An air chamber of volume V has a neck area of cross section A into which a ball of mass m just fits and can move up and down without any friction, figure. Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time periodof oscillations assuming pressure volume variatinos of air to be isothermal.

Explanation

To solve the problem step by step, we will analyze the motion of the ball in the air chamber and derive the expression for the time period of oscillation when the ball is pressed down and released.

### Step 1: Understand the System
We have an air chamber with a volume \( V \) and a neck with a cross-sectional area \( A \). A ball of mass \( m \) fits into the neck and can move up and down without friction. When the ball is pressed down a distance \( Y \) and released, we need to show that it executes Simple Harmonic Motion (SHM).

### Step 2: Change in Volume
When the ball is pressed down by a distance \( Y \), the volume of air in the chamber decreases. The change in volume \( \Delta V \) can be expressed as:
\[
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