Shape sequence of rope coiling on a rotating plane

Physics Department, Thammasat University - Klong Luang, Pathumthani 12120, Thailand

Received: 15 January 2019
Accepted: 7 March 2019

Abstract

We study the equilibrium shapes of rope coiling on a rotating plane. For a rope fed from a height with slow feeding velocities, the shape is always circular, merely with different radii, regardless of the plane frequency. For moderate feeding velocities, the epitrochoid sets in at low plane frequencies and is deformed to a circle at high plane frequencies. For fast feeding velocities, which deposit a large amount of length rapidly on the plane, the shape sequence as the plane frequency increases comprises a hypotrochoid, an epitrochoid and a circle. The hypotrochoid-to-epitrochoid transition is associated with a reversal of the direction of the angular momentum. The criteria for the epitrochoid-to-circle transition are an equality between the primary and secondary radii of the epitrochoid, and the number of secondary loops reducing to zero. Motivated by the trajectory of a charged particle in a Penning trap being an epitrochoid, we extend the Penning trap model to give a hypotrochoid by applying the electric potential whose equipotential surface is an ellipsoid, unlike a hyperboloidal equipotential surface of an epitrochoid. The Penning trap analogue provides a unified way to rationalise the coiling frequencies and the angular momenta of both an epitrochoid and a hypotrochoid. The phase diagram for all the shapes is experimentally constructed as a function of the feeding velocity and the plane frequency.