A body moving along a circular path of radius R with velocity V , has centripetal acceleration a If its velocity is made equal to 2v , then its centripetal acceleration is

Options

A. 4a
B. 2a
C. `a/4`
D. `a/2`

Explanation

To solve the problem step by step, we need to understand the relationship between velocity, radius, and centripetal acceleration in circular motion.

### Step-by-Step Solution:

1. **Understanding Centripetal Acceleration**:
The formula for centripetal acceleration \( a \) of an object moving in a circular path is given by:
\[ a = \frac{v^2}{R} \]
where \( v \) is the linear velocity and \( R \) is the radius of the circular path.

2. **Given Conditions**:
Initially, the body is moving with velocity \( v \) and has centripetal acceleration \( a \). Therefore, we can express this as:
\[ a = \frac{v^2}{R} \]

3. **New Velocity**:
Now, the velocity of the body is changed to \( 2v \). We need to find the new centripetal acceleration \( a_1 \) when the velocity is \( 2v \).

4. **Calculating New Centripetal Acceleration**:
Using the formula for centripetal acceleration again for the new velocity:
\[ a_1 = \frac{(2v)^2}{R} \]

5. **Expanding the Expression**:
Simplifying the expression for \( a_1 \):
\[ a_1 = \frac{4v^2}{R} \]

6. **Relating New Acceleration to Initial Acceleration**:
We know from the initial condition that:
\[ a = \frac{v^2}{R} \]
Therefore, we can express \( a_1 \) in terms of \( a \):
\[ a_1 = 4 \left(\frac{v^2}{R}\right) = 4a \]

7. **Final Answer**:
Thus, when the velocity is increased to \( 2v \), the new centripetal acceleration \( a_1 \) is:
\[ a_1 = 4a \]

### Conclusion:
The centripetal acceleration when the velocity is doubled is four times the initial centripetal acceleration.