# Investigating the "Spinning Coin" (Euler Disk) Problem

Most everyone has spun a coin on its edge on a table top, and many find the result quite fascinating.  The coin gradually begins to fall on its side while spinning, makes a whirring sound with increasing frequency the longer it spins, and then abruptly stops.  The Swiss physicist, Leonhard Euler, studied this back in the 1700's.  An educational toy, referred to as Euler's disk can now be purchased on-line and in hobby shops specializing in science.  Such disks have been carefully engineered to spin for a much longer time than a coin.

While the physics of a spinning coin is similar to that of a gyroscope, the physics can get very complicated.  Complex mathematical theories have been deducted, under a variety of simplifying assumptions, to explain the fascinating behavior of a spinning coin.  Never-the-less, the spinning coin provides for interesting student discussion at all grade levels.

As the coin spins, its center-of-mass lowers.  This results in a loss of gravitational potential energy, which is converted into kinetic energy of rotation.  There are also dissipative forces including (1) rolling friction that affects the precession rate and (2) air resistance that becomes especially important just before the coin stops spinning.

OK, we can't very well attach PocketLab to a coin, but we can attach it to other larger disks that behave much like a coin.  The wood disk shown in the figure below was found to work very well with PocketLab.  It is about 5" in diameter and 3/4" thick, and was purchased at a local hobby store.  PocketLab was attached to the center of the disk using some Scotch Removable Poster Tape. The Z-axis is perpendicular to the face of the wood disk, while the X-axis and Y-axis lie on a plane parallel to the face of the wood disk.

PocketLab was set to provide angular velocity measurements at the highest rate permitted by the device running the PocketLab app.  The magnetic field was zeroed, and the wood disk was given a spin on a level surface.  The video below was obtained from the PocketLab app and shows the X, Y, and Z angular velocity vs. time graphs superimposed on a movie of the spinning disk.

Another movie, taken with an iPhone in SLO-MO mode, makes it much easier to study the motion of the spinning disk.  The movie appears below.

A graph of X, Y, and Z angular velocity from one run of the experiment is shown in the figure below.  The graph was created in Excel from data provided by the PocketLab app.
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The green curve shows how the rate of precession slows down as time progresses.  The red and blue curves show the speed-up of the wobbling that occurs as time progresses.  The horizontal portion of the red curve at the bottom left of the graph is created when the Y angular velocity exceeds the 2000 degrees/sec limit of the PocketLab.  The spinning stops abruptly at approximately 11.2 seconds into the movie.

What a great exercise for students to perform with PocketLab!  Set up, data collection, and entry-level discussion can easily be accomplished in one class period.

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Rich- Your lab made me think of a dynamics problem I had in grad school. I dug up my old textbook so I could share. We analyzed the motion of a similar spinning disk. It was one of the toughest dynamics problems I ever had to solve.

You have to solve for a system of equations of the angular acceleration of the spinning disk and the linear acceleration of the disk's center of mass. Eventually the system of equations gets too cumbersome to solve by-hand and so we used a program called MotionGenesis to simulate the motion.

Here's how we defined the spinning disk system:

We simulated for the "lean angle", L of the disk and then the path of the disk's center of mass, Do.

Qualitatively analyzing the video and the gyroscope data, it looks like it agrees with our simulation. Someday, I'll try solving the equations of motion with the Euler disk experimental data and see how it matches the simulation.

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