Five and a half years ago I saw a simpler version of this question on the physics quiz Caltech sends to incoming freshman so they can prove how smart they are before arriving. This generalization appears in the first chapter of Motion Mountain, a free PDF textbook by Christoph Schiller, which purports to cover all of modern physics, integrating the theoretical and empirical viewpoints. I was linked here by a blog post from by Bee at Backreaction.
You have a spool of thread resting on a table, like so.
You pool on the thread very gently. What is the behavior of the spool as a function of the angle the thread makes with the horizontal? Assume the spool rolls without slipping. You should find two separate domains of behavior (one where to spool rolls forward, one where it rolls backward). What is the critical angle separating them? Provide a geometric interpretation of this result. Or, if you are sufficiently clever (I was not), skip right to the geometric interpretation, and use this to derive the critical angle.
What if the table is inclined with respect to the horizontal?
I only read as far as this first problem, so I don’t know yet whether the book actually provides the answer or simply asks the question. I don’t mind if you cheat, honestly. So long as you think the answer is pretty, however you get it.