Euler's Disk

Euler's Disk, invented between 1987 and 1990 by Joseph Bendik,is a trademarked scientificeducational toy.It is used to illustrate and study thedynamic systemof a spinning and rolling disk on a flat or curved surface. It has been the subject of several scientific papers.Bendik named the toy after mathematicianLeonhard Euler.
Joseph Bendik first noted the interesting motion of the spinning disk while working atHughes Aircraft(Carlsbad Research Center) after spinning a heavy polishing chuck on his desk at lunch one day.
The apparatus is 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 axialprecession.
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, its rotation about the vertical axis slows, while its contact point oscillation increases.  Lit from above, its contact point and nearby lower edge in shadow, the diskappearsto levitate before halting.
The commercial toy consists of a heavy, thick chrome-plated steel disk and a rigid, slightlyconcave, mirrored base. Includedholographicmagnetic stickers can be attached to the disk, to enhance the visual effect of wobbling. These attachments may make it harder to see and understand the processes at work, however.
When spun on a flat surface, the disk exhibits a spinning/rolling motion, slowly progressing through varying rates and types of motion before coming to rest. Most notably, theprecessionrate of the disk'saxis of symmetryincreases as the disk spins down. The mirror base provides a low-friction surface; its slight concavity keeps the disk from "wandering" off the surface.
Any disk, spun on a reasonably flat surface (such as acoinspun on a table), will exhibit essentially the same type of motion as an Euler Disk, but for a much shorter time. Commercial disks provide a more effective demonstration of the phenomenon, having an optimizedaspect ratioand a precision polished, slightly rounded edge to maximize the spinning/rolling time.
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 velocityω{\displaystyle \omega }.   If the motion is non-dissipative (frictionless),ω{\displaystyle \omega }is constant, and the motion persists forever; this is contrary to observation, sinceω{\displaystyle \omega }is not constant in real life situations. In fact, the precession rate of the axis of symmetry approaches afinite-time singularitymodeled by apower lawwith exponent approximately −1/3 (depending on specific conditions).
There are two conspicuous dissipative effects:rolling frictionwhen the disk slips along the surface, andair dragfrom the resistance of air. Experiments show thatrolling frictionis mainly responsible for the dissipation and behavior—experiments in avacuumshow that the absence of air affects behavior only slightly, while the behavior (precession rate) depends systematically oncoefficient of friction. In the limit of small angle (i.e. immediately before the disk stops spinning), air drag (specifically,viscous dissipation) is the dominant factor, but prior to this end stage, rolling friction is the dominant effect.
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 axis3^{\textstyle {\widehat {\mathbf {3} }}}. Since the center of the disk and the point of contact are instantaneously at rest (assuming there is no slipping) axis3^{\displaystyle {\widehat {\mathbf {3} }}}is the instantaneous axis of
rotation. The angular momentum isL=kMa2ω3^{\textstyle \mathbf {L} =kMa^{2}\omega {\widehat {\mathbf {3} }}}which holds for any thin, circularly symmetric disk with massM{\displaystyle M};k=1/2{\displaystyle k=1/2}for a disk with mass concentrated at the rim,k=1/4{\displaystyle k=1/4}for a uniform disk (like Euler disk),a{\displaystyle a}is the radius of the disk, andω{\displaystyle \omega }is the angular velocity along3^{\displaystyle {\widehat {\mathbf {3} }}}.
