Energy in Simple Harmonic Motion

An undamped oscillator continuously exchanges kinetic and potential energy while their total stays constant.

At maximum displacement — the oscillator is momentarily at rest, so the kinetic energy is zero and all the energy is potential.

At the equilibrium position — the displacement is zero, so the potential energy is zero and all the energy is kinetic, with the speed at its maximum.

In between — energy is shared between the two, always summing to the same total.

Total energy —

E = ½ m ω² A²

which follows from E = ½mv²_max with v_max = ωA.

Energy depends on the square of the amplitude — doubling the amplitude quadruples the total energy. This is the same A² relationship that governs wave intensity, and for the same underlying reason.

Energy against displacement — plotting against x gives two parabolas and a horizontal line:

Potential energy rises as x², a parabola with its minimum at the equilibrium position.

Kinetic energy is the mirror image, a downward parabola peaking at x = 0.

Total energy is a horizontal line at the top, since the two always add to the same value.

Energy against time — both kinetic and potential energy vary at twice the frequency of the oscillation. This catches people out. The oscillator passes through equilibrium twice per cycle, so the kinetic energy peaks twice per cycle, and energy does not care about direction: moving left at speed v and right at speed v give the same kinetic energy.

Example: if an oscillator has period 2.0 s, its displacement repeats every 2.0 s but its kinetic energy repeats every 1.0 s. Sketching the two graphs one above the other makes the doubling obvious.

With damping — the total energy falls over time as work is done against resistive forces, so the amplitude decays. Since E ∝ A², the energy falls faster than the amplitude does.