Study of normal modes and symmetry breaking in a two-dimensional pendulum

Abstract: We present an experimental setup to demonstrate normal modes and symmetry breaking in a two-dimensional pendulum. In our experiment we have used two modes of a single oscillator to demonstrate normal modes, as opposed to two single oscillators used in standard setups of two-dimensional pendulums. Breaking of the cylindrical symmetry of the pendulum is achieved by attaching a spring in the suspension. This leads to interesting visual patterns in the motion, wherein the plane of the oscillator shifts with time, the motion then becomes elliptical, shifts back again to planar, before finally returning to planar motion in the original plane. The symmetry breaking leads to non-degenerate normal modes of oscillation, whose interplay gives rise to the observed motion patterns. This also explains why for a real pendulum, the plane of motion always shifts, unlike the ideal two-dimensional pendulum where the plane of oscillation is supposed to remain fixed. This curious fact also contributes to the difficulties involved in building a Foucault's pendulum, where the plane of rotation due to Coriolis force needs to be accurately measured. The strength of the symmetry breaking in our system can be quantified by a parameter the "return time", which is defined as the time over which the pendulum returns to its original motion pattern. We propose this setup as a pedagogical tool to introduce the concepts of normal modes and symmetry breaking in a physics laboratory. The motion patterns that emerge have a high visual impact and we have also described in detail the quantitative observations can be made with this setup.