A suspended rope wrapped around a cylinder

The problem of a suspended rope wrapped around a fixed cylinder is studied. If the suspension force is larger than a certain threshold (which is larger than the weight of the rope), the rope would remain tightly wrapped around the cylinder. For suspension forces smaller than that threshold but larger than another threshold, the rope becomes loose (loses contact with the cylinder) at some points, but still remains at rest. Theses thresholds are obtained when there is no friction. The system is then analyzed also with friction, and the threshold of tight-wrapping is obtained for that case as well.