A heavy uniform rope of mass M and length 4a has one end connected to a fixed point on a smooth horizontal table by light elastic spring of natural length a and modulus ½ Mg, while the other end hangs down over the edge of the table.
When the spring has its natural length, the free end of the rope hangs a distance a vertically below the level of the table top. The system is released from rest in this position. Show that the free end of the rope executes simple harmonic motion, and find its period and amplitude.