The dynamics of a massless hoop

Abstract

For many years it was believed that a rigid, massless hoop which is loaded with a heavy particle on its rim and rolling on a rough surface would hop when the normal reaction became zero. This is a hypothetical model that raises a number of very interesting and subtle aspects of dynamics. In this paper the motion of such hoops is analyzed in detail, and a new motion with unusual characteristics is found.

Gliding motions of a rigid body: the curious dynamics of Littlewood's rolling hoop

“…The most comprehensive treatment of Littlewood's hoop in the literature to date is by Theron & du Plessis [17]. These authors corrected errors in the earlier works [12,13,16].…”

“…In 1997, Tokieda [12] was the first to observe that gliding motions occur. The first detailed analysis of these motions was published four years later by Theron and du Plessis [17]. An alternative method to establish that N must vanish is to consider the moment balance: Iθ = (π P × (F f + N)) • E 3 .…”

Dynamics of a light hoop with an attached heavy disk: Inside an interaction pulse

“…The attached disk has radius ρ and mass m. The mass center of the system is at the point G such that C G = ϑ R 0 , where ϑ = (1 − k)m/M, k = ρ/R 0 , and M = m + m h is the total mass of the system. and Reeve [1966], and which has been a subject of several recent publications addressing the question of whether the hoop lifts off the horizontal surface during its motion [Tokieda 1997;Butler 1999;Pritchett 1999;Theron 2000;2002;Theron and du Plessis 2001;Liu and Yun 2004]. This hopping of the hoop dilemma was associated with the uncertainties in the description of motion of a hypothetical dynamic system which engages in rotation but does not possess any rotational inertia, which can become dynamically indeterminate.…”

“…The dynamics of motion of a light hoop with an attached disk was studied in detail in [Pritchett 1999;Theron 2000;Theron and du Plessis 2001], with the goal of addressing the question of possible hopping of the hoop and the dynamic indeterminacy in some stages of motion of an idealized system of a massless hoop with an attached point mass. In this and the subsequent section we extend this analysis to describe the interaction pulses and the stick/slip transitions throughout the motion of the hoop, for various mass ratios and the geometric and friction parameters.…”

“…For µ min s < tan(ϕ 0 /2), a massless hoop with an attached point mass released from rest with a point mass at an angle ϕ 0 skims the surface without interaction with it (N = F = 0,ẍ = 0,ÿ = −g) until the impact of the point mass with the surface at ϕ = π [Theron and du Plessis 2001]. For µ min s > tan(ϕ 0 /2), the rolling stage of the motion of a massless hoop with an attached point mass is specified by (5) with ς = 0, which has a closed form solution sin(ϕ/2) = sin(ϕ 0 /2) cosh (g/4R) 1/2 t , with the corresponding normal force N = mg 2 cos 2 (ϕ/2) − cos 2 (ϕ 0 /2) .…”