Thanks to everybody who replied! I ended up toughing it out with a partial solution of a differential equation (not to be confused with the solution of a partial differential equation).
I'm pretty sure my answer is equivalent to this. Here is my solution:
I set \theta as the CCW angular...
Homework Statement
(Important disclaimer: I came up with this problem out of curiosity in a classical mechanics course, so it might not be in a textbook, in the right place, or even solvable for that matter. However, I feel like it should be solvable, and most likely with elementary methods...