$A$ cylindrical piece of cork of density $\rho$, base area $A$, and height $h$ floats in a liquid of density $\rho_{l}$. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period $T=2 \pi \sqrt{\frac{h \rho}{\rho_{l} g}}$. (Ignore damping due to viscosity of the liquid).

(N/A) Base area of the cork $= A$
Height of the cork $= h$
Density of the liquid $= \rho_{l}$
Density of the cork $= \rho$
In equilibrium, the weight of the cork equals the weight of the liquid displaced by the floating cork.
Let the cork be depressed slightly by a distance $x$. As a result, an additional volume of liquid is displaced, creating an extra upward buoyant force (up-thrust) which acts as the restoring force.
Restoring force $F = -(\text{Weight of extra liquid displaced})$
$F = -(A \cdot x \cdot \rho_{l} \cdot g)$
According to the simple harmonic motion force law, $F = -kx$, where $k$ is the force constant.
Comparing the two expressions, $k = A \rho_{l} g$.
The time period of oscillation is given by $T = 2 \pi \sqrt{\frac{m}{k}}$, where $m$ is the mass of the cork.
Mass of the cork $m = \text{Volume} \times \text{Density} = (A \cdot h) \cdot \rho$.
Substituting $m$ and $k$ into the time period formula:
$T = 2 \pi \sqrt{\frac{A h \rho}{A \rho_{l} g}} = 2 \pi \sqrt{\frac{h \rho}{\rho_{l} g}}$.