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 RR always stops sliding and spinning at the same instant. Moreover, under arbitrary nonzero initial values of translational speed vv and angular rotation rate ωω, the terminal value of the speed ratio ϵ0=v/Rωϵ0=v/Rω 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=R1/R2η=R1/R2 stops sliding and spinning at the same time, but with a terminal speed ratio dependent on ηη. For a two-tier disk with lower tier of thickness H1H1 and radius R1R1 and upper tier of thickness H2H2 and radius R2R2, the motion depends on both ηη and the thickness ratio λ=H1/H2λ=H1/H2. While translation and rotation stop simultaneously, their terminal ratio ϵ0ϵ0 either vanishes when k>2/3, is a nonzero constant when 1/2