  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.
