an interesting one dimensional system is the simple pendulum consisting of a point mass m fixed to the end of a mass less rod (length l) whose other end is pivoted from the ceiling to let it swing freely in a vertical plane . the pendulums position can be specified by its angle @ from the equilibrium position a) prove that the pendulums potential energy is U(@)=mgl([email protected]) write down the total energy e as a function of @ and @^. B) show that by differentiation your expression for E with respect to t you can get the equation of motion for @ and that the equation of motion is just the familiar t=Ia t is torque I is moment of inertial a is angle acceleration C) assuming that the angle @ remains small solve for @(t) and show that motion is periodic with period t=2pie(1/g)^1/2