Air Drag Controls the Finite-Time Singularity of Euler's Disk

TL;DR

This study reveals that air drag primarily causes the finite-time singularity in Euler's disk, with rolling friction dominating earlier, providing clarity on the dissipation mechanisms involved.

Contribution

It demonstrates that viscous air drag governs the late-time dynamics of Euler's disk near the singularity, a novel insight supported by experiments with varying parameters.

Findings

Air drag dominates near the singularity.

Rolling friction is dominant at earlier times.

Mass dependence confirms air drag's role.

Abstract

The motion of a disk spinning to rest after being tipped on its side is a classic example of a finite-time singularity, yet the dominant dissipation mechanism governing this process remains debated. Using stereoscopic high-speed imaging, we study the dynamics of disks with varying mass and radius on different surfaces. We show that the late-time motion near the singularity is governed by viscous air-drag arising from shear in the boundary layer beneath the disk, as evidenced by the mass dependence of the dynamics, measurements in a partial vacuum, and a geometric control using a steel ring. At earlier times, dissipation is dominated by rolling friction, which on glass exhibits an unexpected sublinear scaling with disk mass, suggesting an adhesion-based rolling resistance. These results clarify the dissipation mechanisms underlying the singularity of Euler's disk and have broader implications for rolling-contact systems operating under low loads on smooth surfaces.