- Thread starter
- #1

"A particle of mass m moves under the influence of a central force $\textbf{F}(\textbf{r}) =−mf(r)e_r$, in the orbit

$r = c\theta^2$, (1)

where c > 0 and (r, θ) and er , eθ are the polar co-ordinates and corresponding basis vectors in the plane of motion of the particle. Show that:

\[

f(r)=-h^2(\frac{6c}{r^4}+\frac{1}{r^3})

\]

where $r^2\dot{\theta}=h$ is constant

\[

f(r)=-h^2(\frac{6c}{r^4}+\frac{1}{r^3})

\]

where $r^2\dot{\theta}=h$ is constant

[Hint: Use the substitution $u(\theta)=\frac{1}{r(\theta}$ to write the radial equation $\ddot{r}-r\dot{\theta}^2=-f(r)$ in terms of u(θ), and then determine f using this equation and (1).]"