1 problem found
Two small beads, \(A\) and \(B\), each of mass \(m\), are threaded on a smooth horizontal circular hoop of radius \(a\) and centre \(O\). The angle \(\theta\) is the acute angle determined by \(2\theta = \angle AOB\). The beads are connected by a light straight spring. The energy stored in the spring is \[ mk^2 a^2(\theta - \alpha)^2, \] where \(k\) and \(\alpha\) are constants satisfying \(k>0\) and \(\frac \pi 4< \alpha<\frac\pi2\). The spring is held in compression with \(\theta =\beta\) and then released. Find the period of oscillations in the two cases that arise according to the value of \(\beta\) and state the value of \(\beta\) for which oscillations do not occur.