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Simple harmonic motion (SHM) occurs when a restoring force is proportional to displacement from equilibrium: F = −kx. The resulting motion is sinusoidal: x(t) = A cos(ωt + φ), where A is amplitude, ω = √(k/m) is angular frequency, and φ is phase. Period T = 2π/ω depends only on system parameters (mass, spring constant), not amplitude. SHM is the mathematical archetype for all oscillatory behavior.
Derive the equations by applying F = ma to F = −kx: m(d²x/dt²) = −kx, then verify that x = A cos(ωt) is a solution. Connect SHM to circular motion: projecting uniform circular motion onto one axis produces SHM.
Simple harmonic motion begins with a single law about forces: the restoring force on an object is proportional to its displacement from equilibrium and directed back toward it, F = −kx. The negative sign is essential — it means the force always pushes opposite to the displacement, so an object displaced to the right is pushed left, and one displaced to the left is pushed right. Any system satisfying this force law will oscillate.
To find *how* it oscillates, apply Newton's second law: F = ma becomes m(d²x/dt²) = −kx. Rearranging gives d²x/dt² = −(k/m)x. This is a second-order ODE saying: the acceleration is proportional to the position, with a negative constant. From your work on differential equations, you know that functions whose second derivative is a negative constant multiple of themselves are sines and cosines. Substituting x(t) = A cos(ωt + φ) and differentiating twice yields d²x/dt² = −ω²A cos(ωt + φ) = −ω²x, which matches if ω = √(k/m). The general solution is x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase (set by initial conditions).
The period T = 2π/ω = 2π√(m/k) is one of the most counterintuitive results in introductory physics: it does not depend on amplitude. A pendulum swung through a small arc takes the same time to complete a cycle as one swung through a larger arc (for small angles). A spring-mass system with a 1 cm amplitude oscillates at the same frequency as one with a 10 cm amplitude, as long as mass and spring constant are the same. This is only true because the restoring force grows in exact proportion to displacement — if you go twice as far, you are pulled back twice as hard, so the system compensates perfectly.
Be careful with the two measures of frequency. Angular frequency ω is in radians per second and appears naturally in the equations of motion. Ordinary frequency f = ω/(2π) is in Hz (cycles per second) and is what you would measure with a stopwatch. The period T = 1/f = 2π/ω connects all three. A common error is plugging ω directly into a formula expecting f, or reading "2π rad/s" as "2π oscillations per second" — remember that one full oscillation is 2π radians, so ω = 2π means f = 1 Hz.
SHM is the archetype for oscillatory behavior across all of physics: electromagnetic oscillations in circuits (LC circuits obey q'' = −q/LC), quantum mechanical ground states, molecular vibrations, and acoustic resonance all reduce to the same equation under the right conditions. Whenever you see a restoring force that is linear in displacement, you are looking at SHM, and you immediately know the solution is sinusoidal with ω determined by the ratio of the restoring strength to the inertia of the system.