The Pendulum in Science: From Galileo to Foucault

What a pendulum is to a physicist

To a physicist, a pendulum is any weight hanging from a fixed point that can swing back and forth under gravity. The simplest version, the simple pendulum, pretends all the mass sits in a single point at the end of a string that weighs nothing and never stretches.

Pull the weight to one side and let go, and gravity pulls it back down. It's moving fastest at the bottom, overshoots, climbs the other side, slows, stops for an instant and comes back. That back-and-forth is an oscillation, and the time for one full round trip is called the period.

It's the same object you hold when you use a crystal or metal pendulum on a chain. Only the question changes: physics wants to know how long a swing takes and why, while someone using a pendulum for reflection watches which way it goes.

Galileo and isochronism

Isochronism is the property that fascinated Galileo Galilei: a pendulum takes almost the same time for every swing, whether it moves a lot or a little. His first biographer, Vincenzo Viviani, wrote that young Galileo noticed it in the cathedral of Pisa, watching a lamp sway and timing it against his own pulse.

It's a wonderful scene, but handle it with care. Viviani wrote it down decades later, and the Museo Galileo in Florence notes that historians have questioned it as a possible legend. What is documented is that Galileo described isochronism in a letter in 1602, and that in his Two New Sciences, published in 1638, he explained that the period depends on the length of the string, not on the weight hanging from it.

Galileo believed isochronism held exactly, even for wide swings. We now know it's an excellent approximation only for small swings, and it took another scientist, a few decades later, to work out the fix.

Huygens and the pendulum clock

The pendulum clock was born in the Netherlands, the work of Christiaan Huygens, who finished his first model at the end of 1656. His idea was to marry the pendulum's steady beat to a clock mechanism, so that each swing let the gears advance by one exact step.

A privilege to manufacture the clocks was granted in 1657, and the Hague clockmaker Salomon Coster built the first ones. A Coster clock dated 1657 survives in Leiden. In 1658 Huygens published a short booklet called Horologium describing the invention.

His major treatise, Horologium Oscillatorium, followed in 1673. In it he showed that if the bob travels along a curve called a cycloid instead of a circular arc, the period stays exactly the same no matter how wide the swing. That solved the flaw Galileo had missed, and for close to three centuries pendulum clocks kept time more accurately than anything before them.

The period formula, without the fear

The simple pendulum formula says that, for small swings, the period is T = 2π√(L/g). T is the time for one round trip, in seconds; L is the length from the pivot to the center of the weight, in meters; and g is the acceleration of gravity, about 9.81 meters per second squared at the Earth's surface.

The surprise is what's missing: neither the mass nor the width of the swing shows up. A lead bob and a wooden bob on the same length of string take the same time, as long as the air doesn't slow them very differently.

For wide swings the formula falls short. At an opening of about 20 degrees, the real period is close to 1% longer; at 60 degrees, it's about 7% longer.

• On a string 1 meter long (a little over 3 feet), the period is about 2 seconds: one second out, one second back.
• At 25 centimeters (about 10 inches), the period drops to roughly 1 second.
• To double the period, you have to make the string four times longer, because L sits inside a square root.
• On the Moon, where gravity is about one sixth as strong, the same pendulum would swing roughly two and a half times more slowly.

Foucault: the pendulum that showed the Earth turning

Foucault's pendulum is the experiment that made the Earth's rotation visible. In February 1851, the French physicist Léon Foucault showed a pendulum about 11 meters long in the Meridian Room of the Paris Observatory, and its plane of swing slowly turned as the hours passed.

A few weeks later, in March 1851, he hung a 28-kilogram bob from a 67-meter wire under the dome of the Panthéon in Paris. Visitors could watch its tip trace lines in sand on the floor, each pass pointing in a slightly different direction.

The pendulum isn't changing course: the floor, and the whole building, turns beneath it. In Paris the plane of swing makes a full circle in about 32 hours, roughly 11 degrees an hour. At the poles it would take one day; at the equator it wouldn't turn at all. The original bob is kept at the Musée des Arts et Métiers in Paris, and since 1995 a replica has been swinging under the Panthéon's dome again.

Why a pendulum eventually stops

A real pendulum stops because it loses a little energy on every swing: air drags on it and the pivot soaks up some of the motion. Physicists call this loss damping, and it's why clocks need a spring or a falling weight to give the pendulum a tiny push each cycle.

That push works because of resonance: tiny nudges, timed to the pendulum's natural rhythm, add up and make the swing grow. It's the same thing that happens when you push a kid on a swing at just the right moment instead of off the beat.

What physics says about the pendulum in your hand

Physics also explains why a hand-held pendulum moves even when you're sure you're holding still. Your hand is never perfectly still, and the slight, involuntary movements that come from what you expect or fear, known as the ideomotor effect, travel down the chain. Because a pendulum responds strongly to small pushes timed to its rhythm, those micro-movements turn into a swing you can see.

No mysterious force is needed to make it move, and that takes nothing away from it: the pendulum amplifies what your body is already expressing. That's why it's good for listening to yourself, not for predicting what hasn't happened yet.

Seeing the physics on Paths

On Paths you can watch these ideas at work. The physics engine simulates a pendulum with gravity and friction: you give it a push, and it swings until it settles on an answer on whichever board you've picked.

The your hand engine goes a step further. You hold your phone like a pendulum, or on a computer the pendulum hangs from your mouse pointer, and your own micro-movements move it. Try it with the yes or no board and notice something Galileo would have loved: the longer the imaginary string, the slower and calmer the swing.