Terminal Motion of Sliding and Spinning Disks with Coulomb Friction

Prof. Patrick D. Weidman, Department of Mechanical Engineering, University of Colorado at Boulder

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. [{\it Phys. Rev. Lett.}, {\bf 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 $\omega$, the terminal value of $\epsilon = v/R\omega$, the ratio of translational to peripherial speed, is always 0.653. Motivated by table top experiments with an eyeglass case, we investigate alternative disk geometries. For an annular disk it is shown that, irrespective of the radius ratio $\eta = R_1/R_2$, the sliding and spinning motions again stop simultaneously. For a composite two-tier disk with lower disk of thickness $H_1$ and radius $R_1$ bonded to an upper concentric disk of thickness $H_2$ and radius $R_2$, the terminal motion can, depending on the height ratio $\lambda = H_1/H_2$ and radius ratio $\eta$, be of three types: (i) both the translational and rotational motion cease simultaneously, or (ii) the the translational motion stops first, or (iii) the rotational motion stops first. Using the normalized composite disk radius of gyration $k$, we show that all terminal motions can be understood in terms of this single variable.