Relations among different energy dissipations of Euler disk

Abstract

The energy dissipation of a disc spinning on a horizontal plane is studied, as the angle α of the coin made with the horizontal plane decreases, while the angular velocity Ω of the point of contact increases. Effect of the ratio x between the thickness and diameter of an Euler disc and the α on the energy dissipation is studied. We find, by using numerical simulation, that when x is small enough, the lose of the kinetic energy and the gravitational potential energy of the mass center is dominant in energy dissipations; when x>0.4142, the rotational kinetic energy dissipation of the disc around the axis that is parallel to the disc surface, is the leading factor. The requirements in which thickness can be neglected are also obtained, and they can give some hints to the relevant theories and experiments. Our results show that when α≥10° and b/a[26] data very well. We also discuss the main energy dissipation distributed among different forms: variation of rolling friction and viscous shear of the air with x and α, also show their transition in the process of the motion. Furthermore, we find that the pure rolling friction is the unique dissipation as x=0.1733 and α>18°, which improves the results obtained before. We speculate that the dominant dissipation is the gliding friction in the final stage of the motion, because when the disc is motionless, one face of the disc lies absolutely in contact with the horizontal surface just before the disc halts. One can assume that they are in contact completely but the disc does not halt, thus axis 1 and axis Z are almost in the same direction. In this case, the energy dissipation of the Euler disc is due to the gliding friction. To some extent, this accounts for the disc final halt.