An ant is sitting at the edge of a rotating disc. If the ant reaches the other end, after moving along the diameter, the angular velocity of the disc will:-

Options

A. remain constant
B. first decreases and then increases
C. first increases, then decrease
D. Iincrease continuously

Answer

To solve the problem, we need to analyze the motion of the ant on the rotating disc and the implications for angular momentum and angular velocity.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a rotating disc with an ant sitting at its edge. The ant moves along the diameter of the disc towards the center and then to the other edge.

2. **Conservation of Angular Momentum**:
- Since there are no external forces acting on the system (disc + ant), the angular momentum of the system is conserved. This means that the initial angular momentum must equal the final angular momentum.

3. **Angular Momentum Expression**:
- The angular momentum \( L \) of the system can be expressed as:
\[ L = I \cdot \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.

4. **Moment of Inertia**:
- The moment of inertia \( I \) of the system consists of two parts: the moment of inertia of the disc and the moment of inertia of the ant.
- For the disc (assuming it has mass \( M \) and radius \( R \)):
\[ I_{\text{disc}} = \frac{1}{2} M R^2 \]
- For the ant (assuming it has mass \( m \) and is at a distance \( r \) from the center):
\[ I_{\text{ant}} = m r^2 \]
- Thus, the total moment of inertia when the ant is at the edge is:
\[ I_{\text{total}} = \frac{1}{2} M R^2 + m R^2 \]

5. **Effect of Ant Moving Towards the Center**:
- As the ant moves towards the center, its distance \( r \) decreases, leading to a decrease in the moment of inertia \( I \).
- Since angular momentum \( L \) is conserved, if \( I \) decreases, \( \omega \) must increase to keep \( L \) constant.

6. **Effect of Ant Moving from Center to Edge**:
- When the ant reaches the center and then starts moving towards the other edge, \( r \) increases again, which increases the moment of inertia \( I \).
- To conserve angular momentum, as \( I \) increases, \( \omega \) must decrease.

7. **Conclusion**:
- Therefore, the angular velocity of the disc first increases as the ant moves towards the center and then decreases as the ant moves from the center to the other edge.

### Final Answer:
The angular velocity of the disc will first increase and then decrease.