A particle of mass m revolves in a circle of radius a about the Z...

The centrifugal force is a pseudo force that arises when observations are made from a rotating reference frame.

The magnitude of the centrifugal force acting on a particle of mass $m$ at a perpendicular distance $r$ from the axis of rotation is given by $F_{c} = m \omega_{\text{frame}}^2 r$, where $\omega_{\text{frame}}$ is the angular speed of the rotating reference frame.

In the given problem, the reference frame rotates with an angular speed $\omega_0$ about the $Z$-axis, and the particle is at a distance $a$ from the $Z$-axis.

Therefore, the centrifugal force acting on the particle is $m \omega_0^2 a$.

The actual angular speed $\omega$ of the particle does not affect the centrifugal force.