Oscillations and Simple Harmonic Motion

Push almost anything slightly away from equilibrium — a pendulum, a guitar string, an atom in a crystal — and the restoring force is, to first approximation, proportional to the displacement. That single fact makes the sine wave the universal motion of the gently disturbed world.

Simple harmonic motion (SHM) occurs when the restoring force is proportional to displacement:

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

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

Solution

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

Energy in SHM — Total energy oscillates between kinetic and potential:

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

At $x=0$: all kinetic. At $x=\pm A$: all potential.

Common oscillators

• Mass-spring: $T = 2\pi\sqrt{m/k}$
• Simple pendulum: $T = 2\pi\sqrt{L/g}$ (for small angles $\theta \ll 1$)

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

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

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 — that is what makes harmonic oscillators good clocks.