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To find out the total mechanical energy of a particle undergoing simple harmonic motion let us consider the linear velocity of the particle to be:
Then the kinetic energy (K) of the particle is given by,
Now the elastic potential energy (U) stored in the particle having displacement x is given by,
where the displacement of the particle is given by,
Now, the total mechanical energy of the simple harmonic oscillator is given as:
This will reduce to
That is the total mechanical energy of a simple harmonic oscillator is a constant of the motion and is proportional to the square of the amplitude. The variation of energy with time is given as follows:
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Waves and Optics