How does friction act on a body, if only 2 regions on it are rough? [closed]

While tackling an Olympiad question, it came to my mind that friction need not act in the same direction at all points on a body. I thought of using integration to evaluate the net frictional force, but was stopped by this statement.

Frictional force between 2 bodies, doesn't depend on the surface area in contact.

If that is the case, then how does the force act on a body like the following, where only the narrow strip marked is rough and hence friction can act only there? What are the individual frictional forces at the points A and B?

The image shows a hollow cylinder, mass $M$ and radius $R$ from top view. The cylinder is placed with its flat end in contact with the ground. Only the gray shaded line is rough, id est, friciton acts only there. The cylinder is rotating about the topmost point. Ignore the $v$ in the diagram.

closed as unclear what you're asking by ja72, Kyle Kanos, ACuriousMind♦, Neuneck, DanuAug 26 '15 at 9:48

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• Your question is unclear to me. Friction acts on each little patch of contact in a way that always opposes motion. – ja72 Aug 21 '15 at 18:06
• Yes, I understand that. What I don't get, is this. If suppose the little patch is of area $dA$ and has height $h$ and density $\rho$, then will the friction force acting on that small patch be $F_{fr} = \mu dm g = \mu g \rho h dA$ ? If yes, then that would imply that the force IS dependent on the surface area in contact. If no, then how can the force be evaluated? – Aritra Das Aug 21 '15 at 18:20
• @ja72 or would the force at either point A and B be dependent on the mass of the entire cylinder? – Aritra Das Aug 21 '15 at 18:44

As for the case you represent, the cylinder will perceive friction forces opposed to its rotation movement, which will be tangential to the circle, and opposed to the linear velocities in these points. So it will either stay static, or it will rotate w.r.t its center, depending on the actual values of $v$ and $\mu$.