Editing Euler's Disk

{{Short description|Scientific educational toy}}
[[File:Euler's disk.png|thumb|right|Computer rendering of Euler's Disk on a slightly concave base]]


'''Euler's Disk''', invented between 1987 and 1990 by Joseph Bendik,<ref>{{cite web
| url = http://discovermagazine.com/1996/dec/playthingsofscie946/
| title = Playthings of Science
| author = Fred Guter
| date = December 1, 1996
| publisher = [[Discover (magazine)|Discover]]
| quote = As Bendik played with the disk, he thought, Perhaps it would make a good toy. 
| access-date = 2018-11-23}}</ref> is a trademarked scientific [[educational toy]].<ref>{{cite web
| url = http://tmsearch.uspto.gov/bin/showfield?f=doc&state=4801:gk6zro.2.1
| title = Trademarks > Trademark Electronic Search System (TESS) > Euler's Disk
| date = September 21, 2010
| publisher = United States Patent and Trademark Office
| quote = Live/Dead Indicator: LIVE
| access-date = 2018-11-23
}}{{Dead link|date=December 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> It is used to illustrate and study the [[dynamic system]] of a spinning and rolling disk on a flat or curved surface. It has been the subject of several scientific papers.<ref>{{cite web|title=Publications|url=http://eulersdisk.com/pubs.html|publisher=eulersdisk.com}}</ref> Bendik named the toy after mathematician [[Leonhard Euler]].<ref name=Moffatt-2000/>

==Discovery==
{{unreferenced section|date=May 2019}}
Joseph Bendik first noted the interesting motion of the spinning disk while working at [[Hughes 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 axial [[precession]].<ref>{{Cite web |last=Hybiske |first=Kevin |title=Euler's Disk Publications |url=http://eulersdisk.com/pubs.html |access-date=2023-05-15 |website=eulersdisk.com}}</ref>

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 disk [[persistence_of_vision|appears]] to levitate before halting.

The commercial toy consists of a heavy, thick chrome-plated steel disk and a rigid, slightly [[concave mirror|concave]], mirrored base. Included [[hologram|holographic]] magnetic 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, the [[precession]] rate of the disk's [[axis of symmetry]] increases 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 a [[coin]] spun 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 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 <math>\omega</math>.   If the motion is non-dissipative (frictionless),  <math>\omega</math> is constant, and the motion persists forever; this is contrary to observation, since <math>\omega</math> 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 &minus;1/3 (depending on specific conditions).

There are two conspicuous dissipative effects: [[rolling friction]] when the disk 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<ref>{{Cite journal | last1 = Easwar | first1 = K. | last2 = Rouyer | first2 = F. | last3 = Menon | first3 = N. | title = Speeding to a stop: The finite-time singularity of a spinning disk | doi = 10.1103/PhysRevE.66.045102 | journal = Physical Review E | volume = 66 | issue = 4 | pages = 045102 | year = 2002 | pmid =  12443243|bibcode = 2002PhRvE..66d5102E }}</ref>—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 disk stops spinning), air drag (specifically, [[viscosity|viscous dissipation]]) 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.<ref>{{cite arXiv |eprint=physics/0008227 |last1= McDonald |first1=Alexander J. |title=The Rolling Motion of a Disk on a Horizontal Plane |last2=McDonald |first2=Kirk T. |year=2000 }}</ref> Let the line from the center of the disk to the point of contact with the plane be called axis <math display="inline">\widehat{\mathbf{3}}</math>. Since the center of the disk and the point of contact are instantaneously at rest (assuming there is no slipping) axis <math>\widehat{\mathbf{3}}</math> is the instantaneous axis of
rotation. The angular momentum is <math display="inline">\mathbf{L} =kMa^2\omega\widehat{\mathbf{3}}</math> which holds for any thin, circularly symmetric disk with mass <math>M</math>; <math>k=1/2</math> for a disk with mass concentrated at the rim, <math>k=1/4</math> for a uniform disk (like Euler disk), <math>a</math> is the radius of the disk, and <math>\omega</math> is the angular velocity along <math>\widehat{\mathbf{3}}</math>.

The contact force <math display="inline">\mathbf{F}</math> is <math display="inline">M g \widehat{\mathbf{z}}</math> where <math>g</math> is the gravitational acceleration and <math>\widehat{\mathbf{z}}</math> is the vertical axis pointing upwards. The torque about the center of mass is <math display="inline">\mathbf{N}=a \widehat{\mathbf{3}} \times Mg\widehat{\mathbf{z}}=\frac{d\mathbf{L}}{dt}</math> which we can rewrite as <math display="inline">\frac{d\mathbf{L}}{dt}= \boldsymbol{\Omega}\times\mathbf{L}</math> where <math display="inline">\boldsymbol{\Omega} = - \frac{g}{ak\omega} \widehat{\mathbf{z}}</math>. We can conclude that both the angular momentum <math display="inline">\mathbf{L}</math>, and the disk are precessing about the vertical axis <math display="inline">\widehat{\mathbf{z}}</math> at rate {{NumBlk||<math display="block">\Omega=\frac{g}{ak\omega}</math>|{{EquationRef|1}}}} At the same time <math>\Omega</math> is the angular velocity of the point of contact with the plane. With axis <math display="inline">\widehat{\mathbf{1}}</math> defined to lie along the symmetry axis of the disk and pointing downwards, it holds that <math display="inline">\widehat{\mathbf{z}} = - \cos \alpha \widehat{\mathbf{1}} - \sin \alpha \widehat{\mathbf{3}}</math>, where <math>\alpha</math> 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 <math display="inline">\omega\widehat{\mathbf{3}} = \Omega \widehat{\mathbf{z}} + \omega_\text{rel} \widehat{\mathbf{1}} </math>, where <math display="inline">\omega_\text{rel}</math> is the angular velocity of the disk along its symmetry axis. From the geometry it is easily concluded that:

