The position $\mathbf{x}=(x, y, z)$ and velocity $\dot{\mathbf{x}}$ of a particle of mass $m$ are measured in a frame which rotates at constant angular velocity $\omega$ with respect to an inertial frame. The particle is subject to a force $\mathbf{F}=-9 m|\boldsymbol{\omega}|^{2} \mathbf{x}$. What is the equation of motion of the particle?

Find the trajectory of the particle in the coordinates $(x, y, z)$ if $\boldsymbol{\omega}=(0,0, \omega)$ and at $t=0, \mathbf{x}=(1,0,0)$ and $\dot{\mathbf{x}}=(0,0,0)$.

Find the maximum value of the speed $|\dot{\mathbf{x}}|$ of the particle and the times at which it travels at this speed.

[Hint: You may find using the variable $\xi=x+i y$ helpful.]