In general the total energy of a harmonic oscillator consists of two parts,  potential energy (P.E) and kinetic energy (K.E.), the former being due to its displacement from the mean position and latter due to its velocity. Since the position and velocity of the harmonic oscillator are continuously changing, P.E. and K.E. also change but their sum i.e., the total energy (T.E) must have the same value at all times.

(i) Potential Energy: The simple restoring force acting on the harmonic oscillator is given by

Now if the oscillator is displaced through a further displacement dx opposite the force, work done in displacing the object is given by

Hence the net work done in displacing the object from mean position (x=0) to (x=x) is given by

By convention, P.E. at the mean position is given as zero. Hence, above relation gives the magnitude of P.E. of harmonic oscillator at a distance x from the mean position i.e.,

              . . . (i)

This shows the P.E. is proportional to the square of the displacement and graph showing the variation of potential energy with the displacement will be a parabola given by continuous lines in the figure. P.E. is maximum at maximum distance and is given by

(ii) Kinetic Energy: Speed of harmonic oscillator is given by equation as

Hence kinetic power of the oscillator is provided by

                          . . . (ii)

                         (iii)

Thus total energy is independent of the distance. It has constant throughout the motion of the oscillator. Also the net energy is same to maximum value of either K.E. or P.E.

 (iii) Average Value of P.E.  and K.E.: By equation (i) P.E. at distance x is given by

The average value of P.E. for one complete oscillation is given by

Because the average magnitude of sine or of cosine function for the complete cycle is equal to zero.

Now K.E. at x is given by

The average value of K.E. for one complete cycle

KE_average  = 

Thus average values of P.E. and K.E. of harmonic oscillator are equal and each is equal to one fourth of the total energy. 

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