Euler's disk and its finite-time singularity

Air viscosity makes the rolling speed of a disk go up as its energy goes down.

Abstract

It is a fact of common experience that if a circular disk (for example, a penny) is spun upon a table, then ultimately it comes to rest quite abruptly, the final stage of motion being characterized by a shudder and a whirring sound of rapidly increasing frequency. As the disk rolls on its rim, the point P of rolling contact describes a circle with angular velocity Ω. In the classical (non-dissipative) theory¹, Ω is constant and the motion persists forever, in stark conflict with observation. Here I show that viscous dissipation in the thin layer of air between the disk and the table is sufficient to account for the observed abruptness of the settling process, during which, paradoxically, Ω increases without limit. I analyse the nature of this ‘finite-time singularity’, and show how it must be resolved.