A particle is moving in a circular path ofradius rat a constant speed v. Which one of the following graphs correctly represents its acceleration a?

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the motion of a particle moving in a circular path at a constant speed and determine the nature of its acceleration.

### Step-by-Step Solution:

1. **Understanding Circular Motion**:
- A particle moving in a circular path at a constant speed is undergoing uniform circular motion. In this type of motion, the speed (magnitude of velocity) remains constant, but the direction of the velocity changes continuously.

2. **Identifying the Type of Acceleration**:
- In uniform circular motion, the particle experiences centripetal acceleration, which is directed towards the center of the circular path. This acceleration is responsible for changing the direction of the velocity vector.

3. **Formula for Centripetal Acceleration**:
- The formula for 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.

4. **Analyzing the Relationship**:
- From the formula \(a_c = \frac{v^2}{r}\), we can see that the centripetal acceleration is inversely proportional to the radius \(r\). This means that as the radius \(r\) increases, the centripetal acceleration \(a_c\) decreases, provided that the speed \(v\) remains constant.

5. **Choosing the Correct Graph**:
- Given that \(a_c\) is inversely proportional to \(r\), the graph that represents this relationship will show a decreasing trend as \(r\) increases.
- Therefore, the correct graph will depict acceleration \(a\) on the y-axis and radius \(r\) on the x-axis, showing that as \(r\) increases, \(a\) decreases.

6. **Conclusion**:
- Based on the analysis, the correct option is the one that shows a decreasing curve, which corresponds to the fourth option.