Central Forces and particle of mass

[roeb]

Homework Statement

A particle of mass m moves under the influence of a central force given by F(r) = -k/r^n. If the particle's orbit is circular and passes through the force center r = 0. Show n = 5. Find the radius of the circular orbit.

Homework Equations

The Attempt at a Solution

I was able to find n = 5 fairly easily with the orbit equation, however I am having difficulties with the radius.
so U(r) = -k/r^4

The orbit equation:
L/mr^2 dr/dphi = 1/m sqrt( 2m{E-U(r)} - L^2/r^2 )

According to my professor, E = 0.
Ueff = Lz^2 / (2mr^2) + -k/r^4
I plotted Ueff and it's certainly different from Kepler's problem. I can kind of see how the parabola (E = 0) in Kepler's problem is equivalent to the circle in this problem but I can't mathematically prove it..

So my question is:
1) How do I really show that E = 0 for a circular orbit with this potential. (Integrating the orbit equation with this potential is not possible in terms of elementary functions as far as I know)
I was under the impression that dUeff/dr = 0 for any central force potential, from plotting Ueff I can see that this point wouldn't yield a circle.

 

[NateD]

2) How do I calculate the radius of the orbit?I'd really appreciate it if someone could help me out here.