<math display="block">\begin{align}
  \omega &= -\Omega \sin \alpha, \\
  \omega_\text{rel} &= \Omega \cos \alpha\\
\end{align}</math>

Plugging <math>\omega = -\Omega \sin \alpha</math> into equation ({{EquationNote|1}}) finally gets 
{{NumBlk||<math display="block">\Omega^2=\frac{g}{ak\sin \alpha}</math>|{{EquationRef|2}}}}

As <math>\alpha</math> adiabatically approaches zero, the angular velocity of the point of contact <math>\Omega</math> 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 is <math display="inline">\Omega - \omega_\text{rel} = \Omega(1 - \cos \alpha),</math> approaches zero. The total angular velocity <math display="inline">\omega=-\sqrt{\frac{g \sin \alpha}{a k}}</math> also vanishes as well as the total energy

<math display="block">E=Mga\sin \alpha + \tfrac{1}{2} kMa^2 \omega^2 = Mga\sin \alpha + \tfrac{1}{2} M k a^2 \frac{g \sin \alpha}{a k} = \tfrac{3}{2} M g a \sin \alpha </math>

as <math>\alpha</math> approaches zero, using the equation ({{EquationNote|2}}). 

As <math>\alpha</math> 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 <math display="inline">\frac{\Omega}{2\pi}</math>, which becomes dramatically higher, <math display="inline">\frac{1}{2\pi} \sqrt{\frac{g}{ak}} \sqrt{\frac{1}{\sin \alpha}}</math>, as the figure rotation rate slows, <math display="inline">2 \sqrt{\frac{g}{ak}} \frac{(\sin \frac{\alpha}{2})^2}{\sqrt{\sin \alpha}}</math>, until the sound abruptly ceases.

=== Levitation illusion ===

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.  See [[persistence 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.

==== US Quarter example ====

A clean [[US 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 and [[heads or tails]] is too foreshortened to show rotation.

== History of research ==
=== Moffatt ===
In the early 2000s, research was sparked by an article in the April 20, 2000 edition of ''[[Nature (journal)|Nature]]'',<ref name=Moffatt-2000>{{Cite journal |last=Moffatt |first=H. K. |date=20 April 2000 |title=Euler's disk and its finite-time singularity |url=https://www.nature.com/articles/35009017 |journal=Nature |language=en |volume=404 |issue=6780 |pages=833–834 |bibcode=2000Natur.404..833M |doi=10.1038/35009017 |issn=1476-4687 |pmid=10786779 |s2cid=197644581|url-access=subscription }}</ref> where [[Keith Moffatt]] showed that [[viscosity|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 <math>t</math> approaches a particular time <math>t_0</math> (which is mathematically a [[integral|constant of integration]]), the viscous dissipation approaches [[infinity]]. The [[Mathematical singularity|singularity]] that this implies is not realized in practice, because the magnitude of the vertical acceleration cannot exceed the acceleration due to [[g-force|gravity]] (the disk loses contact with its support surface).  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[\left(\frac{2a}{9g}\right)^3 \frac{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 Earth's gravity, <math>\mu</math> the [[dynamic viscosity]] of [[air]], and <math>M</math> the mass of the disk.  For the commercially available Euler's Disk toy (see link in "External links" below), <math>\tau</math> is about <math>10^{-2}</math> seconds, at which time the angle between the coin and the surface, <math>\alpha</math>, is approximately 0.005 radians and the rolling angular velocity, <math>\Omega</math>, is about 500 Hz.

Using the above notation, the total spinning/rolling time is:

:<math>t_0 = \frac{\alpha_0^3 M}{2\pi\mu a}</math>

where <math>\alpha_0</math> is the initial inclination of the disk, measured in [[radian]]s.  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>

===Experimental results===
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 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.<ref>{{cite journal | last=Van den Engh | first=Ger |author2=Nelson, Peter |author3=Roach, Jared  | title=Analytical dynamics: Numismatic gyrations | journal=Nature | date=30 November 2000 | volume=408 | issue=6812 | page=540 | doi=10.1038/35046209| pmid=11117733 |bibcode = 2000Natur.408..540V | s2cid=4407382 | doi-access=free }}</ref>  Van den Engh used a [[rijksdaalder]], a [[Netherlands|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 <math>\alpha</math> (i.e., just before the disk settles).<ref>{{cite journal | last=Moffatt | first=H. K. | title=Reply: Numismatic gyrations | journal=Nature | date=30 November 2000 | volume=408 | issue=6812 | page=540 | doi=10.1038/35046211|bibcode = 2000Natur.408..540M | s2cid=205011563 | doi-access=free }}</ref> 

