Disk Jockey - Ant on Rotating Disk · USAPhO 2021 #B1
Physics · mechanics
Problem
A disk of uniform mass density, mass M, and radius R sits at rest on a frictionless floor. The disk is attached to the floor by a frictionless pivot at its center, which keeps the center of the disk in place, but allows the disk to rotate freely. An ant of mass m << M is initially standing on the edge of the disk; you may give your answers to leading order in m/M.
a. The ant walks an angular displacement θ along the edge of the disk. Then it walks radially inward by a distance h << R, tangentially through an angular displacement −θ, then back to its starting point on the disk. Assume the ant walks with constant speed v.
[Figure: Diagram showing rectangular path with angular displacement θ and radial distance h]
Through what net angle does the disk rotate throughout this process, to leading order in h/R?
b. Now suppose the ant walks with speed v along a circle of radius r, tangent to its starting point.
[Figure: Diagram showing circular path of radius r]
Through what net angle does the disk rotate?