Consider the following setup:
A (homogeneous) pulley of mass $M>0$ is placed in the vertical plane, where it is fixed at its center but can freely rotate. An ideal rope is placed around the pulley and two point masses are attached to the ends of the rope. Assume, moreover, that the rope does not slip over the pulley.
Now, what I want to do is to find all the forces acting on the pulley. Surely, we have the gravitational force $Mg$, pointing downwards and acting on the center of mass of the pulley, and a constraint force $V$ which prevents it from falling.
However, these cannnot be all the forces. In fact, there is a force of static friction between the rope and the pulley which allows the former not to slip and the latter to rotate. (I imagine one should also take into account the "weight" of the rope "pressing" on the pulley, but since the rope is ideal, hence massless, this force should be null. Correct me if I'm wrong.)
My question, now, is
How do we find the magnitude, the direction and the point of application of this friction force so that we can write down the equations of motion?
Perhaps a line integral is required? This seems overkill...but still, how do we find the remaining forces (i.e. the forces beside gravity and the constraint force) acting on the pulley in a strictly rigorous way?
Please, note that I'm not just interested in the answer itself, but on how one arrives at the answer as well!