Chapter 9.5, Problem 14P

2A hoop of mass $M$ and radius $a$ rolls without slipping down an inclined plane of angle $\alpha .$ Find the Lagrangian and the Lagrange equation of motion. Hint: The kinetic energy of a body which is both translating and rotating is a sum of two terms: the translational kinetic energy $\frac{1}{2}M{v}^2$ where $v$ is the velocity of the center of mass, and the rotational kinetic energy $\frac{1}{2}I{\omega }^2$ where $\omega$ is the angular velocity and $I$ is the moment of inertia around the rotation axis through the center of mass.