Chapter 9.5, Problem 11P

A yo-yo (as shown) falls under gravity. Assume that it falls straight down, unwinding as it goes. Find the Lagrange equation of motion. Hints: The kinetic energy is the sum of the translational energy $\frac{1}{2}m{\stackrel{\dot{}}{z}}^2$ and the rotational energy $\frac{1}{2}I{\stackrel{\dot{}}{\theta }}^2$ where $I$ is the moment of inertia. What is the relation between $\stackrel{\dot{}}{z}$ and $\stackrel{\dot{}}{\theta }$? Assume the yo-yo is a solid cylinder with inner radius $a$ and outer radius $b$.