Consider an inertial frame of reference $S$ and a frame of reference $S^{\prime}$ which is rotating with constant angular velocity $\boldsymbol{\omega}$ relative to $S$. Assume that the two frames have a common origin $O$.

Let $\mathbf{A}$ be any vector. Explain why the derivative of $\mathbf{A}$ in frame $S$ is related to its derivative in $S^{\prime}$ by the following equation

$$\left(\frac{d \mathbf{A}}{d t}\right)_{S}=\left(\frac{d \mathbf{A}}{d t}\right)_{S^{\prime}}+\omega \times \mathbf{A} .$$

[Hint: It may be useful to use Cartesian basis vectors in both frames.]

Let $\mathbf{r}(t)$ be the position vector of a particle, measured from $O$. Derive the expression relating the particle's acceleration as observed in $S,\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S}$, to the acceleration observed in $S^{\prime},\left(\frac{d^{2} \mathbf{r}}{d t^{2}}\right)_{S^{\prime}}$, written in terms of $\mathbf{r}, \boldsymbol{\omega}$ and $\left(\frac{d \mathbf{r}}{d t}\right)_{S^{\prime}}$

A small bead of mass $m$ is threaded on a smooth, rigid, circular wire of radius $R$. At any given instant, the wire hangs in a vertical plane with respect to a downward gravitational acceleration $\mathbf{g}$. The wire is rotating with constant angular velocity $\boldsymbol{\omega}$ about its vertical diameter. Let $\theta(t)$ be the angle between the downward vertical and the radial line going from the centre of the hoop to the bead.

(i) Show that $\theta(t)$ satisfies the following equation of motion

$$\ddot{\theta}=\left(\omega^{2} \cos \theta-\frac{g}{R}\right) \sin \theta .$$

(ii) Find any equilibrium angles and determine their stability.

(iii) Find the force of the wire on the bead as a function of $\theta$ and $\dot{\theta}$.