Chapter 6.3 Energies of harmonic oscillator (Waves and Oscillations)

Total energy of a body executing S.H.M.:

The mechanical energy E of a particle executing simple harmonic motion is partly kinetic and partly potential. If no non-conservative forces, such as the force of friction act on the particle, the sum of its kinetic energy (K) and potential energy (U) remains constant. Therefore, total energy, E = K + U = constant.

As the displacement increases, the potential energy increases and the kinetic energy decreases and vice versa. But the total energy E is conserved.

Therefore the total energy of the system, as expected, is constant because it has the same as the maximum value of any one of the two forms of energy  The variation of the potential and kinetic energies is shown in Fig. 5 (b). Therefore, at any position the kinetic and potential energies each contributes energy whose sum is always 1/2 k. a^2. 

Average value of kinetic and potential energies of a harmonic oscillator:

Let the displacement of a particle executing simple harmonic motion at any instant be y. If the mass of the particle be m and its velocity at that instant be v, then the kinetic energy is . The potential energy of the particle at the same instant is the amount of work that must be done in overcoming the force through a displacement y and is given by  where F is the force required to maintain the displacement and dy is a small displacement. Therefore, we can write

Potential energy of the particle, U=∫_0^y 〖F.dy〗