Past exam of the theoretical physics I course of the University of Cambridge / 2022 / 1 / b / Solution

Let $\theta$ and $\phi$ be the lower and upper azimuthal angles, measured from a common fixed direction. The squared spring length isBy Hooke's law, a spring with zero natural length has potential energy $k{d}^2/2$. ThereforeConstant gravitational energies can be omitted. It is useful to define the moments of inertia $I_1=m{a}^2$, $I_2=M{A}^2$ and the angular coupling $\kappa =kaA$. The part of the potential that affects motion is simply $-\kappa \cos (\theta -\phi )$. In particular, $h$ changes the constant spring energy but not the angular equations, because the spring has zero natural length.