Paper 4, Section II, B

Let $(r, \theta)$ be polar coordinates in the plane. A particle of mass $m$ moves in the plane under an attractive force of $m f(r)$ towards the origin $O$. You may assume that the acceleration a is given by

$\mathbf{a}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+\frac{1}{r} \frac{d}{d t}\left(r^{2} \dot{\theta}\right) \hat{\theta}$

where $\hat{\mathbf{r}}$ and $\hat{\theta}$ are the unit vectors in the directions of increasing $r$ and $\theta$ respectively, and the dot denotes $d / d t$.

(a) Show that $l=r^{2} \dot{\theta}$ is a constant of the motion. Introducing $u=1 / r$ show that $\dot{r}=-l \frac{d u}{d \theta}$ and derive the geometric orbit equation

$l^{2} u^{2}\left(\frac{d^{2} u}{d \theta^{2}}+u\right)=f\left(\frac{1}{u}\right)$

(b) Suppose now that

$f(r)=\frac{3 r+9}{r^{3}}$

and that initially the particle is at distance $r_{0}=1$ from $O$, moving with speed $v_{0}=4$ in a direction making angle $\pi / 3$ with the radial vector pointing towards $O$.

Show that $l=2 \sqrt{3}$ and find $u$ as a function of $\theta$. Hence or otherwise show that the particle returns to its original position after one revolution about $O$ and then flies off to infinity.

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