The position $\mathbf{x}$ and velocity $\dot{\mathbf{x}}$ of a particle of mass $m$ are measured in a frame which rotates at constant angular velocity $\boldsymbol{\omega}$ with respect to an inertial frame. Write down the equation of motion of the particle under a force $\mathbf{F}=-4 m \omega^{2} \mathbf{x}$.

Find the motion of the particle in $(x, y, z)$ coordinates with initial condition

$$\mathbf{x}=(1,0,0) \quad \text { and } \quad \dot{\mathbf{x}}=(0,0,0) \quad \text { at } t=0$$

where $\boldsymbol{\omega}=(0,0, \omega)$. Show that the particle has a maximum speed at $t=(2 n+1) \pi / 4 \omega$, and find this speed.

[Hint: you may find it useful to consider the combination $\zeta=x+i y$.]