(a) Consider an inertial frame $S$, and a frame $S^{\prime}$ which rotates with constant angular velocity $\boldsymbol{\omega}$ relative to $S$. The two frames share a common origin. Identify each term in the equation

$$\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}=\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}-2 \boldsymbol{\omega} \times\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}-\boldsymbol{\omega} \times(\boldsymbol{\omega} \times \mathbf{r})$$

(b) A small bead $P$ of unit mass can slide without friction on a circular hoop of radius $a$. The hoop is horizontal and rotating with constant angular speed $\omega$ about a fixed vertical axis through a point $O$ on its circumference.

(i) Using Cartesian axes in the rotating frame $S^{\prime}$, with origin at $O$ and $x^{\prime}$-axis along the diameter of the hoop through $O$, write down the position vector of $P$ in terms of $a$ and the angle $\theta$ shown in the diagram $\left(-\frac{1}{2} \pi \leqslant \theta \leqslant \frac{1}{2} \pi\right)$.

(ii) Working again in the rotating frame, find, in terms of $a, \theta, \dot{\theta}$ and $\omega$, an expression for the horizontal component of the force exerted by the hoop on the bead.

(iii) For what value of $\theta$ is the bead in stable equilibrium? Find the frequency of small oscillations of the bead about that point.