Solve Power-Law Spiral With Angular Momentum L

[simms_mj]

Homework Statement

Given L (angular momentum), find the form of V (r) so that the path of a particle is given by the spiral r = C(theta)^k, where C and k are constants

Homework Equations

L = mr²(theta)'

The Attempt at a Solution

I know I have to find a expression for r' using no theta's and insert it into the energy formula E = K + V(r) but I am lost on how to apply it.

 

[Dr.D]

Is that prime you have written on theta' in the angular momentum expression the time derivative of theta?

If so, doesn't that give you a relation between r and theta-dot?
The required trajectory is a relation between r and theta. Could you eliminate r between these two and do anything with that?