Equation of Motion

The force on an object may be determined using Newton's second law of motion $F = ma$.

There is an elastic force restoring the mass to its equilibrium position, given by Hooke's law $F = -Kx$, where $K$ is a constant describing the stiffness of the spring.

Since Newton's third law of motion states that ``for every action there is an equal and opposite reaction', these forces are equal, yielding

$$m\frac{d^2x}{dt^2} = -Kx \quad \longrightarrow \quad \frac{d^2x}{dt^2} + \omega_0^2x = 0,$$

where $\omega_0=\sqrt{K/m}$.

Recall that $x=A\cos(\omega_0t + \phi)$ is a solution to this equation.