Oscillations and Periodic Motion

Motion That Repeats

Oscillation is motion that repeats around an equilibrium, from pendulums to atoms. Its ideal form is simple harmonic motion (SHM), which occurs when the restoring force is proportional to displacement: Hooke's law, $F = -kx$.

Combining Hooke's law with Newton's second law yields the differential equation $\ddot{x} = --\omega^2 x$, where angular frequency is $\omega = \sqrt{k/m}$. Its solution is a sinusoid, $x(t) = A\cos(\omega t + \phi)$, where $A$ is the maximum displacement.

The Core Law & Real Limits

Common pitfall: Thinking a larger amplitude makes an ideal oscillator take longer per cycle. In SHM, the period is independent of amplitude: a bigger swing feels a proportionally bigger restoring force, covering extra distance in the same time.

Real oscillators experience damping (friction), which shrinks amplitude over time. When an oscillator is driven by an external periodic force matching its natural frequency $\omega$, resonance occurs: energy accumulates and the amplitude grows dramatically.

Resonance powers musical instruments and tuned circuits, but can destroy structures if left unchecked.