Euler's disk
Encyclopedia
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. This phenomenon has been studied since Leonard Euler in the 18th century, hence the name.
It is manufactured and distributed by the Toysmith Group of the Damert Company. It was invented by Joe Bendik, and was originally marketed by the Tangent Toy Company.
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.
. If the motion is non-dissipative, 
is constant and the motion persists forever, contrary to observation (since 
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).
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.
, 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
approaches a particular time 
(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
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 commercial toy (see link below), 
is about 
seconds, at which 
and the rolling angular velocity 
.
Using the above notation, the total spinning time is
where
is the initial inclination of the disk. Moffatt also showed that, if 
, the finite-time singularity in 
is given by
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
(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 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.
Coin
A coin is a piece of hard material that is standardized in weight, is produced in large quantities in order to facilitate trade, and primarily can be used as a legal tender token for commerce in the designated country, region, or territory....
), and has been the subject of a number of scientific papers. This phenomenon has been studied since Leonard Euler in the 18th century, hence the name.
It is manufactured and distributed by the Toysmith Group of the Damert Company. It was invented by Joe Bendik, and was originally marketed by the Tangent Toy Company.
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 (or spolling, a blend of "spin" and "roll"), slowly moving through different rates and types of motion before coming to rest – for example, the precessionPrecession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
rate of the axis of symmetry accelerates as the disk spins down. Euler’s Disk has an optimized aspect ratio
Aspect ratio
The aspect ratio of a shape is the ratio of its longer dimension to its shorter dimension. It may be applied to two characteristic dimensions of a three-dimensional shape, such as the ratio of the longest and shortest axis, or for symmetrical objects that are described by just two measurements,...
and a precision polished, slightly rounded edge to maximize the spinning/rolling time. A coin
Coin
A coin is a piece of hard material that is standardized in weight, is produced in large quantities in order to facilitate trade, and primarily can be used as a legal tender token for commerce in the designated country, region, or territory....
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. If the motion is non-dissipative, is constant and the motion persists forever, contrary to observation (since is not constant in real life situations). In fact, precessionPrecession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
rate of the axis of symmetry approaches a finite-time singularity modeled by a power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
with exponent approximately −1/3 (depending on specific conditions).
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
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
) is the dominant factor, but prior to this rolling friction is the dominant effect.
Moffatt
Recent research was sparked by an article in the April 20, 2000 edition of NatureNature (journal)
Nature, first published on 4 November 1869, is ranked the world's most cited interdisciplinary scientific journal by the Science Edition of the 2010 Journal Citation Reports...
, where Keith Moffatt
Keith Moffatt
Henry Keith Moffatt FRS FRSE is a Scottish applied mathematician with principal research interests in the field of fluid dynamics.-Research:...
shows that viscous dissipation
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
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
approaches a particular time (which is mathematically a constant of integrationIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
), 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
G-force
The g-force associated with an object is its acceleration relative to free-fall. This acceleration experienced by an object is due to the vector sum of non-gravitational forces acting on an object free to move. The accelerations that are not produced by gravity are termed proper accelerations, and...
in magnitude. Moffatt goes on to show that the theory breaks down at a time
before the final settling time , given bywhere
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 commercial toy (see link below), is about seconds, at which and the rolling angular velocity .Using the above notation, the total spinning time is
where
is the initial inclination of the disk. Moffatt also showed that, if , the finite-time singularity in is given byRebuttals
Moffatt's work inspired several other workers to investigate the dissipative mechanism of a spinning/rolling disk, which 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 photographyHigh speed photography
High speed photography is the science of taking pictures of very fast phenomena. In 1948, the Society of Motion Picture and Television Engineers defined high-speed photography as any set of photographs captured by a camera capable of 128 frames per second or greater, and of at least three...
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
(angle between the coin and the surface, i.e., just before the coin settles).Van den Engh used a rijksdaalder, a Dutch
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
coin, whose magnetic properties allowed it to be spun at a precisely determined rate.
Later work at the University of Guelph
University of Guelph
The University of Guelph, also known as U of G, is a comprehensive public research university in Guelph, Ontario, Canada. It was established in 1964 after the amalgamation of Ontario Agricultural College, the Macdonald Institute, and the Ontario Veterinary College...
by D. Petrie and coworkers showed that carrying out the experiments in a vacuum (pressure 0.1 pascal
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
) 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
2007–2008 Writers Guild of America strike
The 2007–2008 Writers Guild of America strike, more commonly referred to as simply the Writers' Strike, was a strike by the Writers Guild of America, East and the Writers Guild of America, West ....
Conan O'Brien
Conan O'Brien
Conan Christopher O'Brien is an American television host, comedian, writer, producer and performer. Since November 2010 he has hosted Conan, a late-night talk show that airs on the American cable television station TBS....
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.
External links
- http://eulersdisk.com
- http://physicsweb.org/article/news/4/4/12
- http://tam.cornell.edu/~ruina/hplab/Rolling%20and%20sliding/Andy_on_Moffatt_Disk.pdf
- http://xxx.lanl.gov/pdf/physics/0008227