A reference frame $S^{\prime}$ rotates with constant angular velocity $\boldsymbol{\omega}$ relative to an inertial frame $S$ that has the same origin as $S^{\prime}$. A particle of mass $m$ at position vector $\mathbf{x}$ is subject to a force $\mathbf{F}$. Derive the equation of motion for the particle in $S^{\prime}$.

A marble moves on a smooth plane which is inclined at an angle $\theta$ to the horizontal. The whole plane rotates at constant angular speed $\omega$ about a vertical axis through a point $O$ fixed in the plane. Coordinates $(\xi, \eta)$ are defined with respect to axes fixed in the plane: $O \xi$ horizontal and $O \eta$ up the line of greatest slope in the plane. Ensuring that you account for the normal reaction force, show that the motion of the marble obeys

$$\begin{aligned} \ddot{\xi} &=\omega^{2} \xi+2 \omega \dot{\eta} \cos \theta, \\ \ddot{\eta} &=\omega^{2} \eta \cos ^{2} \theta-2 \omega \dot{\xi} \cos \theta-g \sin \theta \end{aligned}$$

By considering the marble's kinetic energy as measured on the plane in the rotating frame, or otherwise, find a constant of the motion.

[You may assume that the marble never leaves the plane.]