Euler's disc and its finite-time singularity.
By: Kevin T. Kilty, 2000
Note to readers. This contribution was in response to a Letter to Nature, 20 April, 2000, p.855, by Dr. Keith Moffatt, who was gracious enough to correspond with me on the topic. Nature's reviewer stated that it was "beneath the dignity" of Nature to consider publishing such trivia. Nature published the "trivia" that prompted it, however. Moffat has proposed that air friction brings the disc to a halt in a time that is independent of the surface on which it spins. My intention was to show that a dependence on surface properties results in the same dynamic behavior.
The correspondence regarding Euler's disc (April 20) was very interesting. However, over the years I have taken countless coins from my pocket and tossed them onto sales counters, thrown them onto furniture, or dropped them on floors. All of this experience has proved two things. First, that a coin shudders to a stop within a few seconds even when it has begun its spin completely upright. The equivalent time based on the theory of your correspondence would be orders of magnitude longer. Second, a coin shudders to a halt more quickly on a rough, soft surface, such as varnished fir, than it does on a smooth hard surface such as a glass sales counter.
I propose a modification of the theory. Let me refer to the analysis in your brief communication and diverge from it only after the first two equations. Equation 1, W 2a=4g/a, relates the rate of spin to the angle that the disc makes with the surface (a ), the acceleration of gravity (g) and the disc radius (a). Equation 2, E=1.5Mga· sina , where (M) is the mass of the disc, represents total mechanical energy.
Consider rolling friction rather than air viscosity as the source of dissipation. When an elastic disc rolls on an elastic surface both the disc and the surface deform. This produces a small component of force that resists the rolling motion. Typically we write this force as being proportional to the weight of the disc, which is f=m Mg, where m is called the coefficient of rolling friction, as long as the acceleration of a is small.. We call this rolling friction even though it has little to do with friction in the usual sense. If we multiply this force by the speed of the point of contact between the disc and surface, W a, the result is a rate of dissipation equal to -m W Mga. Therefore, Equation 3 becomes
1.5Mga(da /dt)=- m W Mga (3)
which integrates to
a 3/2(t)= (t0-t)/t1; where, t1 = 0.5 (a/g)1/2/m (4)
This equation displays a finite-time singularity. The disc will settle to the surface in a time t0 =a 03/2 t1, which is the correct order of magnitude for a coin begun at a 0=0.5 or greater as long as m is about 0.001. Gravity acting on the mass of the disc provides only limited torque, which limits the acceleration of a to a value less than 4g/5a. This has the further interesting consequence that as the coin settles, the vertical reaction at the point of contact diminishes and so does the rolling friction. In other words, before the coin reaches its limiting acceleration viscous dissipation has truly become the dominant effect, and the coin is now so nearly flat to the surface that an escaping cushion of air limits its fall. Thus, the coin follows an evolving dynamic that begins with rolling friction, passes through a phase of viscous braking, and ends with a cushioned fall to the surface.
Very truly yours,
Kevin T. Kilty
Factory Automation Engineering
SEH America, Inc.
Vancouver, WA