Ivars Peterson's MathTrek

June 26, 2000

# Spinning to a Rolling Stop

Spin a coin on a tabletop. As it loses energy and tips toward the surface, the coin begins to roll on its rim, wobbling faster and faster and faster. Toward the end, the coin generates a characteristic rattling sound of rapidly increasing frequency until it suddenly stops with a distinctive shudder.

Mathematician H. Keith Moffatt of the Isaac Newton Institute for Mathematical Sciences in Cambridge, England, has come up with an explanation of why this motion ends so abruptly instead of lingering as the coin keeps on rolling faster.

The culprit is the thin layer of air trapped between the tipping coin and the table, Moffatt reports in the April 20 Nature. As its tilt becomes more pronounced, the rolling coin squeezes and swirls the air beneath. The flowing air takes up energy, tipping the coin even closer to the surface. At some point, the coin's edge finally loses its grip on the table and falls flat.

The larger and heavier the disk, the more dramatic is the effect, Moffatt notes. Such long-lived behavior can be observed in a commercial toy called Euler's disk--a thick, 400-gram, chrome-plated steel disk, 3.75 centimeters in diameter, with a rounded edge to help keep it in motion for remarkably long periods of time on a circular, slightly concave platform with a mirror finish (see http://www.tangenttoy.com/euler/).

Moffatt's mathematical model correctly predicts that Euler's disk would typically spin about 100 seconds before it finally hums to a stop. In fact, it was this toy that originally inspired Moffatt to delve into the underlying mathematics of spinning disks. He had come across it in a toy catalog that he was perusing to find gifts for his grandchildren and, intrigued, had to order it.

In his analysis, Moffatt focused on the role of air viscosity. What would a spinning disk do in a vacuum, where there would be no air cushion?

Other dissipative mechanisms, such as rolling friction and vibration of the supporting table, would take over, Moffatt says. The disk would still tip and its rolling speed increase, though at a different rate, before stopping. "I know of at least two experiments that are now under way to investigate this," he notes.

These other dissipative mechanisms are much more difficult to quantify, however. Rolling friction, for example, depends on the physical properties of both the disk and the surface on which it rolls.

Interestingly, a spinning ring also collapses with a characteristic rattling sound, much like a spinning disk, even though there is much less aerodynamic drag. In this case, rolling friction probably makes a significant contribution to the observed effect.

"The problem again is to provide a quantitative theory for the dissipation of energy associated with rolling friction," Moffatt remarks. "I don't think such a theory exists at present."

Mathematical investigations of the coin-rattling phenomenon may provide insights into turbulence, Moffatt suggests. Aided by this simple, tabletop model of what mathematicians describe as a finite-time singularity, researchers can take a fresh look at the yet-unresolved question of whether such singularities can occur in the interior of a fluid in turbulent motion.

References:

2000. Old problem, new spin. Plus (June). See http://www.pass.maths.org.uk/issue11/news/spin/index.html.

Chang, K. 2000. Simple spinning toy poses a mathematical challenge. New York Times (April 25).

Moffatt, H.K. 2000. Euler's disk and its finite-time singularity. Nature 404(April 20):833.

Peterson, I. 2000. Spinning to a rolling stop. Science News 157(May 6):303.

Keith Moffatt has a Web page at http://www.newton.cam.ac.uk/keith_biog.html.

Information about the Euler's disk toy can be found at http://www.tangenttoy.com/euler/.