Please help the asker edit the question so that it asks about the underlying physics concepts instead of specific computations. If there are any complete answers, please flag them for moderator attention.
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My son is practicing with his high school Science Olympiad team and came across this question:
His source includes an answer, without explanation, thatthe coefficient of friction is$\mu\approx3.3$.
I tried solving this myself. I don't think the question is well-posed (and that the question doesn't ask for theminimum coefficient of friction makes more suspicious). I made the assumptions that (1) the wheel is a solid disc and hence has moment of inertia$I = \frac{1}{2} m r^2$, and (2) that the rope is horizontal where it last makes contact with the wheel.
We can set up force / torque equations:$$\sum F_y = 0$$$$\sum F_x = m a_x$$$$\sum \tau = I \alpha_x$$where$m$ is the mass of the wheel,$x$ is oriented up the incline, and$y$ is oriented normal to the incline. Here's a crude free-body diagram I made with Gemini:
The above assumptions allow us to rewrite$I= \frac{1}{2} m r^2$ and to orient the tension force acting on the top of the wheel. We further assume rolling without slipping so that$a_x = r \alpha_x$. From here it's just algebra: solve the first equation to find the normal force between the plane and the wheel, then solve the other two equations to find the necessary friction force between the wheel and the plane. In the end I find thatthe minimum coefficient of friction is$\mu_{\rm{min}} = 1/\sqrt{3}$.
My question is whether I've modeled this situation correctly and whether there's any other reasonable way to model this with the information given?
