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

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.

Figures (8)

• Figure 1: Left: sketch of the Euler's disk system. The precession frequency $\Omega=\dot{\psi}$ corresponds to the frequency of the contact point. Right:$84$ g aluminum disk on steel surface.
• Figure 2: A)$\theta$ vs. $t_f-t$ for filleted disks of varying mass and constant diameter on glass. Three repetitions per disk. Inset: Corresponding $\Omega$ vs. $\sin\theta$ data with best fit $\Omega\propto(\sin\theta)^{-.49}$. B) The exponent $n$ from Eq. \ref{['theta']} fitted to the final 0.1 s of motion for disks of varying mass for both sharp-edged and filleted disks on different surfaces. C) Prefactor $A$ from fits to Eq. \ref{['theta']} scaled by $R^{-7/9}$ for the last 0.1s of motion. Inset: $A$ vs. $R$ for sharp disks with varying radius; scaled by $m^{4/9}$.
• Figure 3: A)$\theta$ vs. $t_f-t$ for a 445 g steel disk on glass in ambient conditions and in a partial vacuum of 0.1 atm. Dashed lines are solutions to Eq. \ref{['dragfric']} with fitted $\mu=10^{-4}$, as well as $\rho_{amb.}=1.18$ kg/m3 and $\rho_{vac}=0.118$ kg/m3. Inset: Enlarged section. B)$\theta$ vs. $t_f-t$ on a glass plate for a 445 g steel disk and a 740 g steel annulus. A power-law fit is shown in red as a visual aid. Frequency ($\Omega$) data are shown in Fig. S4.
• Figure S1: A)$\theta$ vs. $t_f-t+0.01$ for filleted disks of varying mass on glass. The data in this graph is translated by $10$ ms to show the motion that occurs after $t_f$. The noise floor for $\theta$ is visible as the leftmost data $\sim 10^{-4}$ rad. Inset: $\theta$ vs. $t_f-t+0.01$ for the steel ring. B) Spin frequency $\Omega$ vs. tilt angle $\theta$ for the data shown in A.
• Figure S2: Top:$\theta$ vs. $t_f-t$ for filleted disks of varying mass and constant diameter on glass, aluminum, and steel sheets. Three repetitions per disk. Bottom: Precession frequency $\Omega$ vs. $\sin\theta$ for the data shown above.