Potential Energy using effective potential

[derrickb]

Homework Statement

For a given angular momentum L, find the potential energy function U(r) that leads to a spiral path of the form r=r₀θ^k. Choose the total energy E to be zero. Hint: obtain an expression for r' that is only a function of r, not θ, and then use the energy equation with effective potential

Homework Equations

E=.5μ(r')²+(L²/2μr²)+U(r)
r=r₀θ^k

The Attempt at a Solution

I can't seem to figure out where to start. I don't want a full solution, but if someone could point me in the right direction, I will try to solve it from there.

 

[AvroArrow]

I understand that I have to solve for r' and then use the energy equation, but I'm not sure how to go about doing that.