An Experimental and Theoretical Study of Euler’s Disk: Material Parameter Dependence and Motion after Falling Flat

Euler’s disk is a heavy, polished steel disk that can spin on a concave base-plate. As it spins, the contact point of the disk with the base-plate describes a circle, and its center of mass slowly descends towards the base-plate, after which the disk falls flat to a sudden halt. We present an extended review of the experimental and theoretical literature. We describe a series of experiments on a commercially available Euler’s disk spinning on different base-plates (flat and concave). We show for the first time that the material properties (Poisson’s ratio and Young’s modulus) of flat base-plates affect the motion of the disk. We show that the disk can continue to rotate about a vertical axis for a short time after it falls flat, an effect (which has not been seen before) exaggerated when the base-plate is lubricated with general purpose oil. We solve the equations of motion numerically for the disk, subject to both Coulomb friction and contour friction. Qualitative agreement with our experimental data was found. An additional multiplicative factor applied to contour friction gave much better quantitative agreement. The discontinuity in contour friction provides us with a mechanism for the continued rotation of the disk after it falls flat, giving good agreement with experiments. This part of the motion has two phases for the flat base-plate: a contactless phase, where the disk is supported by a thin layer of fluid (air or oil), followed by a contact phase across the face of the disk where Coulomb friction brings it to a halt. For a concave base-plate, there is no contactless phase and the circumference of the disk contacts the base-plate and Coulomb friction brings it to a halt.