For a particle performing uniform circular motion, choose the incorrect statement from the following.

Options

A. Magnitude of particle velocity (speed) remains constant
B. Particle velocity remains directed perpendicular to radius vector.
C. Direction of acceleration keeps changing as particle moves.
D. Magnitude of acceleration does not remain constant.

Answer

To solve the problem of identifying the incorrect statement regarding a particle performing uniform circular motion, we can analyze the properties of such motion step by step.

### Step-by-Step Solution:

1. **Understanding Uniform Circular Motion**:
- In uniform circular motion, a particle moves along a circular path with a constant speed. This means that while the speed (magnitude of velocity) remains constant, the direction of the velocity vector changes continuously.

2. **Centripetal Force and Acceleration**:
- For a particle to maintain uniform circular motion, a centripetal force must act on it, directed towards the center of the circular path. This force is responsible for changing the direction of the velocity vector, keeping the particle in circular motion.

3. **Angular Velocity and Acceleration**:
- In uniform circular motion, the angular velocity (ω) is constant. Since angular velocity is constant, the angular acceleration (α), which is the rate of change of angular velocity, is zero.

4. **Tangential Acceleration**:
- Tangential acceleration (a_t) is given by the formula \( a_t = r \cdot \alpha \). Since α is zero in uniform circular motion, the tangential acceleration is also zero. This means there is no change in the speed of the particle along the circular path.

5. **Centripetal Acceleration**:
- The centripetal acceleration (a_c) is given by \( a_c = \frac{v^2}{r} \), where v is the constant speed of the particle and r is the radius of the circular path. While the magnitude of centripetal acceleration remains constant, its direction continuously changes as it always points towards the center of the circle.

6. **Analyzing the Statements**:
- Now, we need to evaluate the statements provided in the question:
- **Statement A**: The magnitude of the particle's velocity remains constant. (True)
- **Statement B**: The particle's velocity remains directed perpendicular to the radius vector. (True)
- **Statement C**: The direction of acceleration keeps changing as the particle moves. (True)
- **Statement D**: The magnitude of the acceleration does not remain constant. (False)

7. **Conclusion**:
- The incorrect statement is **Statement D**, as the magnitude of centripetal acceleration remains constant, even though its direction changes.