"Consider the system consisting of a bead of mass m sliding on a smooth circular wire hoop of mass 2m and radius R in a vertical plane, and the vertical plane containing the hoop free to rotate about the vertical axis. Let φ denote the angle between the plane containing the hoop and a fixed horizontal direction.
(You may assume that the moment of inertia of the hoop about an axis which contains a diameter of the hoop is equal to mR^2.)"
I need to find the lagrangian for the above problem. I was thinking of using spherical coordinates with constant radius for the particle but because the hoop has mass I'm a little confused. Do i need to use the additional coordinate of the centre of mass of the hoop, or am I overthinking the problem?