I have been working on the following AP Physics 1 free-response question:
1974M2. The moment of inertia of a uniform solid sphere (mass $M$, radius $R$) is $\frac{2}{5}MR^2$. The sphere is placed on an inclined plane (angle $θ$) and released from rest. Determine the minimum coefficient of friction $μ$ between the sphere and plane with which the sphere will roll down the incline without slipping.
I understand that the torque in this case is a result of static friction. I have performed the following work so far:
$$\sum I = \frac{2}{5}MR^2$$ $$\sum \tau = I\alpha = \frac{2}{5}MRa_t$$ $$\sum \tau = \mu F_NR = \mu Mgcos{\theta}R$$ $$a_t = \frac{5}{2}\mu gcos{\theta}$$
I know that the next step is to substitute $a_t$ into $\sum F = Ma_t$ but am struggling when it comes to defining what forces make up $\sum F$. My first instinct was for $$\sum F = Mgsin{\theta}$$
but the answer key that I looked at also took into account the frictional force in computing $$\sum F = Mgsin{\theta} - \mu Mgcos{\theta}$$
This doesn't make sense to me because I would effectively be solving for the kinetic frictional coefficient but the question is asking for the friction that causes the sphere to roll, which I believe to be static. I understand how there would be kinetic friction, but don't see how it relates back to the question.
I think there's clearly something I'm misunderstanding. Could someone help clarify what's going on here?
