# Euler's Disk

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Euler's Disk is a scientific educational toy, used to illustrate and study the dynamic system of a spinning disk on a flat surface (such as a spinning coin), and has been the subject of a number of scientific papers.[1] This phenomenon has been studied since Leonhard Euler in the 18th century, hence the name.

## Components and use

The toy consists of a heavy, thick chrome plated steel disk, a slightly concave, mirrored base, and holographic magnetic stickers which can be placed on the disk. The disk, when spun on the mirror, exhibits a spinning/rolling motion, slowly moving through different rates and types of motion before coming to rest – for example, the precession rate of the axis of symmetry accelerates as the disk spins down. Euler’s Disk has an optimized aspect ratio and a precision polished, slightly rounded edge to maximize the spinning/rolling time. A coin spun on a table, as with any disk spun on a relatively flat surface, exhibits essentially the same type of motion, but is normally more constrained in the length of time before stopping. The Disk provides a more effective demonstration of the phenomenon than more commonly found items.

## Physics

A spinning/rolling 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 $\omega$. If the motion is non-dissipative, $\omega$ is constant and the motion persists forever, contrary to observation (since $\omega$ is not constant in real life situations). In fact, precession rate of the axis of symmetry approaches a finite-time singularity modeled by a power law with exponent approximately −1/3 (depending on specific conditions).[2]

There are two conspicuous dissipative effects: rolling friction when the coin slips along the surface, and air drag from the resistance of air. Experiments show that rolling friction is mainly responsible for the dissipation and behavior – experiments in a vacuum show that the absence of air affects behavior only slightly, while the behavior (precession rate) depends systematically on coefficient of friction. In the limit of small angle (i.e., immediately before the coin stops spinning), air drag (specifically, viscous dissipation) is the dominant factor, but prior to this rolling friction is the dominant effect.

## History of research

### Moffatt

Recent research was sparked by an article in the April 20, 2000 edition of Nature,[3] where 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. This theoretical hypothesis has been contradicted by subsequent research, which shows that rolling friction is the dominant factor.

Moffatt shows that, as time $t$ approaches a particular time $t_0$ (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 acceleration due to gravity in magnitude. Moffatt goes on to show that the theory breaks down at a time $\tau$ before the final settling time $t_0$, given by

$\tau \simeq \left[\left(\frac{2a}{9g}\right)^3 \frac{2\pi\mu a}{M}\right]^\frac{1}{5}$

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

Using the above notation, the total spinning time is

$t_0 = \frac{\alpha_0^3 M}{2\pi\mu a}$

where $\alpha_0$ is the initial inclination of the disk. Moffatt also showed that, if $t_0-t>\tau$, the finite-time singularity in $\Omega$ is given by

$\Omega\sim(t_0-t)^{-\frac{1}{6}}$

### Rebuttals

Moffatt's work inspired several other workers to investigate the dissipative mechanism of a spinning/rolling disk, whose results contradicted his explanation. These used spinning objects and surfaces of various geometries (disks and rings), with varying coefficients of friction, both in air and in a vacuum, and used tools such as high speed photography to quantify the phenomenon.

In the 30 November 2000 issue of Nature, physicists Van den Engh and coworkers discuss experiments in which disks were spun in a vacuum. They found that slippage between the disk and the surface could account for observations, and the presence or absence of air affected the disk's behaviour only slightly. They pointed out that Moffatt's analysis would predict a very long wobbling time for a disk in a vacuum, which was not observed.

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 $\alpha$ (angle between the coin and the surface, i.e., just before the coin settles).

Van den Engh used a rijksdaalder, a Dutch 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[4] 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°.

On several occasions during the 2007–2008 Writers Guild of America strike Conan O'Brien would spin his wedding ring on his desk, trying to spin the ring for as long as possible. The quest to achieve longer and longer spin times led him to invite MIT professor Peter Fisher on to the show to experiment with the problem. Ring spinning in a vacuum had no identifiable effect while a Teflon spinning surface gave a record time of 51 seconds, corroborating the claim that rolling friction is the primary mechanism for kinetic energy dissipation.