The contact forceF{\textstyle \mathbf {F} }isMgz^{\textstyle Mg{\widehat {\mathbf {z} }}}whereg{\displaystyle g}is the gravitational acceleration andz^{\displaystyle {\widehat {\mathbf {z} }}}is the vertical axis pointing upwards. The torque about the center of mass isN=a3^×Mgz^=dLdt{\textstyle \mathbf {N} =a{\widehat {\mathbf {3} }}\times Mg{\widehat {\mathbf {z} }}={\frac {d\mathbf {L} }{dt}}}which we can rewrite asdLdt=Ω×L{\textstyle {\frac {d\mathbf {L} }{dt}}={\boldsymbol {\Omega }}\times \mathbf {L} }whereΩ=−gakωz^{\textstyle {\boldsymbol {\Omega }}=-{\frac {g}{ak\omega }}{\widehat {\mathbf {z} }}}. We can conclude that both the angular momentumL{\textstyle \mathbf {L} }, and the disk are precessing about the vertical axisz^{\textstyle {\widehat {\mathbf {z} }}}at rate
At the same timeΩ{\displaystyle \Omega }is the angular velocity of the point of contact with the plane. With axis1^{\textstyle {\widehat {\mathbf {1} }}}defined to lie along the symmetry axis of the disk and pointing downwards, it holds thatz^=−cos⁡α1^−sin⁡α3^{\textstyle {\widehat {\mathbf {z} }}=-\cos \alpha {\widehat {\mathbf {1} }}-\sin \alpha {\widehat {\mathbf {3} }}}, whereα{\displaystyle \alpha }is the inclination angle of the disc with respect to the horizontal plane. The angular velocity can be thought of as composed of two partsω3^=Ωz^+ωrel1^{\textstyle \omega {\widehat {\mathbf {3} }}=\Omega {\widehat {\mathbf {z} }}+\omega _{\text{rel}}{\widehat {\mathbf {1} }}}, whereωrel{\textstyle \omega _{\text{rel}}}is the angular velocity of the disk along its symmetry axis. From the geometry it is easily concluded that:
ω=−Ωsin⁡α,ωrel=Ωcos⁡α{\displaystyle {\begin{aligned}\omega &=-\Omega \sin \alpha ,\\\omega _{\text{rel}}&=\Omega \cos \alpha \\\end{aligned}}}
Pluggingω=−Ωsin⁡α{\displaystyle \omega =-\Omega \sin \alpha }into equation (1) finally gets
Asα{\displaystyle \alpha }adiabatically approaches zero, the angular velocity of the point of contactΩ{\displaystyle \Omega }becomes very large, making a high-frequency sound associated with the spinning disk. However, the rotation of the figure on the face of the coin, whose angular velocity isΩ−ωrel=Ω(1−cos⁡α),{\textstyle \Omega -\omega _{\text{rel}}=\Omega (1-\cos \alpha ),}approaches zero. The total angular velocityω=−gsin⁡αak{\textstyle \omega =-{\sqrt {\frac {g\sin \alpha }{ak}}}}also vanishes as well as the total energy
E=Mgasin⁡α+12kMa2ω2=Mgasin⁡α+12Mka2gsin⁡αak=32Mgasin⁡α{\displaystyle E=Mga\sin \alpha +{\tfrac {1}{2}}kMa^{2}\omega ^{2}=Mga\sin \alpha +{\tfrac {1}{2}}Mka^{2}{\frac {g\sin \alpha }{ak}}={\tfrac {3}{2}}Mga\sin \alpha }
asα{\displaystyle \alpha }approaches zero, using the equation (2).
Asα{\displaystyle \alpha }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 frequencyΩ2π{\textstyle {\frac {\Omega }{2\pi }}}, which becomes dramatically higher,12πgak1sin⁡α{\textstyle {\frac {1}{2\pi }}{\sqrt {\frac {g}{ak}}}{\sqrt {\frac {1}{\sin \alpha }}}}, as the figure rotation rate slows,2gak(sin⁡α2)2sin⁡α{\textstyle 2{\sqrt {\frac {g}{ak}}}{\frac {(\sin {\frac {\alpha }{2}})^{2}}{\sqrt {\sin \alpha }}}}, until the sound abruptly ceases.
As a circularly symmetric disk settles, the separation between a fixed point
on the supporting surface and the moving disk above oscillates at increasing frequency, in sync with the rotation axis angle off vertical.
The levitation illusion results when the disk edge reflects light when tilted slightly up above the supporting surface, and in shadow when tilted slightly down in contact.  The shadow is not perceived, and the rapidly flashing reflections from the edge above supporting surface are perceived as steady elevation.  Seepersistence of vision.
