One point on a horizontal circular wire \(\mathrm{C}\) of radius \(R\) is attached to a thin, vertical axle which turns at constant angular velocity \(\Omega\) about the vertical axis, causing \(\mathrm{C}\) to turn around with it. A bead of mass \(m\) moves without friction on C.

(a) Show that relative to \(\mathrm{C}\) the bead oscillates like a pendulum.

(b) Find the frequency of small-amplitude oscillations of the bead in terms of any or all of \(R, \Omega\), and \(m\).