An air chamber of volume `V` has a long of cross sectional area `A`. A ball of mass `m` is fixed sympthlly in the track The ball modulus of air is `B` ball is pressed down slightly and released, the time period of the oscillation is

Options

A. `2pi sqrt((mV)/(2BA^(2)))`
B. `pi sqrt((2mV)/(BA^(2)))`
C. `2pi sqrt((mV)/(BA^(2)))`
D. `(pi)/(2) sqrt((m)/(BA^(2)))`

Explanation

To find the time period of oscillation of the ball in the air chamber, we can follow these steps:

### Step 1: Understand the system
We have a ball of mass `m` that is fixed in a track within an air chamber of volume `V` and cross-sectional area `A`. The ball is pressed down slightly and released, leading to oscillation.

### Step 2: Identify the restoring force
When the ball is pressed down, it compresses the air in the chamber. The restoring force acting on the ball is due to the change in pressure of the air. The pressure change can be related to the volume change of the air chamber.

### Step 3: Use the Bulk Modulus
The bulk modulus of air, denoted as `B`, relates the change in pressure to the change in volume. The relationship is given by:
\[ B = -\frac{dP}{\frac{dV}{V}} \]
Where \( dP \) is the change in pressure and \( dV \) is the change in volume.

### Step 4: Relate volume change to displacement
When the ball is displaced by a small distance \( x \), the change in volume \( dV \) can be expressed as:
\[ dV = A \cdot x \]
Where \( A \) is the cross-sectional area of the chamber.

### Step 5: Calculate the restoring force
The change in pressure \( dP \) can be expressed as:
\[ dP = \frac{B \cdot dV}{V} = \frac{B \cdot A \cdot x}{V} \]
The restoring force \( F \) acting on the ball is given by:
\[ F = A \cdot dP = A \cdot \frac{B \cdot A \cdot x}{V} = \frac{B \cdot A^2 \cdot x}{V} \]

### Step 6: Apply Hooke's Law
According to Hooke's Law, the restoring force is also related to the displacement by:
\[ F = -k \cdot x \]
Where \( k \) is the effective spring constant. By equating the two expressions for the restoring force, we have:
\[ -k \cdot x = \frac{B \cdot A^2 \cdot x}{V} \]
Thus, we can identify:
\[ k = -\frac{B \cdot A^2}{V} \]

### Step 7: Find the time period of oscillation
The time period \( T \) of a mass-spring system is given by:
\[ T = 2\pi \sqrt{\frac{m}{k}} \]
Substituting the expression for \( k \):
\[ T = 2\pi \sqrt{\frac{m \cdot V}{B \cdot A^2}} \]

### Final Answer
Thus, the time period of the oscillation is:
\[ T = 2\pi \sqrt{\frac{m \cdot V}{B \cdot A^2}} \]

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