Hopping Hoops Don't Hop

Abstract

The dynamical behavior of a massless hoop with an attached point mass under the influence of gravity is an old chestnut, with some surprising features, including the question of whether or not it can hop. It appears in Littlewood's Miscellany [1] and most recently in this MONTHLY [2]. That there are many interesting aspects of this problem was suggested to me many years ago by the late Prof. J.C. Miller of Pomona College, who may have gotten a hint of the trickier parts directly from Littlewood himself. The purpose of this Note is to show that Littlewood's and Tokeida's asserted is wrong, even with a more realistic hoop and more realistic friction, and to suggest some approaches to a self consistent investigation of the rolling hoop. In its simplest form, the problem asks for the behavior of a massless hoop of radius R with a point mass M attached to its rim, rolling in a vertical plane on a level floor under gravity. Implicitly in Littlewood's problem, and explicitly in this Note, we understand the concept of massless to be the limit of a positive hoop mass tending to 0; otherwise the rotational behavior of the hoop in free fall is undefined. The idea of rough is central to the problem; it is (roughly) defined as a no-slip constraint at the point of contact of the hoop and the floor. Let i E [O, 7r ] be the angle from the radius vector to the mass, measured from vertical; other ranges of 1# are not considered in this Note. If the total energy is equal to the gravitational potential energy of the mass one diameter above the floor, the solution to the problem is the assertion by Littlewood that the hoop lifts off the ground or by Tokeida that the mass pulls the hoop up at i# = 7r/2; this particular value of i# depends on the total energy, but is especially simple in the case cited. Denote the normal force conferred by the floor on the hoop by n. The hopping conclusion is alleged to follow from the observation that, for zero kinetic energy at i# = 0 and for the rim constrained to be in no-slip contact with the floor, n > 0 for i# 7r/2. The evaluation of n(0) is elementary using Newton's second law and conservation of energy (the condition is conservative). An equivalent argument [2] is that the motion of the point mass either follows a cycloid for 0 < i < 7r/2, or its free fall parabolic preference for 7r/2 < i#; here the upper limit was not specified. This is wrong on both mathematical and physical grounds. To show this, we begin with the equations of motion, with the following notational conventions. Nondimensionalize the problem by measuring distance in units of radius R, mass in units of M, and time in units of ,R/g, where g is gravitational acceleration. In these units, gravitational acceleration is -1. The horizontal and vertical coordinates of the point mass are x = iU + sin iU and y = 1 + cos i0, respectively. The kinetic energy is given by (1/2)(x2 + y2) = p2(1 + COS i9), where the overdot denotes the time derivative, and the potential energy by y = 1 + cos i#. With this notation, Newton's law takes the form