Euler's Disk

submitted by Shepppard
1 decade 3 years 6 months 1 week ago
From youtube: Give the disk a spin and observe the hypnotic display of light and continually changing sound. The movement seems to go forever

33 Comments

BoneRemakesays...

"Euler's Disk" is a trademark for a product manufactured and distributed by the "Damert Company" (Toysmith Group), consisting of a metal disk, a base having an upwards-facing concave mirror, and holographic, patterned magnetic stickers. One or more magnetic stickers may be attached to the top surface of the disk. The disk, when spun on the mirror, exhibits a spinning/rolling motion. Euler’s Disk has an optimized aspect ratio and precision polished, rounded edges to maximize the spinning/rolling time. A coin spun on a table, or any disc spun on a relatively flat surface, exhibits essentially the same type of motion.

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 ω. If the motion is non-dissipative, ω is constant and the motion persists forever, contrary to observation (since ω is not constant in real life situations).

In the April 20, 2000 edition of Nature, 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.

Moffatt shows that, as time t approaches a particular time t0 (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 τ before the final settling time t0, given by

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

where a is the radius of the disk, g is the acceleration due to Earth's gravity, μ the dynamic viscosity of air, and M the mass of the disk. For the commercial toy (see link below), τ 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=\left(\frac{\alpha_0^3}{2\pi}\right)\frac{M}{\mu a}

where α0 is the initial inclination of the disk. Moffatt also showed that, if t0 − t > τ, the finite-time singularity in Ω is given by.....

http://en.wikipedia.org/wiki/Euler%27s_disk

AnimalsForCrackerssays...

>> ^GeeSussFreeK:

Just add LCD and you have days of excitement.


CRT is my drug of choice.

Also wow, the intensity really picked up towards the end. Sounded exactly like the first evolution you have in the game Space Invaders Infinity Gene for the iPhone. They probably used one of these buggers for that sound effect.

therealblankmansays...

>> ^BoneRemake:

"Euler's Disk" is a trademark for a product manufactured and distributed by the "Damert Company" (Toysmith Group), consisting of a metal disk, a base having an upwards-facing concave mirror, and holographic, patterned magnetic stickers. One or more magnetic stickers may be attached to the top surface of the disk. The disk, when spun on the mirror, exhibits a spinning/rolling motion. Euler’s Disk has an optimized aspect ratio and precision polished, rounded edges to maximize the spinning/rolling time. A coin spun on a table, or any disc spun on a relatively flat surface, exhibits essentially the same type of motion.
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 ω. If the motion is non-dissipative, ω is constant and the motion persists forever, contrary to observation (since ω is not constant in real life situations).
In the April 20, 2000 edition of Nature, 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.
Moffatt shows that, as time t approaches a particular time t0 (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 τ before the final settling time t0, given by
\tau\simeq\left(2a/9g\right)^{3/5} \left(2\pi\mu a/M\right)^{1/5}
where a is the radius of the disk, g is the acceleration due to Earth's gravity, μ the dynamic viscosity of air, and M the mass of the disk. For the commercial toy (see link below), τ 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=\left(\frac{\alpha_0^3}{2\pi}\right)\frac{M}{\mu a}
where α0 is the initial inclination of the disk. Moffatt also showed that, if t0 − t > τ, the finite-time singularity in Ω is given by.....
http://en.wikipedia.org/wiki/Euler%27s_disk


Nerd.

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