A pendulum, comprising a string of length L and a small sphere, swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (Fig. P8.72) (a) Show that if the sphere is released from a height below that of the peg, it will return to this height after striking the peg. (b) Show that if the pendulum is released from the horizontal position (θ = 90°) and is to swing in a complete circle centered on the peg then the minimum value of d must be 3L/5.

Peg Figure P8.72