A particle is moving in a circular path with a constant speed. If `theta` Is the angular displacement, then starting from `theta=0` , the maximum and minimum change in the linear momentum will occur when value of `theta` is respectively

Options

A. `45^(@)` and `90^(@)`
B. `90^(@)` and `180^(@)`
C. `180^(@)` and `360^(@)`
D. `90^(@)` and `270^(@)`

Explanation

To solve the problem, we need to analyze the change in linear momentum of a particle moving in a circular path with constant speed as it undergoes angular displacement.

### Step-by-Step Solution:

1. **Understanding Linear Momentum**:
The linear momentum \( P \) of a particle is given by the formula:
\[ P = m \cdot v \]
where \( m \) is the mass of the particle and \( v \) is its speed.

2. **Initial Momentum at \( \theta = 0^\circ \)**:
At \( \theta = 0^\circ \), the particle is at a certain point on the circular path. We can denote the initial momentum as:
\[ P_1 = m \cdot v \]

3. **Final Momentum at \( \theta = 180^\circ \)**:
When the particle moves to \( \theta = 180^\circ \), it is directly opposite to its initial position. The direction of its velocity vector will also be opposite. Therefore, the final momentum can be expressed as:
\[ P_2 = -m \cdot v \]

4. **Calculating Change in Momentum at \( \theta = 180^\circ \)**:
The change in momentum \( \Delta P \) when the particle moves from \( \theta = 0^\circ \) to \( \theta = 180^\circ \) is:
\[ \Delta P = P_2 - P_1 = (-m \cdot v) - (m \cdot v) = -2m \cdot v \]
The magnitude of this change is:
\[ |\Delta P| = 2m \cdot v \]

5. **Final Momentum at \( \theta = 360^\circ \)**:
When the particle completes a full circle and returns to \( \theta = 360^\circ \), its momentum will be the same as at \( \theta = 0^\circ\):
\[ P_2 = m \cdot v \]

6. **Calculating Change in Momentum at \( \theta = 360^\circ \)**:
The change in momentum at this position is:
\[ \Delta P = P_2 - P_1 = (m \cdot v) - (m \cdot v) = 0 \]
Therefore, the magnitude of this change is:
\[ |\Delta P| = 0 \]

7. **Conclusion**:
From the calculations, we find that:
- The maximum change in linear momentum occurs at \( \theta = 180^\circ \) with a value of \( 2m \cdot v \).
- The minimum change in linear momentum occurs at \( \theta = 360^\circ \) with a value of \( 0 \).

### Final Answer:
- Maximum change in linear momentum occurs at \( \theta = 180^\circ \).
- Minimum change in linear momentum occurs at \( \theta = 360^\circ \).