On the terminal motion of sliding spinning disks with uniform Coulomb friction

We review previous investigations concerning the terminal motion of disks sliding and spinning with uniform dry friction across a horizontal plane. Previous analyses show that a thin circular ring or uniform circular disk of radius R always stops sliding and spinning at the same instant. Moreover, under arbitrary nonzero initial values of translational speed v and angular rotation rate ω , the terminal value of the speed ratio ϵ 0 = v / R ω is always 1.0 for the ring and 0.653 for the uniform disk. In the current study we show that an annular disk of radius ratio η = R 1 / R 2 stops sliding and spinning at the same time, but with a terminal speed ratio dependent on η . For a two-tier disk with lower tier of thickness H 1 and radius R 1 and upper tier of thickness H 2 and radius R 2 , the motion depends on both η and the thickness ratio λ = H 1 / H 2 . While translation and rotation stop simultaneously, their terminal ratio ϵ 0 either vanishes when k > 2 / 3 , is a nonzero constant when 1 / 2 < k < 2 / 3 , or diverges when k < 1 / 2 , where k is the normalized radius of gyration. These three regimes are in agreement with those found by Goyal et al. [S. Goyal, A. Ruina, J. Papadopoulos, Wear 143 (1991) 331] for generic axisymmetric bodies with varying radii of gyration using geometric methods. New experiments with PVC disks sliding on a nylon fabric stretched over a plexiglass plate only partially corroborate the three different types of terminal motions, suggesting more complexity in the description of friction.