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The motion of a bead on a rotating circular hoop is investigated usingelementary calculus and simple symmetry arguments. The peculiar trajectories ofthe bead at different speeds of rotation of the hoop are presented. Phaseportraits and nature of fixed points are studied. Bifurcation is observed withchange in the rotational speed of the hoop. At a critical speed of rotation ofthe hoop, there appears an interesting relation between the time period andamplitude of oscillation of the bead. The study introduces several importantaspects of nonlinear dynamics. It is suitable for students having basicunderstanding in elementary calculus and classical mechanics.

Bead on a rotating circular hoop: a simple yet feature-rich dynamical system