In ideal simple harmonic motion, kinetic energy and spring potential energy continually transform into one another while their sum remains constant.

For a mass-spring oscillator,

$$E=K+U =\frac12mv^2+\frac12kx^2.$$

Turning points

At the maximum displacements

$$x=\pm A,$$

the velocity is zero. All the mechanical energy is elastic potential energy:

$$E=\frac12kA^2.$$

Equilibrium

At equilibrium,

$$x=0,$$

so the spring potential energy is zero relative to that reference. The speed and kinetic energy are maximal:

$$E=\frac12mv_{\max}^2.$$

Intermediate positions

At any position between equilibrium and a turning point,

$$\frac12mv^2+\frac12kx^2=\frac12kA^2.$$

As $|x|$ increases, potential energy increases and kinetic energy decreases.

Idealization

This conservation assumes no damping or external energy input. Friction or drag gradually removes mechanical energy, while a driving force can add it.

The ideal oscillator therefore provides a clean picture of periodic energy exchange that later extends to waves, resonant systems and electrical oscillations.