Conservation of mechanical energy applies to a simple harmonic motion system. We should find that the sum of the potential and kinetic energies is constant.

U = ½ kx² = ½ kA² cos²(ωt + φ)

K = ½ mv² = ½ mA²ω² sin²(ωt + φ)

We know that ω² = k/m, so the kinetic energy expression becomes:

K = ½ kA² sin²(ωt + φ)

Because sin²(something) + cos²(something) = 1

U + K = ½ kA² = constant

Also, U + K = ½ mA² ω²

Graphs of potential and kinetic energy as a function of time show that the total energy is constant, and that energies go through two complete cycles for each oscillation of the object.

Graphing the energies as a function of position is also interesting.