How to twirl a hula-hoop

TL;DR

This paper analyzes the dynamics of twirling a hula-hoop with a waist moving along an elliptic path, deriving exact and approximate solutions, and identifying stability conditions and novel effects like inverse twirling.

Contribution

It provides the first analytical solutions for hula-hoop twirling with elliptic waist motion, including stability analysis and phase conditions for successful twirling.

Findings

One family of solutions is stable, the other unstable.

Optimal phase difference for twirling is between c/2 and c.

Inverse twirling occurs when waist moves opposite to hoop rotation.

Abstract

We consider twirling of a hula-hoop when the waist of a sportsman moves along an elliptic trajectory close to a circle. For the case of the circular trajectory, two families of exact solutions are obtained. Both of them correspond to twirling of the hula-hoop with a constant angular speed equal to the speed of the excitation. We show that one family of solutions is stable, while the other one is unstable. These exact solutions allow us to obtain approximate solutions for the case of an elliptic trajectory of the waist. We demonstrate that in order to twirl a hula-hoop one needs to rotate the waist with a phase difference lying between \pi/2 and \pi. An interesting effect of inverse twirling is described when the waist moves in opposite direction to the hula-hoop rotation. The approximate analytical solutions are compared with the results of numerical simulation