A bead of mass m is constrained to move with constant speed 2ms−1 on a horizontal circular wire of radius 6cm. A distant lamp casts the shadow of the bead on a vertical screen parallel to the plane of the circle. The bead passes through the point farthest from the screen at t=0. The position of the shadow measured along a horizontal axis on the screen is denoted by x.
(i) Write the equation of motion of the shadow x(t), clearly indicating its amplitude, angular frequency and initial phase.
(ii) For one complete oscillation, calculate the total time interval during which the kinetic energy of the shadow exceeds its potential energy.
(iii) Show that the average power supplied by the restoring force to the shadow over half a time period is zero.