Oscillations and Simple Harmonic Motion

Oscillations and SHM

Push almost anything slightly away from equilibrium, and the restoring force is proportional to displacement. That single fact makes the sine wave the universal motion of the gently disturbed world.

Simple harmonic motion (SHM) occurs when this restoring force acts:

$F = -kx \quad\Longrightarrow\quad \frac{d^2x}{dt^2} = -\omega^2 x$

where $\omega = \sqrt{k/m}$ is the angular frequency.

General Solution

$x(t) = A\cos(\omega t + \phi)$

Physics Link: SHM is the universal model for small oscillations about any stable equilibrium. Taylor-expand the potential energy to second order and you always get $F \approx -kx$.

Energy and Oscillators

Energy in SHM oscillates between kinetic and potential, keeping total energy constant:

$E = \frac{1}{2}kA^2 = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$

At $x=0$, energy is all kinetic. At $x=\pm A$, energy is all potential.

Common Oscillators

Damped oscillations add a friction term: $\ddot{x} + 2\gamma\dot{x} + \omega_0^{2} x = 0$, producing an exponentially decaying amplitude $Ae^{-\gamma t}$.

Common Pitfall: The period of a mass-spring oscillator does not depend on amplitude. A bigger swing travels farther but moves proportionally faster, and the two effects cancel exactly.