# Force acting on a massive pulley

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!

• If you simplify/idealize the problem from the start by saying you have a non-slip massless rope, I don't see much point in trying to consider things like friction afterwards. In the end it depends on what you want to calculate, but from the text it seems that you should be more concerned about the forces from the point masses than friction or the constraint force. Commented Aug 12, 2017 at 15:49
• @user1583209 there are several problems which involve such a setup, or a similar one, in introductory classical mechanics, and in practically all cases it is needed to find the forces which act on the pulley, or at least those which generate momentum. So yes, there is a point in the question. Commented Aug 12, 2017 at 18:20
• Yes, I am aware of these kind of problems. My point was, that with the initial assumptions, you basically get rid of any discussion of how the rope interacts with the pulley (i.e. friction). So it is kind of strange to me trying to consider friction afterwards. To give you another example: Implicitly you assume that the pulley is rigid (cannot be deformed). That's why you are not considering forces leading to deformation either. Commented Aug 12, 2017 at 20:00