A pendulum with a broken cylindrical symmetry: normal modes and beats

Abstract

Abstract A pendulum, which has the cylindrical symmetry around its suspension wire, has two degenerate normal modes. We break this symmetry by putting a small magnet in the pendulum body and by running the pendulum in the presence of an inhomogeneous magnetic field. The force due to the magnetic field gradient modifies the equilibrium position and the spatial derivatives of this force define the normal modes of the pendulum with two different oscillation frequencies. We describe this system theoretically and we test these results by a series of experiments. This system provides a classical analogue of the degenerate perturbation theory in quantum mechanics.