if speed of an object revolving in a circular path is dobled and angular speed is reduced to half of original value, then centripetal acceleration will becom/remain

Options

A. same
B. double
C. half
D. quadruple

Answer

To solve the problem, we need to analyze how the centripetal acceleration changes when the speed of an object revolving in a circular path is doubled and the angular speed is reduced to half of its original value.

### Step-by-step Solution:

1. **Understand the Formula for Centripetal Acceleration**:
The centripetal acceleration \( a_c \) can be expressed in two ways:
\[ a_c = \frac{v^2}{r} \]
or
\[ a_c = \omega^2 r \]
where \( v \) is the linear speed, \( r \) is the radius of the circular path, and \( \omega \) is the angular speed.

2. **Relate Linear Speed and Angular Speed**:
The relationship between linear speed \( v \) and angular speed \( \omega \) is given by:
\[ v = \omega r \]
This means we can express the radius \( r \) in terms of \( v \) and \( \omega \):
\[ r = \frac{v}{\omega} \]

3. **Substituting in the Centripetal Acceleration Formula**:
We can express centripetal acceleration in terms of \( v \) and \( \omega \):
\[ a_c = \frac{v^2}{r} = \frac{v^2}{\frac{v}{\omega}} = v \cdot \omega \]

4. **Analyzing the Changes**:
- The speed is doubled: \( v' = 2v \)
- The angular speed is halved: \( \omega' = \frac{\omega}{2} \)

5. **Calculate the New Centripetal Acceleration**:
Substitute the new values into the centripetal acceleration formula:
\[ a_c' = v' \cdot \omega' = (2v) \cdot \left(\frac{\omega}{2}\right) \]
Simplifying this gives:
\[ a_c' = 2v \cdot \frac{\omega}{2} = v \cdot \omega \]

6. **Conclusion**:
Since \( a_c' = v \cdot \omega \) is equal to the original centripetal acceleration \( a_c \), we conclude that:
\[ a_c' = a_c \]
Therefore, the centripetal acceleration remains the same.

### Final Answer:
The centripetal acceleration will remain the same.