Paper 4, Section II, C

Dynamics and Relativity | Part IA, 2015

Consider a particle with position vector r(t)r(t) moving in a plane described by polar coordinates (r,θ)(r, \theta). Obtain expressions for the radial (r)(r) and transverse (θ)(\theta) components of the velocity r˙\dot{\mathbf{r}} and acceleration r¨\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/r2-p / r^{2} in the radial direction, where pp 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 kr˙/r2-k \dot{r} / r^{2} where kk 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=r2θ˙h=r^{2} \dot{\theta} is constant.

Obtain the equation of motion

d2udθ2+khdudθ+u=ph2\frac{d^{2} u}{d \theta^{2}}+\frac{k}{h} \frac{d u}{d \theta}+u=\frac{p}{h^{2}}

where u=r1u=r^{-1}, and find its general solution assuming that k/h<2k /|h|<2. Show that so long as the motion remains bounded it eventually becomes circular with radius h2/ph^{2} / p.

Obtain the expression

E=12h2(u2+(dudθ)2)puE=\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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