On the Equilibria of a Heavy Hoop Suspended on a Nail

We consider the plane problem of the equilibrium of a homogeneous heavy thin elliptical hoop suspended on a thin horizontal nail. Under the assumption that a dry friction force acts between the nail and the hoop, the dependence of the set of equilibrium positions on the friction coefficient and the semi-axes of the ellipse is studied. The results obtained apply to the following problem: to describe the equilibrium positions of a heavy rigid body (“gun”) suspended on a nail using a rope, both ends of which are fixed in the body. It is shown how such a distribution of results can be carried out directly in the case when the center of mass of the body is located in the middle between the suspension points.