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 decreases
D. increase continuously

Answer

To solve the problem, we need to analyze the situation of the ant moving along the diameter of a rotating disc and how it affects the angular velocity of the disc.

### Step-by-Step Solution:

1. **Understand the System**:
- We have a rotating disc with an ant sitting at the edge. The disc has a radius \( R \) and the ant has a mass \( m \).
- The moment of inertia of the disc is given by \( I_{disc} = \frac{1}{2} M R^2 \), where \( M \) is the mass of the disc.

2. **Initial Moment of Inertia**:
- When the ant is at the edge of the disc, the moment of inertia of the system (disc + ant) can be calculated as:
\[ I_{initial} = I_{disc} + mR^2 = \frac{1}{2} M R^2 + mR^2 \]

3. **Conservation of Angular Momentum**:
- Since there is no external torque acting on the system, angular momentum is conserved. Therefore:
\[ I_{initial} \cdot \omega_{initial} = I_{final} \cdot \omega_{final} \]
- Here, \( \omega_{initial} \) is the initial angular velocity and \( \omega_{final} \) is the final angular velocity after the ant moves.

4. **Final Moment of Inertia**:
- As the ant moves towards the center, its distance from the axis of rotation decreases. When the ant reaches the center, its contribution to the moment of inertia becomes zero.
- The final moment of inertia when the ant is at the center is:
\[ I_{final} = I_{disc} + m(0)^2 = I_{disc} = \frac{1}{2} M R^2 \]

5. **Analyzing the Change**:
- Initially, the moment of inertia is greater than the final moment of inertia because the ant contributes \( mR^2 \) when it is at the edge.
- As the ant moves from the edge to the center, the moment of inertia decreases.

6. **Effect on Angular Velocity**:
- From the conservation of angular momentum, since \( I_{initial} > I_{final} \), it follows that \( \omega_{final} > \omega_{initial} \).
- This means that as the ant moves towards the center, the angular velocity of the disc increases.

7. **Conclusion**:
- The angular velocity of the disc will initially increase as the ant moves towards the center and then will decrease after the ant crosses the center and continues to the other edge. Therefore, the correct answer is that the angular velocity of the disc will first increase and then decrease.

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