System Equations for Two Spring-Coupled Masses

Figure 7: A short section of a string.

The extensions of the left, middle and right springs are $x_1, x_2-x_1$, and $-x_2$, respectively.

When a spring is extended by $x$, the mass attached to the

• left experiences a positive horizontal restoring force $F_r = Kx$;
• right experiences an equal and opposite force $F_r = -Kx$,

where $K$ is the spring constant.

The equation of motion for the displacement of the first mass:

$$m\ddot{x}_1 = -Kx_1 + K(x_2-x_1),$$

and the second mass,

$$m\ddot{x}_2 = -K(x_2-x_1) + K(-x_2).$$