Euler's disk

From Academic Kids

Euler's disk, named after Leonhard Euler, is a circular disk that spins, without slipping, on a surface. The canonical example is a coin spinning on a table. It is universally observed that a spinning Euler's disk ultimately comes to rest; and it does so quite abruptly, the final stage of motion being accompanied by a whirring sound of rapidly increasing frequency. As the disk rolls, the point P of rolling contact describes a circle that oscillates with a constant angular velocity <math>\omega<math>. If the motion is non-dissipative, <math>\omega<math> is constant and the motion persists forever, contrary to observation.

In the 20 April 2000 edition of Nature, Keith Moffatt shows 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. He also showed that the motion concluded in a finite-time singularity.

Moffatt shows that, as time <math>t<math> approaches a particular time <math>t_0<math> (which is mathematically a constant of integration), the viscous dissipation approaches infinity. The singularity that this implies is not realized in practice because the vertical acceleration cannot exceed the gee in magnitude. Moffatt goes on to show that the theory breaks down at a time <math>\tau<math> before the final settling time <math>t_0<math>, given by

<math>

\tau\simeq\left(2a/9g\right)^{3/5} \left(2\pi\mu a/M\right)^{1/5}<math>

where <math>a<math> is the radius of the disk, <math>g<math> is the acceleration due to gravity, <math>\mu<math> the dynamic viscosity of air, and <math>M<math> the mass of the disk. For the commercial toy (see link below), <math>\tau<math> is about <math>10^{-2}<math> seconds, at which <math>\alpha\simeq 0.005<math> and the rolling angular velocity <math>\Omega\simeq 500\rm Hz<math>.

Using the above notation, the total spinning time is

<math>

t_0=\left(\frac{\alpha_0^3}{2\pi}\right)\frac{M}{\mu a}<math> where <math>\alpha_0<math> is the initial inclination of the disk. Moffatt also showed that, if <math>t_0-t>\tau<math>, the finite-time singularity in <math>\Omega<math> is given by <math>\Omega\sim(t_0-t)^{-1/6}.<math>

Rebuttals

Moffatt's work inspired several other workers to investigate the dissipative mechanism of Euler's disk. In the 30 November 2000 issue of Nature, physicists Van den Engh and coworkers discuss experiments in which coins were spun in a vacuum. They found that slippage between the coin and the surface could account for observations, and the presence or absence of air affected the coin's behaviour only slightly. They pointed out that Moffatt's analysis would predict a very long wobbling time for a coin in a vacuum.

Moffatt responded with a generalized theory that should allow experimental determination of which dissipation mechanism is dominant, and pointed out that the dominant dissipation mechanism would always be viscous dissipation in the limit of small <math>\alpha<math>.

Van den Engh used a Dutch 2.5 guilder coin, whose magnetic properties allowed it to be spun at a precisely determined rate.

Later work at the University of Guelph by D. Petrie and coworkers (American Journal of Physics, 70(10), Oct 2002, p. 1025) showed that carrying out the experiments in a vacuum (pressure 0.1 pascal) did not affect the damping rate. Petrie also showed that the rates were largely unaffected by replacing the disk with a ring, and that the no-slip condition was satisfied for angles greater than 10.

These experiments indicated that rolling friction is mainly responsible for the dissipation, especially in the early stages of motion.

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