The speed of revolution of a particle going around a circle is doubled and its speed is halved, what happens to the centripetal acceleration?

Options

A. Becomes 4 times
B. Doubled
C. Halved
D. Remains unchanged

Answer

To solve the problem, we need to analyze how the centripetal acceleration changes when the speed of revolution of a particle is doubled and then halved. We will use the formula for centripetal acceleration and the relationships between linear speed, angular speed, and radius.

### Step-by-Step Solution

1. **Understand the Formula for Centripetal Acceleration**:
The centripetal acceleration (a_c) is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \(v\) is the tangential speed and \(r\) is the radius of the circular path.

2. **Identify the Changes in Speed**:
We are told that the speed of the particle is first doubled and then halved. Let's denote the original speed as \(v\).
- When the speed is doubled, the new speed \(v' = 2v\).
- When the speed is halved, the new speed becomes \(v'' = \frac{v}{2}\).

3. **Calculate Centripetal Acceleration for Each Case**:
- **Case 1**: When speed is doubled (\(v' = 2v\)):
\[ a_{c1} = \frac{(2v)^2}{r} = \frac{4v^2}{r} \]

- **Case 2**: When speed is halved (\(v'' = \frac{v}{2}\)):
\[ a_{c2} = \frac{\left(\frac{v}{2}\right)^2}{r} = \frac{\frac{v^2}{4}}{r} = \frac{v^2}{4r} \]

4. **Analyze the Changes in Centripetal Acceleration**:
Now, we have two expressions for centripetal acceleration:
- After doubling the speed: \(a_{c1} = \frac{4v^2}{r}\)
- After halving the speed: \(a_{c2} = \frac{v^2}{4r}\)

5. **Conclusion**:
The centripetal acceleration changes based on the speed. When the speed is doubled, the centripetal acceleration increases by a factor of 4. When the speed is halved, the centripetal acceleration decreases to a quarter of the original value.

However, the question asks what happens to the centripetal acceleration after both changes are applied sequentially.

### Final Answer:
The centripetal acceleration will vary based on the sequence of changes applied to the speed. After doubling the speed, the centripetal acceleration increases significantly, and after halving the speed, it decreases significantly. The final centripetal acceleration will depend on the specific values of \(v\) and \(r\).