An inertial reference frame $S$ and another reference frame $S^{\prime}$ have a common origin $O$, and $S^{\prime}$ rotates with angular velocity $\boldsymbol{\omega}(t)$ with respect to $S$. Show the following:

(i) the rates of change of an arbitrary vector a $(t)$ in frames $S$ and $S^{\prime}$ are related by

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

(ii) the accelerations in $S$ and $S^{\prime}$ are related by

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

where $\mathbf{r}(t)$ is the position vector relative to $O$.

A train of mass $m$ at latitude $\lambda$ in the Northern hemisphere travels North with constant speed $V$ along a track which runs North-South. Find the magnitude and direction of the sideways force exerted on the train by the track.