A rod of length L, cross sectional area A and density `rho` is hanging from a rigid support by spring of stiffries k. A very small sphere of mass m is rigidly attached at the bottom of the rod. The rod is partially immersed in a liquid of density `rho`. Find the period of small oscillations.

Explanation

To find the period of small oscillations of the system described, we can follow these steps:

### Step 1: Understand the System
We have a rod of length \( L \), cross-sectional area \( A \), and density \( \rho \) hanging from a rigid support via a spring with stiffness \( k \). A small sphere of mass \( m \) is attached at the bottom of the rod, and the rod is partially immersed in a liquid of density \( \rho_{\text{liquid}} \).

### Step 2: Determine the Forces Acting on the System
When the rod is displaced downward by a small distance \( x \), two forces will act on it:
1. **Buoyant Force**: The increase in buoyant force when the rod is displaced by \( x \) is given by:
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