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# Coupled Oscillators – Two masses, Three Springs

When two or more objects undergoing harmonic motion are connected, we classify them as coupled oscillators.  The resulting motion can become relatively complicated.  However, a few steps can help simplify the the analysis.

## Coordinate System for Two Coupled Oscillators

Coordinate choice for two coupled oscillators.

As usual, when working with an oscillator, the obvious choice of coordinate system is at the equilibrium point.  However, with multiple oscillators, each has its own equilibrium point.  This makes it necessary to measure from multiple locations, something you may not have had to do before.  In the diagram above, we measure the position of each of the masses relative to its equilibrium point.

This allows us to determine the stretch or compression of each spring.  For example, the left hand spring has been stretched by an amount

$$\Delta \ell =\ell_o + x_1$$

The following video goes into a bit more detail.