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In simple harmonic motion, kinetic and potential energy continuously exchange, with total energy E = ½mω²A² constant. The energy oscillates between kinetic and potential at twice the frequency of displacement.
Plot kinetic, potential, and total energy vs. time and position. Verify that maximum kinetic energy equals maximum potential energy. Relate amplitude to total energy.
You already know simple harmonic motion: a restoring force proportional to displacement (F = −kx) produces sinusoidal oscillations described by x(t) = A cos(ωt + φ), with angular frequency ω = √(k/m) and period T = 2π/ω. You also know conservation of energy: in the absence of non-conservative forces, the total mechanical energy of a system is constant. Energy analysis in oscillating systems is what you get when you apply conservation of energy to the specific case of SHM — and the result reveals the oscillation from a completely different angle.
Start with the two forms of energy in a spring-mass system. Kinetic energy is KE = ½mv², which is largest when the mass moves fastest. Potential energy (elastic potential energy stored in the spring) is PE = ½kx², which is largest when the spring is most compressed or stretched. Now use conservation of energy: KE + PE = E (constant). Substituting x(t) = A cos(ωt) gives PE = ½k A² cos²(ωt) and, since v = −Aω sin(ωt), KE = ½mA²ω² sin²(ωt). Because ω² = k/m, both terms simplify to ½kA² times a squared trig function, and sin²(ωt) + cos²(ωt) = 1 ensures the total is always E = ½kA². Total energy depends only on the amplitude — double the amplitude, quadruple the energy.
The exchange between kinetic and potential energy is continuous and perfectly timed. At the turning points (x = ±A), the mass is momentarily at rest: all energy is potential (PE = ½kA², KE = 0). At the equilibrium position (x = 0), the spring is relaxed: all energy is kinetic (KE = ½kA² = ½mv²_max, PE = 0). This means the maximum speed v_max = Aω — larger amplitude or higher frequency gives greater maximum speed. Between these extremes, energy sloshes back and forth between kinetic and potential, always summing to E = ½kA².
Notice that while displacement oscillates at frequency ω (one full cycle per period T), the energy functions oscillate at twice the frequency — sin²(ωt) and cos²(ωt) each complete two full cycles per period T, because squaring doubles the frequency. This means energy is at maximum twice per oscillation cycle: kinetic energy peaks when the mass passes through equilibrium going in either direction, and potential energy peaks at both turning points. This factor-of-two relationship between displacement frequency and energy frequency is a precise mathematical result worth internalizing.
Finally, connect this to amplitude as the energy parameter. When friction or damping is present, energy is gradually removed from the system, and amplitude decreases. The amplitude decay directly tracks the energy loss: since E ∝ A², a 50% reduction in energy corresponds to a 29% reduction in amplitude (since 0.71² ≈ 0.5). This relationship between amplitude and energy is essential for analyzing damped oscillators, resonance, and driven systems in the topics that build on this one.