Paper 4, Section II, C

Consider a particle with position vector $r(t)$ moving in a plane described by polar coordinates $(r, \theta)$. Obtain expressions for the radial $(r)$ and transverse $(\theta)$ components of the velocity $\dot{\mathbf{r}}$ and acceleration $\ddot{\mathbf{r}}$.

A charged particle of unit mass moves in the electric field of another charge that is fixed at the origin. The electrostatic force on the particle is $-p / r^{2}$ in the radial direction, where $p$ is a positive constant. The motion takes place in an unusual medium that resists radial motion but not tangential motion, so there is an additional radial force $-k \dot{r} / r^{2}$ where $k$ is a positive constant. Show that the particle's motion lies in a plane. Using polar coordinates in that plane, show also that its angular momentum $h=r^{2} \dot{\theta}$ is constant.

Obtain the equation of motion

$\frac{d^{2} u}{d \theta^{2}}+\frac{k}{h} \frac{d u}{d \theta}+u=\frac{p}{h^{2}}$

where $u=r^{-1}$, and find its general solution assuming that $k /|h|<2$. Show that so long as the motion remains bounded it eventually becomes circular with radius $h^{2} / p$.

Obtain the expression

$E=\frac{1}{2} h^{2}\left(u^{2}+\left(\frac{d u}{d \theta}\right)^{2}\right)-p u$

for the particle's total energy, that is, its kinetic energy plus its electrostatic potential energy. Hence, or otherwise, show that the energy is a decreasing function of time.

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