A study of the gravitational interaction of a unit point mass moving external to(r>R)but under the influence of a fixed very thin ring of radiusRand mass ofM(M≫m)is presented. We consider the case when the motion of the point mass is restricted to the plane of the ring. With this restriction, it is possible to find trajectories for which the system has a positive total energy, and yet the point mass is bound to, and doomed to collide with, the ring. An example of a numerically iterated trajectory is included to support and illustrate this conclusion.