Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: A heavy disk rolls on a horizontal table.
Figure 2: Euler's disk is a chrome-plated steel disk with one edge machined to a smooth radius.

References

  1. 1

    Pars, L. A. Treatise on Analytical Dynamics (Heinemann, London, 1965).

    MATH  Google Scholar 

  2. 2

    Euler, L. Theoria Motus Corporum Solidorum Seu Rigidorum (Greifswald, 1765).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to H. K. Moffatt.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Moffatt, H. Euler's disk and its finite-time singularity. Nature 404, 833–834 (2000). https://doi.org/10.1038/35009017

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing