Abstract
It is true that there are a number of possible dissipative mechanisms for the rolling disk in addition to viscous dissipation in the surrounding air: vibration of the supporting surface, rolling friction due to plastic deformation at the point of rolling contact, and, as suggested by van den Engh et al. , dissipation due to slipping rather than rolling. The 'adiabatic' equation that I used, relating the precessional angular velocity Ω to the angle α, is valid only under the rolling condition, and experiments indicate that this condition is indeed satisfied for the 'toy' Euler's disk rolling on a flat, smooth horizontal glass plate placed on a firm table (V. A. Vladimirov, personal communication). I believe therefore that slipping does not occur in this case.
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Moffatt replies
The problem really is to identify the dominant dissipative mechanism, for a given disk and a given surface, and then to evaluate the associated rate of dissipation of energy as a function of the angle α (which is proportional to the energy). If this rate of dissipation of energy turns out to be proportional to a power of α, where the exponent of this power, λ say, is less than one, then, under the adiabatic approximation, a finite-time singularity (for which Ω becomes infinite) will occur.
The air-viscosity mechanism I described yields λ = −2 (note that air viscosity is relatively insensitive to pressure, so that partial evacuation of the vessel in which the disk experiment is conducted should have only a small effect). An improved theory that takes account of oscillatory Stokes layers on the disk and supporting surface (L. Bildsten, personal communication) yields λ=−5/4. If 'rolling' friction is assumed to dissipate energy at a rate proportional to Ω, then λ=−1/2. Careful experiments under a variety of conditions should distinguish between these various possibilities.
I chose to focus on viscous dissipation because that is the only mechanism for which a fundamental (rather than empirical) description is available, namely that based on the Navier–Stokes equations of fluid dynamics. The fact that the air-viscosity mechanism exhibits the strongest singularity as α tends to zero suggests that this mechanism will always dominate when α is sufficiently small. For larger α and smaller disks (such as the 2.5-guilder coin), rolling friction is an equally plausible candidate (A. Ruina, personal communication), but determination of the associated rate of dissipation of energy (in terms of the physical properties of the disk and the surface) involves solution of the equations of (possibly plastic) deformation in both solids at the moving point of rolling contact, a difficult problem, which, so far as I am aware, still awaits definitive analysis.
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Moffatt, H. Numismatic gyrations. Nature 408, 540 (2000). https://doi.org/10.1038/35046211
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DOI: https://doi.org/10.1038/35046211
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