  The motion of a bead on a rotating circular hoop is investigated using
elementary calculus and simple symmetry arguments. The peculiar trajectories of
the bead at different speeds of rotation of the hoop are presented. Phase
portraits and nature of fixed points are studied. Bifurcation is observed with
change in the rotational speed of the hoop. At a critical speed of rotation of
the hoop, there appears an interesting relation between the time period and
amplitude of oscillation of the bead. The study introduces several important
aspects of nonlinear dynamics. It is suitable for students having basic
understanding in elementary calculus and classical mechanics.
