12.3 Energy in Simple Harmonic Motion

In an ideal spring-mass oscillator, total mechanical energy stays constant while energy transfers between spring potential store and kinetic store. At turning points (x = +A or x = -A), speed is zero, so kinetic energy is zero and potential energy is maximum. At equilibrium (x = 0), potential energy is minimum and kinetic energy is maximum.

This energy view gives a second route to many kinematics results. Instead of integrating acceleration or differentiating displacement, you can use conservation directly. In exam settings, the fastest path is often whichever form matches the quantity you are given: x, v, or energy fraction.

The Ep-x graph is a parabola opening upward because Ep depends on x squared. The Ek-x graph is an inverted parabola across the allowed displacement interval [-A, +A]. Their sum is a horizontal line at ET. Reading these three together is one of the cleanest SHM diagnostics available.

Students sometimes expect energy graphs to repeat exactly like displacement graphs. They repeat, but with different symmetry and period behavior: because of squared dependence, Ep and Ek repeat every half-cycle in time even though displacement repeats every full cycle.

If x(t) is sinusoidal, then x squared and v squared are sinusoidal-squared. Squaring removes sign, so positive and negative displacements contribute equally to potential energy. That is why Ep(t) has two maxima in one displacement period. The same logic applies to Ek(t).

This half-period repetition is not a mathematical curiosity; it helps when inferring period from an energy graph. If an energy curve repeats every 0.40 s, the displacement period is 0.80 s. Missing this factor-of-two mapping is a very common high-level mistake.