The geometric phase and the dry friction of sleeping tops on inclined planes

TL;DR

This paper investigates the motion of a spinning sleeping top on inclined planes, revealing a complex dry friction law that explains its stable and unstable behaviors depending on the inclination angle.

Contribution

It introduces a phenomenological dry friction law that accounts for the top's motion, stability, and geometric trajectories on inclined planes.

Findings

Sleeping tops are force free below a critical inclination.

Trajectory invariance under frequency rescaling in the adiabatic limit.

Instability and downhill acceleration at larger inclinations.

Abstract

We report on the motion of a spinning sleeping top on an inclined plane. Below a critical inclination angle the sleeping tops are force free. The trajectory of a sleeping top on weakly inclined planes in the adiabatic limit is invariant of the angular frequency of the top and thus invariant under a rescaling of the time, however not invariant under time reversal. The stationary trajectory of the sleeping top is characterized by its angle to the in plane horizontal direction. At larger inclinations of the plane the stationary motion of the top becomes unstable and the top accelerates downhill. The behavior points towards a complex law of dry friction of the contact point between the top tip and the material of the inclined plane that depends on a slip parameter. We propose a phenomenological law of dry friction that can explain the relaxation of the top into the sleeping position, the geometric behavior of the top trajectories, and the instability of the stationary motion at larger inclination angles.