1. An interesting one-dimensional system is the simple pendulum,consisting of a point mass m, fixed to the end of a massless rod(length l), whose other end is pivoted from the ceiling to let itswing freely in a vertical plane, as shown in figure below. Thependulumâs position can be specified by its angle ? from theequilibrium position. (It could be equally specified by itsdistance s from equilibrium, i.e. s = l?, but the angle is moreconvenient).
(a) Prove that the pendulumâs potential energy (measured from theequilibrium level) is
(b) Write down the total energy E as a function of ? and .
(c) Show that by differentiating your expression for E with respectto t you can get the equation of motion for ? and that the equationof motion is just the familiar (where is the torque, I is themoment of inertia, and ? is the angular acceleration)
(d) Assuming that the angle ? remains small throughout the motion,solve for ?(t) and show that the motion is periodic withperiod
1. An interesting one-dimensional system is the simple pendulum,consisting of a point mass m, fixed to the end of a massless rod(length l), whose other end is pivoted from the ceiling to let itswing freely in a vertical plane, as shown in figure below. Thependulumâs position can be specified by its angle ? from theequilibrium position. (It could be equally specified by itsdistance s from equilibrium, i.e. s = l?, but the angle is moreconvenient).
(a) Prove that the pendulumâs potential energy (measured from theequilibrium level) is
(b) Write down the total energy E as a function of ? and .
(c) Show that by differentiating your expression for E with respectto t you can get the equation of motion for ? and that the equationof motion is just the familiar (where is the torque, I is themoment of inertia, and ? is the angular acceleration)
(d) Assuming that the angle ? remains small throughout the motion,solve for ?(t) and show that the motion is periodic withperiod
