A motorcycle of mass $M$ moves on a bowl-shaped surface specified by its height $h(r)$ where $r=\sqrt{x^{2}+y^{2}}$ is the radius in cylindrical polar coordinates $(r, \phi, z)$. The torque exerted by the motorcycle engine on the rear wheel results in a force $\mathbf{F}(t)$ pushing the motorcycle forward. Assuming $\mathbf{F}(t)$ is directed along the motorcycle's velocity and that the motorcycle's vertical velocity and acceleration are small, show that the motion is described by

$$\begin{aligned} \ddot{r}-r \dot{\phi}^{2} &=-g \frac{d h}{d r}+\frac{F(t)}{M} \frac{\dot{r}}{\sqrt{\dot{r}^{2}+r^{2} \dot{\phi}^{2}}} \\ r \ddot{\phi}+2 \dot{r} \dot{\phi} &=\frac{F(t)}{M} \frac{r \dot{\phi}}{\sqrt{\dot{r}^{2}+r^{2} \dot{\phi}^{2}}} \end{aligned}$$

where dots denote time derivatives, $F(t)=|\mathbf{F}(t)|$ and $g$ is the acceleration due to gravity.

The motorcycle rider can adjust $F(t)$ to produce the desired trajectory. If the rider wants to move on a curve $r(\phi)$, show that $\phi(t)$ must obey

$$\dot{\phi}^{2}=g \frac{d h}{d r} /\left(r+\frac{2}{r}\left(\frac{d r}{d \phi}\right)^{2}-\frac{d^{2} r}{d \phi^{2}}\right)$$

Now assume that $h(r)=r^{2} / \ell$, with $\ell$ a constant, and $r(\phi)=\epsilon \phi$ with $\epsilon$ a positive constant, and $0 \leqslant \phi<\infty$ so that the desired trajectory is a spiral curve. Assuming that $\phi(t)$ tends to infinity as $t$ tends to infinity, show that $\dot{\phi}(t)$ tends to $\sqrt{2 g / \ell}$ and $F(t)$ tends to $4 \epsilon M g / \ell$ as $t$ tends to infinity.