A hoop stands in a vertical plane on a rough incline which the plane of the hoop cuts in a line of greatest slope. It is kept in equilibrium by a string fastened to a point in the circumference, wound round it, and fastened to a peg in the incline further up and in the same plane. If \lambda is the angle of friction, \theta the angle the hoop subtends at the peg, and \alpha that of the incline, show that there is limiting equilibrium when \theta = \alpha + \arccos \left[\frac{\sin (a - \lambda)}{\sin \lambda}\right]. What will happen if \theta has a greater value?