Chapter 9.5, Problem 18P

A hoop of mass $m$ in a vertical plane rests on a frictionless table. A thread is wound many times around the circumference of the hoop. The free end of the thread extends from the bottom of the hoop along the table, passes over a pulley (assumed weightless), and then hangs straight down with a mass $m$ (equal to the mass of the hoop) attached to the end of the thread. Let $x$ be the length of thread between the bottom of the hoop and the pulley, let $y$ be the length of thread between the pulley and the hanging mass m, and let $\theta$ be the angle of rotation of the hoop about its center if the thread unwinds. What is the relation between x, y, and $\theta ?$ Find the Lagrangian and Lagrange’s equations for the system. If the system starts from rest, how does the hoop move?