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

Options

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

Answer

To solve the problem, we need to analyze the relationship between the speed, angular speed, radius, and centripetal acceleration of an object moving in a circular path.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Centripetal acceleration (\(a_c\)) is given by the formula:
\[ a_c = \frac{v^2}{r} \]
where \(v\) is the linear speed and \(r\) is the radius of the circular path.

2. **Initial Conditions**:
- Let the initial speed of the object be \(v_1 = v\).
- Let the initial angular speed be \(\omega_1 = \omega\).
- The radius of the circular path is \(r_1 = r\).
- The initial centripetal acceleration is:
\[ a_{c1} = \frac{v^2}{r} \]

3. **New Conditions**:
- The speed is doubled: \(v_2 = 2v\).
- The angular speed is halved: \(\omega_2 = \frac{\omega}{2}\).
- We need to find the new radius \(r_2\). The relationship between linear speed and angular speed is:
\[ v = \omega r \]
- Therefore, for the new conditions:
\[ v_2 = \omega_2 r_2 \]
- Substituting the values:
\[ 2v = \left(\frac{\omega}{2}\right) r_2 \]

4. **Solve for the New Radius**:
- Rearranging the equation gives:
\[ r_2 = \frac{2v}{\frac{\omega}{2}} = \frac{2v \cdot 2}{\omega} = \frac{4v}{\omega} \]
- Since \(r = \frac{v}{\omega}\), we can express \(r_2\) in terms of \(r\):
\[ r_2 = 4r \]

5. **Calculate New Centripetal Acceleration**:
- Now, we can find the new centripetal acceleration \(a_{c2}\):
\[ a_{c2} = \frac{v_2^2}{r_2} = \frac{(2v)^2}{4r} = \frac{4v^2}{4r} = \frac{v^2}{r} \]

6. **Conclusion**:
- Since \(a_{c2} = \frac{v^2}{r} = a_{c1}\), the centripetal acceleration remains the same.

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