Physics Complete: Energy of Simple Hormonic Motion

According to law of conservation of energy total energy of a system always remains constant and the particle in simple harmonic motion is not a exception from it.

A particle executing simple harmonic motion has kinetic and potential energies, both varying between the limits, zero and maximum. Here we will use the equations of a particle's displacement and velocity in SHM.

The velocity of a particle executing SHM, is a periodic function of time. It is zero at the extreme positions of displacement. Therefore, the kinetic energy (K) of such a particle, which is defined as

is also a periodic function of time, being zero when the displacement is maximum and maximum when the particle is at the mean position.Since the sign of v is immaterial in K, the period of K is T/2.
The spring force F = –kx is a conservative force, with associated potential energy U = 1/2 K x^2.
Hence the potential energy of a particle executing simple harmonic motion is,

Thus the potential energy of a particle executing simple harmonic motion is also periodic, with period T/2, being zero at the mean position and maximum at the extreme displacements.

Total Energy

The total mechanical energy of a harmonic oscillator is thus independent of time as expected for motion under any conservative force. The time and displacement dependence of the potential and kinetic energies of a linear simple harmonic oscillator are shown in figure.

In a linear harmonic oscillator, all energies are positive and peak twice during every period. For x = 0, the energy is all kinetic and for x = ± A it is all potential.

In between these extreme positions, the potential energy increases at the expense of kinetic energy. The former stores its potential energy and the latter stores its kinetic energy.