Why the World Swings — Pendulums and Simple Harmonic Motion

There are many things in the world that swing. Playground swings, pendulum clocks, guitar strings, ocean waves, the beating heart, and the quartz crystal that keeps time inside your smartphone. Though they differ wildly in scale and form, they all dutifully repeat a back-and-forth motion. Why is nature so fond of swinging?

Let us look — without diving too deep into calculation — at where this swinging comes from.

The heart of swinging: a "pull-back force"

Swinging things share a common feature. They have a place where they can rest — an equilibrium position — and whenever they are displaced from it, they always try to return. This "force that tries to restore" is called the restoring force.

Picture a spring. Stretch it beyond its natural length and it tries to contract; compress it and it tries to expand. Pull a swing back and it returns forward; lift a pendulum to the side and it returns to straight down. In every case, "the more it is displaced, the more it is pushed back." Moreover, while the displacement is small, the restoring force is roughly proportional to the size of the displacement. Written as a formula, for a position $x$ the force is $-kx$. The leading minus sign expresses the idea of "pulling back in the opposite direction."

But here lies the seed of oscillation. Once drawn back by the restoring force, the object does not come to a neat stop at the equilibrium. It has built up momentum, so it overshoots. Then it is pulled back from the other side. Overshoot, get pulled back, overshoot again — this endless tug-of-war is the true nature of swinging.

Swinging, summed up in one line

Let us write this "overshoot and pull-back" in Newton's language of motion. Force produces acceleration ($F=ma$), so if the restoring force is $-kx$, the acceleration is proportional to it. Tidied up, swinging fits into a single line:

$$\frac{d^2 x}{dt^2} = -\omega^2 x$$

The left side is "the rate of change of the rate of change of position," that is, the acceleration. The right side is "a constant times the current position, with the opposite sign." Read aloud, it makes just one claim: the farther from equilibrium, the more strongly it is pulled back toward the center. Here $\omega$ is the number that sets the pace of the swing; it grows larger the stiffer the spring and the shorter the pendulum.

An equation like this — one that ties an unknown function (here the position $x(t)$) to its own derivatives (its rates of change) — is called a differential equation. In a single line it states how "the present state" and "the way it is changing" are related, and far beyond swinging it is the common language nature uses to describe anything that changes over time — which is exactly why it is the star of this series.

If you actually work it out, the solutions turn out to be the familiar $\sin$ and $\cos$: a wave that glides back and forth, never stopping, forever keeping the same rhythm — that is simple harmonic motion. The detailed methods of solving it are left to the elementary chapter on the harmonic oscillator; what is worth savoring here is rather the fact that such a rich variety of swinging arises from a single local rule of one line.

What Galileo found in church

One of the first to take the wonder of swinging seriously was Galileo Galilei. As the story goes, in his youth he watched a lamp hanging from the ceiling of the Cathedral of Pisa sway gently in the breeze. Timing it against his own pulse, he noticed something: whether the swing is wide or narrow, the time for one round trip barely changes.

This is counterintuitive. A wider swing covers a longer path back and forth, so you might expect it to take longer. But a wider swing also moves faster, and the two effects very nearly cancel out. This property is called isochronism. A rhythm of swinging that stays nearly constant regardless of amplitude is extraordinarily convenient as a "device for keeping time."

Later this discovery bore fruit as the pendulum clock, and humanity held a steady measure of time for the first time. The study of swinging led directly to the mastery of accurate time. It is also striking that the very same mechanism that produces swinging creates motion that neither decays nor grows but repeats endlessly — in contrast to the exponential growth and decay explored in a companion reading.

The same mechanism lurks behind every swing

Once you notice the skeleton of "restoring force + overshoot," swings all over the world begin to look alike.

A guitar string, when plucked, tries to return to its original position, overshoots, and returns again — and so it sounds. Air molecules too, when pushed, try to return and vibrate, reaching the ear as sound waves. Electrons inside atoms and the lattice inside crystals likewise quiver, pulled back toward equilibrium whenever displaced. Even electrical circuits: a coil and a capacitor make electric charge "overshoot and get pulled back," producing a swing — that is, alternating current.

In other words, wherever there is a stable equilibrium, swinging usually lurks. The world's fondness for swinging is no whim but the simple, inevitable result of "a force that restores + momentum that cannot stop." That is exactly why a single equation of simple harmonic motion can bind together so many phenomena.

Closing — carrying the one line of swinging

Behind the commonplace phenomenon of swinging live a humble restoring force that "pulls back by the amount of the displacement" and a momentum that "cannot stop and passes through." Write the tug-of-war between the two as a single line of differential equation, and the pendulum, the string, and alternating current can all be explained at once. The reason the world is so fond of swinging lies, in the end, right there.

Once you are at home with the differential-equation idea that "the present state determines the next step," the next thing to try is actually solving this swing as a formula. At the elementary level, we take up simple harmonic motion and its relative the harmonic oscillator, and savor the rhythm of swinging and the behavior of damping and resonance together with the handling of formulas.