Simple Harmonic Motion

In Simple Harmonic Motion (SHM), the total mechanical energy is conserved and is equally distributed between the kinetic energy ( ) and the potential energy ( ) at any point in time. The total energy of an object performing SHM is given by: At any position during the SHM: 1. Kinetic Energy ( ) is given by: Where is the mass of the object and is its velocity at that position. 2. Potential Energy ( ) is given by: Where is the spring constant and is the displacement from the equilibrium position. At extreme positions (when , the amplitude), the velocity is zero, so all the energy is stored as potential energy, and at the mean position (when ), all the energy is converted to kinetic energy. Now, the average values of kinetic energy and potential energy over a complete cycle of SHM can be derived as: - The average kinetic energy is: - The average potential energy is: Therefore, the ratio of average kinetic energy to average potential energy is: Thus, the ratio of the average kinetic energy to the average potential energy for an SHM is .