Oscillations and waves

Energy in simple harmonic motion

In ideal simple harmonic motion, kinetic and potential energies interchange while total mechanical energy is conserved.

Without friction or drag, an oscillator continually exchanges kinetic and potential energy while the total stays constant. It momentarily stops at each endpoint and moves fastest through equilibrium.

Energy versus phase

Writing $x=A\cos\theta$ and $v=-A\omega\sin\theta$ gives $U/E=\cos^2\theta$ and $K/E=\sin^2\theta$. They always add to one: energy shifts between forms, rather than appearing or disappearing.

Quick check

From $v^2=\omega^2(A^2-x^2)$ we obtain $K=\frac12m\omega^2(A^2-x^2)$ and $U=\frac12m\omega^2x^2$. At $x=0$, all mechanical energy is kinetic; at $x=\pm A$, it is all potential. Between these points the two forms continuously exchange while their sum remains constant without damping.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear. For a spring oscillator, this also shows that total energy is proportional to the square of amplitude.