Simple Harmonic Motion / Oscillations
Simple harmonic motion is the first place force, energy, and graphical analysis from Units 2 and 4 all get applied to *periodic* motion rather than motion that starts and ends. The throughline is the restoring-force condition (F proportional to -x) that makes springs and pendulums behave identically in structure despite looking nothing alike -- once that condition clicks, period, frequency, and the energy-vs-position graphs in the final lesson follow from it rather than needing to be memorized separately.
What you'll learn
- Identify the conditions for SHM (restoring force proportional to displacement).
- Calculate period and frequency for a mass-spring system and a simple pendulum.
- Relate displacement, velocity, and acceleration throughout an SHM cycle.
- Analyze KE ↔ PE energy transformations over an SHM cycle.
- Interpret position-time, velocity-time, and energy-time graphs for oscillators.
- Explain how amplitude, mass, and spring constant (or length and g) affect period.
- Distinguish SHM from other periodic/oscillatory motion.
1. Conditions for Simple Harmonic Motion
Simple harmonic motion (SHM) occurs whenever the restoring force on an object is directly proportional to its displacement from equilibrium and points back toward equilibrium: \(F = -kx\). The negative sign captures "restoring" — the force always opposes the displacement, pulling the object back.
This specific linear relationship (force proportional to displacement) is what produces smooth, sinusoidal motion and, remarkably, a period that is completely independent of amplitude — a large swing and a small swing of the same pendulum take the same time per cycle.
2. Mass-Spring Systems
For a mass on a spring, the period is \(T = 2\pi\sqrt{\dfrac{m}{k}}\) — independent of amplitude, and independent of \(g\) as well. A vertical spring-mass system oscillates with this same period about its new (lower) equilibrium position; gravity shifts where the equilibrium sits but does not change how quickly the mass oscillates around it.
3. The Simple Pendulum
For a simple pendulum swinging through small angles, \(T = 2\pi\sqrt{\dfrac{L}{g}}\) — notice mass does not appear at all. Two pendulums of very different mass but the same length swing with identical periods (at the same location, same \(g\)).
4. Energy in Simple Harmonic Motion
Total mechanical energy in SHM is constant and can be written two equivalent ways: \(E_{total} = \tfrac{1}{2}kA^2\) (all potential, at maximum displacement \(A\)) or \(E_{total} = \tfrac{1}{2}mv_{max}^2\) (all kinetic, at the equilibrium position where speed is greatest). Between these extremes, energy continuously trades between kinetic and potential while the total stays fixed.
5. Graphs of SHM Quantities
Position, velocity, and acceleration in SHM are all sinusoidal, but out of step with each other: velocity is a quarter-cycle ahead of position (maximum speed occurs where displacement is zero), while acceleration is exactly opposite in phase to position (\(a = -\omega^2 x\), maximum magnitude at maximum displacement). Reading a position-time graph, the amplitude is the peak height and the period is the time for one full repeating cycle; from these, \(\omega = 2\pi/T\) unlocks the maximum speed (\(\omega A\)) and maximum acceleration (\(\omega^2 A\)).
Key equations
- F = -kx — Any SHM restoring-force problem
- T = 2π√(m/k) — Mass-spring system period
- T = 2π√(L/g) — Simple pendulum period (small angle)
- Etotal = 1/2 kA^2 = 1/2 mv_max^2 — Total energy in SHM
- f = 1/T — Converting between period and frequency