A cylindrical piece of cork of base area A and height h floats in a liquid of density `rho_(1)`. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
`T=2pisqrt((hrho)/(rho_(1)g))`

Explanation

To solve the problem, we will analyze the forces acting on the cork and derive the expression for the period of oscillation when it is slightly depressed and released.

### Step 1: Understanding the System
We have a cylindrical piece of cork with base area \( A \) and height \( h \) floating in a liquid of density \( \rho_1 \). When the cork is in equilibrium, a portion of it is submerged in the liquid. Let \( y \) be the height of the cork submerged in the liquid.

### Step 2: Applying Archimedes' Principle
According to Archimedes' principle, the buoyant force \( F_b \) acting on the cork is equal to the weight of the liquid displaced by the submerged part of the cork. This can be expressed as:
\[
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