Euler's Disk


Euler's Disk, invented between 1987 and 1990 by Joseph Bendik, is a trademark for a scientific educational toy. It is used to illustrate and study the dynamic system of a spinning and rolling disk on a flat or curved surface, and it has been the subject of a number of scientific papers.

Discovery

Joseph Bendik first noted the interesting motion of the spinning disk while working at Hughes Aircraft after spinning a heavy polishing chuck on his desk at lunch one day. The spinning effect was so dramatic that he immediately called his friend and co-worker Richard Henry Wyles to take a look. He also called his friend Larry Shaw on the phone and had him listen to the sound of the spinning disk. For the next several years, Joe, Rich, and Larry worked to optimize the motion of the disk and develop a commercial version of the toy. All three share work on the patent filed for the toy in 1997.
The apparatus is known as a dramatic visualization of energy exchanges in three different, tightly-coupled processes. As the disk gradually decreases its azimuthal rotation, there is also a decrease in amplitude and increase in the frequency of the disk's axial precession.
The evolution of the disk's axial precession is easily visualized in a slow motion video by looking at the side of the disk following a single point marked on the disk. The evolution of the rotation of the disk is easily visualized in slow motion by looking at the top of the disk following an arrow drawn on the disk representing its radius.
As the disk releases the initial energy given by the user and approaches a halt, the disk seems to defy gravity through these dynamic exchanges of energy. Bendik named the toy after Leonhard Euler, who studied similar physics in the 18th century.

Components and operation

The commercially available toy consists of a heavy, thick chrome-plated steel disk and a rigid, slightly concave, mirrored base. Included holographic magnetic stickers can be attached to the disk, to enhance the visual effect of wobbling. These attachments are strictly decorative and may reduce the capacity to see and understand what processes are truly at work.
The disk, when spun on a flat surface, exhibits a spinning/rolling motion, slowly progressing through different rates and types of motion before coming to rest. Most notably, the precession rate of the disk's axis of symmetry accelerates as the disk spins down. The rigid mirror is used to provide a suitable low-friction surface, with a slight concavity which keeps the spinning disk from "wandering" off a support surface.
An ordinary coin spun on a table, as with any disk spun on a relatively flat surface, exhibits essentially the same type of motion, but does not rotate for anywhere near as long as an Euler's Disk. Commercially available Euler's Disks provide a more effective demonstration of the phenomenon than more commonly found items, having an optimized aspect ratio and a precision polished, slightly rounded edge to maximize the spinning/rolling time.

Physics

A spinning/rolling disk ultimately comes to rest quite abruptly, the final stage of motion being accompanied by a whirring sound of rapidly increasing frequency. As the disk rolls, the point of rolling contact describes a circle that oscillates with a constant angular velocity. If the motion is non-dissipative, is constant, and the motion persists forever; this is contrary to observation, since is not constant in real life situations. In fact, the precession rate of the axis of symmetry approaches a finite-time singularity modeled by a power law with exponent approximately −1/3.
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 depends systematically on coefficient of friction. In the limit of small angle, air drag is the dominant factor, but prior to this end stage, rolling friction is the dominant effect.

Steady motion with the disk center at rest

The behavior of a spinning disk whose center is at rest can be described as follows. Let the line from the center of the disk to the point of contact with the plane be called axis. Since the center of the disk and the point of contact are instantaneously at rest axis is the instantaneous axis of
rotation. The angular momentum is which holds for any thin, circularly symmetric disk with mass ; for a disk with mass concentrated at the rim, for a uniform disk, is the radius of the disk, and is the angular velocity along.
The contact force is where is the gravitational acceleration and is the vertical axis pointing upwards. The torque about the center of mass is which we can rewrite as where. We can conclude that both the angular momentum, and the disk are precessing about the vertical axis at rate At the same time is the angular velocity of the point of contact with the plane. Let's define axis to lie along the symmetry axis of the disk and poiting downwards. Then it holds that, where is the inclination angle of the disc with respect to the horizontal plane. The angular velocity can be thought of as composed of two parts, where is the angular velocity of the disk along its symmetry axis. From the geometry we easily conclude that:
Plugging into equation we finally get
As adiabatically approaches zero, the angular velocity of the point of contact becomes very large, and one hears a high-frequency sound associated with the spinning disk. However, the rotation of the figure on the face of the coin, whose angular velocity is approaches zero. The total angular velocity also vanishes as well as the total energy
as approaches zero. Here we have used the equation.
As approaches zero the disk finally loses contact with the table and the disk then quickly settles on to the horizontal surface. One hears sound at a frequency, which becomes dramatically higher until the sound abruptly ceases.

