Theoretical study of the energy and oscillating period for a uniformly distributed mass spring

Abstract

We theoretically investigate a fundamental problem in the spring oscillator for a uniformly distributed mass. The wave equation and its solution method are derived in detail. The kinetic energy of the spring and oscillator, the elastic potential energy of the spring, and the oscillating period of the system are solved using a wave equation. Our results show that the total energy and each order harmonic energy of the oscillating system are constant. In addition, the oscillating period of the system is also reported, which is equal to one-third of the spring mass m added to the mass M when m ≤ M.