Spinning disks under gravity

In this work I examine the counterintuitive accelerating rotation of a disk-shaped object, such as a coin, when it is released on a smooth surface with a spin about a vertical axis. The only existing theoretical model examines the terminal stages of the motion and proposes air drag as the primary agency behind the sudden stop. Subsequent experimental work has tended to favour rolling friction or sliding friction as the principal motivator. Here I show that a combination of rolling friction and sliding friction is necessary and sufficient to understand the complete motion of the disk. Rolling friction is dominant in the initial stages of the fall but it changes to sliding friction after the disk dips below a certain angle with the horizontal plane. The change occurs as a result of the inability of the normal reaction to provide the torque for the motion. Qualitatively, this model can account for all the observations associated with the disk. Quantitative analysis has been carried out with the friction treated as perturbations on the corresponding steady-state frictionless solutions and a finite-time singularity in the precession rate has been obtained with a critical exponent of 1/2