Problem 1: A heavy ring of mass M can slide on a frictionless rod. The rod is attached to a wall at one end with a bracket and the other end hangs downward, so the angle between the rod and the wall is α. One end of a massless spring of constant k is attached to the bracket and the other end is connected to the ring. Determine the Lagrangian for the system and equation of motion. Problem 2: Obtain the equations of motion for a spherical pendulum, that is, a pendulum that is not constrained to oscillate in a plane.