The Importance of Simple Harmonic Motion

Any vibration where the force exerted on a body is proportion to the displacement from equilibrium is simple harmonic.
This includes all systems obeying Hook'e's Law,  \(\mathbf{F}=-k \mathbf{x}\).
By letting  \(\mathbf{F} = m \mathbf{a}\)  we get  \(m \mathbf{a} - k \mathbf{x} \rightarrow a = - \frac{k}{m} \mathbf{x} = - \omega^2 x\).
Actually, a system does not need to obey Hooke's Law to any great extent to execute simple harmonic motion. If the vibrations are small enough, most vibrations are approximately simple harmonic. Often this can produce important results. By assuming that the Vibrations of atoms in a crystal lattice are simple harmonic we obtain the important equation  \(E=mc \theta\)