The levitation illusion can be enhanced by optimizing the curve of the lower edge so the shadow line remains high as the disk settles.  A mirror can further enhance the effect by hiding the support surface and showing separation between
moving disk surface and mirror image.
Disk imperfections, seen in shadow, that could hamper the illusion, can be hidden in a skin pattern that blurs under motion.
A cleanUS Quarter(minted 1970–2022), rotating on a flat hand mirror, viewed from the side near the mirror surface, demonstrates the phenomenon for a few seconds.
Lit by a point source directly over the center of the soon to settle quarter,
side ridges are illuminated when the rotation axis is away from the viewer,
and in shadow when the rotation axis is toward the viewer.  Vibration blurs the ridges andheads or tailsis too foreshortened to show rotation.
In the early 2000s, research was sparked by an article in the April 20, 2000 edition ofNature,whereKeith Moffattshowed thatviscous dissipationin the thin layer ofairbetween 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 afinite-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 timet{\displaystyle t}approaches a particular timet0{\displaystyle t_{0}}(which is mathematically aconstant of integration), the viscous dissipation approachesinfinity. Thesingularitythat this implies is not realized in practice, because the magnitude of the vertical acceleration cannot exceed the acceleration due togravity(the disk loses contact with its support surface).  Moffatt goes on to show that the theory breaks down at a timeτ{\displaystyle \tau }before the final settling timet0{\displaystyle t_{0}}, given by
wherea{\displaystyle a}is the radius of the disk,g{\displaystyle g}is the acceleration due to Earth's gravity,μ{\displaystyle \mu }thedynamic viscosityofair, andM{\displaystyle M}the mass of the disk.  For the commercially available Euler's Disk toy (see link in "External links" below),τ{\displaystyle \tau }is about10−2{\displaystyle 10^{-2}}seconds, at which time the angle between the coin and the surface,α{\displaystyle \alpha }, is approximately 0.005 radians and the rolling angular velocity,Ω{\displaystyle \Omega }, is about 500 Hz.
Using the above notation, the total spinning/rolling time is:
whereα0{\displaystyle \alpha _{0}}is the initial inclination of the disk, measured inradians.  Moffatt also showed that, ift0−t>τ{\displaystyle t_{0}-t>\tau },  the finite-time singularity inΩ{\displaystyle \Omega }is given by
Moffatt's theoretical work inspired several other scientists 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 (disks and rings), with varying coefficients of friction, both in air and in a vacuum, and used instrumentation such ashigh speed photographyto quantify the phenomenon.
In the 30 November 2000 issue ofNature, physicists Van den Engh, Nelson and Roach discuss experiments in which disks were spun in a vacuum.Van den Engh used arijksdaalder, aDutchcoin, whosemagneticproperties 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 smallα{\displaystyle \alpha }(i.e., just before the disk settles).
Later work at theUniversity of Guelphby Petrie, Hunt and Grayshowed that carrying out the experiments in a vacuum (pressure 0.1pascal) did not significantly affect the energy dissipation rate.  Petrieet al.also showed that the rates were largely unaffected by replacing the disk with aringshape, and that the no-slip condition was satisfied for angles greater than 10°. Another work by Caps, Dorbolo, Ponte, Croisier, and Vandewallehas 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 the2007–2008 Writers Guild of America strike, talk show hostConan O'Brienwould spin hiswedding ringon his desk, trying to spin the ring for as long as possible.  The quest to achieve longer and longer spin times led him to inviteMITprofessor Peter Fisher onto the show to experiment with the problem.  Spinning the ring in a vacuum had no identifiable effect, while aTeflonspinning 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 Leinewho 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.
Euler's Disks appear in the 2006 filmSnow Cakeand in the TV showThe Big Bang Theory, season 10, episode 16, which aired February 16, 2017.
The sound team for the 2001 filmPearl Harborused 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.
The principles of the Euler Disk were used with specially made rings on a table as a futuristic recording medium in the 1960 movieThe Time Machine.