A particle of mass m rotates with a uniform angular speed `omega`. It is viewed from a frame rotating about the Z-axis with a uniform angular speed `omega_0`. The centrifugal force on the particler is

Options

A. `momega^2a`
B. `momega^2_0a`
C. `m((omega+omega_0)/2)^2a`
D. `momega omega_0a`

Answer

To solve the problem of finding the centrifugal force on a particle of mass \( m \) rotating with a uniform angular speed \( \omega \) as viewed from a frame rotating about the Z-axis with a uniform angular speed \( \omega_0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Centrifugal Force**:
- Centrifugal force is a pseudo force that appears when observing motion from a rotating frame of reference. It acts outward on a mass moving in a circular path.

2. **Identify the Acceleration of the Rotating Frame**:
- The acceleration experienced by an object in a rotating frame is given by the formula:
\[ a = \omega_0^2 r \]
where \( r \) is the radius of the circular path of the particle.

3. **Calculate the Centrifugal Force**:
- The centrifugal force \( F_c \) can be calculated using the formula:
\[ F_c = m \cdot a \]
Substituting the expression for acceleration from step 2, we get:
\[ F_c = m \cdot (\omega_0^2 r) \]

4. **Final Expression**:
- Thus, the centrifugal force acting on the particle is:
\[ F_c = m \omega_0^2 r \]

### Summary of the Solution:
The centrifugal force on the particle, as viewed from a rotating frame with angular speed \( \omega_0 \), is given by:
\[ F_c = m \omega_0^2 r \]