On the equilibria and uniform rotations of a dumbbell-shaped body on a rough horizontal plane with two contact points

A problem of motion of a dumbbell-shaped body on a horizontal rough plane is considered. It is assumed that the dumbbell is a weightless inextensible rod, with masses being concentrated at two points of it, and there is dry friction between these points and the plane. It is also assumed that a constant force acts perpendicular to the rod on some fixed point on it. The conditions under which the rod is at rest, as well as the conditions under which the rod uniformly rotates around one of its points of support, are determined. The relationship between the magnitude of the angular velocity of uniform rotation and the force providing such a rotation is revealed. Bifurcation diagrams are constructed and analyzed.