Later work at the [[University of Guelph]] by Petrie, Hunt and Gray<ref>{{cite journal|last=Petrie|first=D.|author2=Hunt, J. L. |author3=Gray, C. G. |s2cid=28497371|title=Does the Euler Disk slip during its motion?|journal=American Journal of Physics|date=2002|volume=70|issue=10|pages=1025–1028|doi=10.1119/1.1501117|bibcode = 2002AmJPh..70.1025P }}</ref> showed that carrying out the experiments in a vacuum (pressure 0.1 [[pascal (unit)|pascal]]) 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 [[torus|ring]] shape, and that the no-slip condition was satisfied for angles greater than 10°. Another work by Caps, Dorbolo, Ponte, Croisier, and Vandewalle<ref>{{cite journal | last1=Caps |first1=H. |last2=Dorbolo |first2=S. |last3=Ponte |first3=S.|last4=Croisier| first4=H |first5=N. |last5=Vandewalle | title=Rolling and slipping motion of Euler's disk | journal=Phys. Rev. E |  date=May 2004 | volume=69 | issue=5 | pages=056610| doi=10.1103/PhysRevE.69.056610 |pmid=15244966 |arxiv=cond-mat/0401278 |bibcode=2004PhRvE..69e6610C |hdl=2268/18081 |s2cid=118949289 |url=http://eulersdisk.com/PRE56610.pdf}}</ref> 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. 

Leine studied multiple types of dissipation or friction for the Euler disk motinn. He has shown that experimental observations are consistent with two different kinds of ''contour friction.'' Contour friction acts against the motion of the contact point along the rim of the disk. One form dominates during early parts of the rolling motion but the other "viscous" contour friction operates in the last second or two.<ref>{{cite journal|last=Leine|first=R.I.|title=Experimental and theoretical investigation of the energy dissipation of a rolling disk during its final stage of motion|journal=Archive of Applied Mechanics|date=2009|volume=79|issue=11|pages=1063–1082|doi= 10.1007/s00419-008-0278-6|bibcode=2009AAM....79.1063L|hdl=20.500.11850/12334|s2cid=48358816|url=http://doc.rero.ch/record/321780/files/419_2008_Article_278.pdf|hdl-access=free}}</ref><ref>{{Cite journal |last1=Borisov |first1=Alexey V. |last2=Mamaev |first2=Ivan S. |last3=Karavaev |first3=Yury L. |date=March 2015 |title=On the loss of contact of the Euler disk |url=http://link.springer.com/10.1007/s11071-014-1811-5 |journal=Nonlinear Dynamics |language=en |volume=79 |issue=4 |pages=2287–2294 |doi=10.1007/s11071-014-1811-5 |bibcode=2015NonDy..79.2287B |issn=0924-090X|url-access=subscription }}</ref>

== 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 sound team for the 2001 film ''[[Pearl Harbor (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.<ref>{{Citation|title=Black Hawk Down and Pearl Harbor Win Sound and Sound Effects Editing: 2002 Oscars| date=22 January 2014 |url=https://www.youtube.com/watch?v=962hWbdSdGo|language=en|access-date=2022-01-09}}</ref>

The principles of the Euler Disk were used with specially made rings on a table as a futuristic recording medium in the 1960 movie ''[[The Time Machine (1960 film)|The Time Machine]]''.

== See also ==

* [[List of topics named after Leonhard Euler]]
* [[Tippe top]] – another spinning physics toy that exhibits surprising behavior

==References==
{{reflist}}

== External links ==
* [http://eulersdisk.com Eulersdisk.com]
* [https://web.archive.org/web/20040617014034/http://physicsweb.org/article/news/4/4/12 The physics of a spinning coin (April 20, 2000) PhysicsWeb]
* [http://www.inm.uni-stuttgart.de/mitarbeiter/leine/papers/journal_publications/Leine_-_Experimental_and_theoretical_investigation_of_the_energy_dissipation_of_a_rolling_disk_during_its_final_stage_of_motion.pdf Experimental and theoretical investigation of the energy dissipation of a rolling disk during its final stage of motion (December 12, 2008) Arch Appl Mech]
* [https://web.archive.org/web/20031102033740/http://tam.cornell.edu/~ruina/hplab/Rolling%20and%20sliding/Andy_on_Moffatt_Disk.pdf Comment on Moffat’s Disk (March 31, 2002)]
* {{cite web|title=Euler's Disk|url=http://www.real-world-physics-problems.com/eulers-disk.html|website=Real World Physics Problems|publisher=real-world-physics-problems.com|access-date=2014-07-11}} Detailed mathematical physics analysis of disk motion
* [https://www.youtube.com/watch?v=L3o0R2hStiY A YouTube video of an Euler's Disk in action]

[[Category:Dynamical systems]]
[[Category:Educational toys]]
[[Category:Spinning tops]]
[[Category:Novelty items]]