Problems

A thin circular disc of mass \(m\), radius \(r\) and with its centre of mass at its centre \(C\) can rotate freely in a vertical plane about a fixed horizontal axis through a point \(O\) of its circumference. A particle \(P\), also of mass \(m,\) is attached to the circumference of the disc so that the angle \(OCP\) is \(2\alpha,\) where \(\alpha\leqslant\pi/2\).

1. In the position of stable equilibrium \(OC\) makes an angle \(\beta\) with the vertical. Prove that \[ \tan\beta=\frac{\sin2\alpha}{2-\cos2\alpha}. \]
2. The density of the disc at a point distant \(x\) from \(C\) is \(\rho x/r.\) Show that its moment of inertia about the horizontal axis through \(O\) is \(8mr^{2}/5\).
3. The mid-point of \(CP\) is \(Q\). The disc is held at rest with \(OQ\) horizontal and \(C\) lower than \(P\) and it is then released. Show that the speed \(v\) with which \(C\) is moving when \(P\) passes vertically below \(O\) is given by \[ v^{2}=\frac{15gr\sin\alpha}{2(2+5\sin^{2}\alpha)}. \] Find the maximum value of \(v^{2}\) as \(\alpha\) is varied.