Simple Harmonic Motion — Springs, Pendulums and x = A cos(ωt)

What makes motion 'simple harmonic'

Many systems oscillate — but simple harmonic motion (SHM) is the special, mathematically clean case defined by one rule:

$$F = -kx$$

The restoring force is proportional to the displacement and points back toward equilibrium (the minus sign). Stretch a spring twice as far, get twice the pull-back. This single condition guarantees sinusoidal motion — and an astonishing number of systems obey it, at least approximately.

The spring-mass system

Hang a mass $m$ on a spring of stiffness $k$, pull it down, and release. Hooke's law ($F = -kx$) is exactly the SHM condition, giving:

$$T = 2\pi\sqrt{\frac{m}{k}}$$

Heavier mass → slower oscillation. Stiffer spring → faster. Notice what's missing: the amplitude. Pull it 2 cm or 10 cm — the period is identical (ideally). That amplitude-independence is SHM's signature and why springs make good timekeepers.

The simple pendulum

A simple pendulum — mass on a light string of length $L$ — also qualifies, but only for small angles, where the restoring component of gravity is approximately proportional to displacement:

$$T = 2\pi\sqrt{\frac{L}{g}} \qquad (\text{small angles})$$

Longer string → slower swing. Mass cancels out entirely. Clockmakers exploited this for centuries: to slow a clock, lengthen the pendulum.

Position as a cosine

Solving $F = -kx$ with $F = ma$ yields a differential equation whose solution is:

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

• $A$: amplitude — maximum displacement
• $\omega = 2\pi f$: angular frequency — how fast phase advances
• $\phi$: phase constant — where in the cycle you start

This is why SHM and uniform circular motion share mathematics: cosine is literally the shadow of circular motion projected onto a line.

Energy sloshing back and forth

In SHM, energy continuously trades forms:

• At maximum displacement: all potential ($\tfrac{1}{2}kA^2$), instantaneously at rest.
• At equilibrium: all kinetic ($\tfrac{1}{2}mv_{\max}^2$), moving fastest.
• Total mechanical energy stays $\tfrac{1}{2}kA^2$ (no friction).

This exchange is why the mass moves fastest through the middle and momentarily stops at the turnarounds — the velocity profile is itself sinusoidal.

Small angles and idealizations

Real springs aren't perfectly Hookean at large stretch, real pendulums aren't point masses on massless strings, and air drag always steals energy. The SHM formulas are idealizations — superbly accurate for small oscillations, progressively worse beyond.

Physicists love SHM anyway because it's the universal first approximation: nearly any stable equilibrium, examined closely enough, looks like $F = -kx$. Molecules vibrating, buildings swaying, LC circuits ringing — all SHM at heart.

Variables and units

Common mistakes

Real-world applications

• Pendulum clocks kept the best time for 300 years because a pendulum's period is nearly independent of swing size.

• Car suspensions are tuned spring-mass-damper systems — the spring sets the natural frequency, the shock absorber kills the bounce.

• Seismometers detect earthquakes with a mass on a spring: the ground moves, inertia keeps the mass still, and the relative motion is recorded.