Problem 1.265

A small body \(A\) is fixed to the inside of a thin rigid hoop of radius \(R\) and mass equal to that of the body \(A\). The hoop rolls without slipping over a horizontal plane; at the moments when the body \(A\) gets into the lower position, the centre of the hoop moves with velocity \(v_{0}\) (Fig. 1.68). At what values of \(v_{0}\) will the hoop move without bouncing?