Question

a particle in a central force field discribed by the spiral orbit r=r0e^(theta). how to show that the force law is inverse cube and that (theta) veries logrithmically with t?.

Answer 1

$r=r_0e^{b\theta},$

$u=\frac 1r=\frac{e^{-b\theta}}{r_0},$

$\frac{d^2u}{d\theta^2}+u=\frac{F}{mr_0^2u^2},$

$(b^2 \frac{e^{-b\theta}}{r_0}+\frac{e^{-b\theta}}{r_0})\cdot \frac 1{r^2}=\frac F{mr_0^2},$

$(1+b^2)\frac{e^{-b\theta}}{r_0}\cdot \frac 1{r^2}=\frac F{mr_0^2},$

$(1+b^2)\frac 1{r^3}=\frac F{mr_0^2},$

$F\sim \frac 1{r^3}.$