= Bead on a horizontal hoop rotating about its rim
{title2=$\ddot\theta=-\omega^2\sin\theta\cos\theta$}
For a smooth horizontal circular hoop of radius $a$, rotated at constant <angular velocity> $\omega$ about a vertical axis through a rim point, measure $\theta$ at that pivot from the diameter toward the bead. In rotating <Cartesian coordinate system> coordinates, $\mathbf r=a(1+\cos2\theta,\sin2\theta,0)$. The tangential component of the <centrifugal acceleration> gives the displayed equation; the <Coriolis acceleration> is normal to the hoop. The farthest point $\theta=0$ is stable for $\omega\ne0$, with <small oscillation> angular frequency $|\omega|$. The pivot itself is unstable. This geometry differs from a vertical rotating hoop and has no tangential gravitational term.
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