A body is moving in a circular path with acceleration a. If its speed gets doubled, find the ratio of centripetal acceleration after and before the speed is changed

Options

A. `1:4`
B. `1:2`
C. `2:1`
D. `4:1`

Explanation

To solve the problem, we need to calculate the ratio of centripetal acceleration before and after the speed of the body is doubled.

### Step-by-Step Solution:

1. **Understand Centripetal Acceleration**:
Centripetal acceleration (a_c) is given by the formula:
\[ a_c = \frac{V^2}{R} \]
where \( V \) is the speed of the body and \( R \) is the radius of the circular path.

2. **Define Initial Conditions**:
Let the initial speed of the body be \( V \). Therefore, the initial centripetal acceleration (before the speed is doubled) is:
\[ a_{before} = \frac{V^2}{R} \]

3. **Define Final Conditions**:
When the speed is doubled, the new speed becomes \( 2V \). The centripetal acceleration after the speed is changed is:
\[ a_{after} = \frac{(2V)^2}{R} \]

4. **Calculate \( a_{after} \)**:
Substitute \( 2V \) into the centripetal acceleration formula:
\[ a_{after} = \frac{(2V)^2}{R} = \frac{4V^2}{R} \]

5. **Find the Ratio of Centripetal Accelerations**:
Now, we need to find the ratio of the centripetal acceleration after the speed change to that before the speed change:
\[ \text{Ratio} = \frac{a_{after}}{a_{before}} = \frac{\frac{4V^2}{R}}{\frac{V^2}{R}} \]

6. **Simplify the Ratio**:
The \( R \) and \( V^2 \) terms cancel out:
\[ \text{Ratio} = \frac{4V^2}{V^2} = 4 \]

7. **Final Answer**:
The ratio of centripetal acceleration after to before the speed is changed is:
\[ \text{Ratio} = 4:1 \]

### Conclusion:
Thus, the final answer is that the ratio of centripetal acceleration after the speed is doubled to that before the speed change is \( 4:1 \).