The terminal motion of sliding and spinning disks with Coulomb friction
P. D. Weidman and C. P. Malhotra
Mechanical Engineering
Analysis of the frictional motion of a uniform circular disk of radius R sliding and spinning on a horizontal table reported by Farkas, et al. Phys. Rev. Lett., 90, 2003) shows that the disk always stops sliding and spinning at the same instant. Moreover, under arbitrary non-zero initial values of translational speed v and angular velocity ω, the terminal ratio &epsilon = v/R ω of translational to peripherial speed is 0.653. Motivated by the motion of a sliding and spinning eyeglass case, we investigate alternative disk geometries. One geometry is the annular disk with single radius ratio parameter η = R1/R2. In this case spinning and sliding stop simultaneously for all η. The second is the composite two-tier disk described by two parameters, its radius ratio η = R1/R2 and it thickness ratio &lambda= H1/H2 where subscripts 1 and 2 denote the lower and upper disk, respectively. Results for this problem can be described by a single parameter, the normalized radius of gyration √I/m /R1, and the terminal motion is one of three types: (i) both translation and rotation stop simultaneously, (ii) the translational motion stops first, (iii) the rotational motion stops first. Videos of experiments confirming the terminal motion states will be presented.