History of research

Moffatt

In the early 2000s, research was sparked by an article in the April 20, 2000 edition of Nature, where Keith Moffatt showed that viscous dissipation in the thin layer of air between the disk and the table would be sufficient to account for the observed abruptness of the settling process. He also showed that the motion concluded in a finite-time singularity. His first theoretical hypothesis was contradicted by subsequent research, which showed that rolling friction is actually the dominant factor.
Moffatt showed that, as time approaches a particular time , the viscous dissipation approaches infinity. The singularity that this implies is not realized in practice, because the magnitude of the vertical acceleration cannot exceed the acceleration due to gravity. Moffatt goes on to show that the theory breaks down at a time before the final settling time, given by:
where is the radius of the disk, is the acceleration due to Earth's gravity, the dynamic viscosity of air, and the mass of the disk. For the commercially available Euler's Disk toy, is about seconds, at which time the angle between the coin and the surface,, is approximately 0.005 radians and the rolling angular velocity,, is about 500 Hz.
Using the above notation, the total spinning/rolling time is:
where is the initial inclination of the disk, measured in radians. Moffatt also showed that, if, the finite-time singularity in is given by

Experimental results

Moffatt's theoretical work inspired several other workers to experimentally investigate the dissipative mechanism of a spinning/rolling disk, with results that partially contradicted his explanation. These experiments used spinning objects and surfaces of various geometries, with varying coefficients of friction, both in air and in a vacuum, and used instrumentation such as high speed photography to quantify the phenomenon.
In the 30 November 2000 issue of Nature, physicists Van den Engh, Nelson and Roach discuss experiments in which disks were spun in a vacuum. Van den Engh used a rijksdaalder, a Dutch coin, whose magnetic properties allowed it to be spun at a precisely determined rate. They found that slippage between the disk and the surface could account for observations, and the presence or absence of air only slightly affected the disk's behavior. They pointed out that Moffatt's theoretical analysis would predict a very long spin 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 .
Later work at the University of Guelph by Petrie, Hunt and Gray showed that carrying out the experiments in a vacuum did not significantly affect the energy dissipation rate. Petrie et al. also showed that the rates were largely unaffected by replacing the disk with a ring shape, and that the no-slip condition was satisfied for angles greater than 10°. Another work by Caps, Dorbolo, Ponte, Croisier, and Vandewalle has concluded that the air is a minor source of energy dissipation. The major energy dissipation process is the rolling and slipping of the disk on the supporting surface. It was experimentally shown that the inclination angle, the precession rate, and the angular velocity follow the power law behavior.
On several occasions during the 2007–2008 Writers Guild of America strike, talk show host 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 onto the show to experiment with the problem. Spinning the ring in a vacuum had no identifiable effect, while a Teflon spinning support surface gave a record time of 51 seconds, corroborating the claim that rolling friction is the primary mechanism for kinetic energy dissipation.
Various kinds of rolling friction as primary mechanism for energy dissipation have been studied by Leine who confirmed experimentally that the frictional resistance of the movement of the contact point over the rim of the disk is most likely the primary dissipation mechanism on a time-scale of seconds.

In popular culture

Euler's Disks appear in the 2006 film Snow Cake and in the TV show The Big Bang Theory, season 10, episode 16, which aired February 16, 2017.
The toy was featured by Michael Stevens in an episode of Michael's Toys on his educational YouTube channel D!NG, which is a spin-off from Vsauce, his original channel. In the ~25 minute episode Stevens tries to go into details and explain how it works.
The sound team for the 2001 film Pearl Harbor used a spinning Euler's Disk as a sound effect for torpedoes. A short clip of the sound team playing with Euler's Disk was played during the Academy Awards presentations.