Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 1 / a / Solution

Measure both angles from the downward vertical, as in the source diagram. With upward $y$, the cylinder centres areThe large hoop has central inertia $4M{a}^2$ and angular velocity $\dot{\theta }$, while the small hoop has no spin in this part. Translation and rotation therefore giveThus $L=T-V$. Counting each angle and velocity as first order, replacing the cosine in the velocity product by one changes only fourth-order terms. Expanding the potential and dropping its constant gives