Rolling and Slipping of Euler’s Disk – Spin a coin and watch it roll!
Rama told the class about this interesting thing that happens when a coin is spun onto a table: the precession rate of the angular velocity vector diverges just before the coin stops spinning – there is a finite-time singularity in the physics of the problem. This is observed even when the coefficient of friction is finite. This phenomenon is known as the Euler’s-disk problem.
I read this delightful paper where an experimental verification of this phenomenon and an assessment of the important mechanism of energy dissipation – the energy of the spinning coin that eventually stops has to go somewhere – are reported. The researchers use lasers and cameras to precisely measure the rates of rotation and precession. They use this to show that the friction due to viscous drag is less important than the friction from the surface on which the coin is spun.
The setup of the problem is quite simple: a circular disk is spun onto a table, and observed using the video. There are three variables of interest – the angle of inclination, the angular velocity, and the precession rate – that can be determined using this setup. The time required for the disk to come to a stop is also measured. Here’s a sketch of the setup, from the paper:[Fig.1] [/Fig.1]
A laser is pointed at the face of the disk and a camera captures the reflection. The path of the image is an ellipse whose dimensions can be used to determine the angle of inclination. A laser beam that skims the surface of the table and is cut (twice for) every time the coin does one rotation about the vertical is used to measure the precession rate. The angular velocity is measured by counting the rotations of the coin – there are two diametrically opposite white spots on the face of the coin.
The angle of inclination, precession rate, and angular velocity are all observed to follow power laws in their evolution. However, whereas the angular velocity and the angle of inclination decay with increasing time, the precession rate in found to, as I’ve already mentioned, blow up as one gets to the collapse-time. The paper lists exponents for these power-laws for a variety of cases – disks of two materials and a torus, and three types of surfaces, for a total of nine cases.
The precession rate is found to increase more rapidly with increasing friction coefficient. The angle of inclination falls off more rapidly as the friction coefficient increases. The angular velocity decays faster as well. However, it shows a much stronger variation with friction coefficient than the other two. This stronger variation turns out to be important, as we shall see next.
These power laws show decisively that the viscous drag from air is not the most important energy dissipation mechanism in the problem. That, however, does not resolve the problem of what the major energy dissipation mechanism is. Here’s why: If the disk was rolling without slipping, theoretical predictions show some agreement with these power-law fits; however, the stronger variation for the angular velocity isn’t predicted. This suggests that the disk is also slipping as it rolls on the surface.
This combination of slipping as opposed to rolling is particularly important near the time of collapse. The authors show this by recording the motion of the centre of mass of the disk, and showing that there are quasi-periodic ‘excursions’ from a circular path, indicative of the presence of slipping in the motion of the disk. Incidentally, viscous drag is found to stabilise the circular motion of the centre of mass in the initial stages of motion.
The paper argues that since earlier theoretical analyses ignored possible slipping of the disk, they got the energy dissipation rates wrong, and hence got their predictions for the time of collapse wrong. Both energy dissipation mechanisms – rolling friction and kinetic friction – have to be taken into account to predict the time of collapse accurately.
Caps, H., Dorbolo, S., Ponte, S., Croisier, H., & Vandewalle, N. (2004). Rolling and slipping motion of Euler’s disk Physical Review E, 69 (5) DOI: 10.1103/PhysRevE.69.056610
[End. Fini. Kaputski. Euler!]