\rho_1$:

$T \propto \dfrac{1}{\sqrt{\rho_1}}$

For the second liquid of density $\rho_2$:

$2T \propto \dfrac{1}{\sqrt{\rho_2}}$

Dividing the two equations:

$\dfrac{2T}{T} = \sqrt{\dfrac{\rho_1}{\rho_2}}$

$2 = \sqrt{\dfrac{\rho_1}{\rho_2}}$

Squaring both sides:

$4 = \dfrac{\rho_1}{\rho_2}$

$\dfrac{\rho_2}{\rho_1} = \dfrac{1}{4}$

Answer: $1/4$