1. Problem: Determine the final translational speed of a spinning disc that is dropped on an infinite horizontal plane. Think of spinning up an automobile wheel suspended over a road (wheel spinning normal to the road), then dropping the wheel on the road. When the wheel rim touches the road, friction will create a torque on the wheel that will cause it to begin rolling down the road. It will accelerate to some translational velocity. How fast will it go? Everything is ideal: uniform solid disc, no air friction, no loss of energy through frictional heating at the disc/plane contact, no wear, etc. 2. Relevant equations The disc has a mass (m), a radius (r) and an initial spin rate Wi (Omega initial). The disc has an initial kinetic energy KEi = IWi^2/2, where I is the disc moment of inertia, mr^2. At equilibrium, the disc has a translational velocity (v), and a final spin rate Wf (Omega final). Its translational kinetic energy is mv^2/2. Its final spin KE is IWf^2/2. 3. The attempt at a solution I'm modeling this under these assumptions: a. The system's initial kinetic energy of spin is the only energy that is available to accelerate the wheel along the plane. b. As frictional torque accelerates the disc in translation, the disc spin rate decreases and the disc's spin kinetic energy decreases, as the disc's translational kinetic energy increases. c. When the disc's translational kinetic energy = the disc's remaining spin kinetic energy, translational acceleration stops and the system is in its final equilibrium state. d. The sum of the final translational KE and spin KE = the initial spin KE. Under these assumptions, the final translational KE (mv^2/2) = the final spin KE (IWf^2/2) and each of these is 1/2 of the initial spin KE (IWi^2/2). Substituting mr^2 for I, the final KE equivalence is mv^2/2 = mr^2Wf^2/2. From this, the translational velocity v should = rWf, or the disc radius times its final angular rate. My uncertainty is in assumption (c.). I assume that the initial disc spin KE decreases to as the disc transnational KE increases, and that when they are equal, no further KE transfer occurs. Is this correct? If not, what determines the value of either spin KE or translational KE at which no further change in KE occurs? Thank you.