On an apparent paradox in the motion of a smoothly constrained rod

The motion in a vertical plane of a straight rod constrained to move between two frictionless concentric circles is investigated. An apparent paradox arises because the reaction forces at the contact points are purely radial, and together with the rod's weight, which acts at the center of mass, are uncapable of supplying the torque about the center of mass necessary for the rod's rotation. A tentative explanation has been given by Carroll [Am. J. Phys. 52, 1010 (1984)], where a mysterious tangential contact force arising from an infinite radial contact force is invoked. We disagree with these conclusions. In postulating that the motion takes place according to certain prescribed geometrical constraints, one must always make certain that the laws of physics are respected, and the model must be chosen accordingly. A simple model consisting of an H-shaped frame with four contact points is used to determine the nature of the contact forces and the origin of the torque. As the link between the two half-rods of the H frame is reduced, a single rod is obtained which can be thought of as the limit of a sector whose angular width is made vanishingly small. The dilemma is resolved by showing that a singular angular density of contact force distribution prevails, but that the contact force itself is finite.