Mechanical Simple Harmonic Oscillators (SHO)

References

Examples of Mechanical Simple Harmonic Oscillators
(source)	Simple Pendulum	(source)
(source)                              Spectrum of a Tuning Fork	(source)

An important calculation - the resonant frequency of a simple harmonic oscillator:

Consider a specific kind of resonator or harmonic oscillator - e.g., a pendulum or a tuning fork - and suppose that

Newton tells us that for any mechanical system

However, harmonic oscillators follow Hooke's law so that

Putting these two relationship together we have

Symbolically

where

We now "guess" the solution

	(click on image to enlarge)
Phasor Representation (source)                   See Phasors, Vectors and Complex Notation

so that

Thus, this solution works if