How Does a Pendulum Work?

A grandfather clock keeps time with a simple swinging weight. But change one pendulum to two, and the result is completely unpredictable chaos. The same physics leads to both perfect timekeeping and the butterfly effect.

Galileo's Discovery

Around 1582, Galileo Galilei reportedly watched a lamp swinging in the Pisa cathedral and noticed something remarkable: whether the swing was wide or narrow, it seemed to take the same time to complete one back-and-forth cycle. He timed it against his own pulse.

This property — that the period is independent of the amplitude (for small swings) — is called isochronism. It's why pendulums were used in clocks for 300 years.

Galileo's idea for a pendulum clock was not built in his lifetime. Christiaan Huygens created the first working pendulum clock in 1656, reducing timekeeping error from ~15 minutes per day to under 15 seconds.

The Period Formula

For a simple pendulum with a small swing angle, the period (time for one complete swing and return) is:

T = 2π × √(L / g)

Where:

T — period in seconds
L — length of the pendulum in metres
g — gravitational acceleration ≈ 9.81 m/s²

Notice: the mass of the bob does not appear in the formula. A heavy pendulum and a light pendulum of the same length swing at exactly the same rate. This is because gravity accelerates all objects equally regardless of mass.

To get a 1-second half-swing (ticking once per second), you need: L = g/(4π²) ≈ 0.248 m — about 25 cm. A 1-metre pendulum has a full period of about 2 seconds.

Energy in a Pendulum

A pendulum continuously converts between two forms of energy:

Gravitational potential energy (GPE): at the top of each swing, the bob is highest, moving slowest. All energy is stored as GPE = mgh where h is the height above the lowest point.
Kinetic energy (KE): at the bottom of the swing, the bob is lowest, moving fastest. All energy has converted to KE = ½mv².

In a perfect (frictionless) pendulum, this conversion would continue forever. In practice, air resistance and friction at the pivot steal energy, so real pendulums gradually slow and stop unless driven by a mechanism (like the escapement in a clock).

Resonance and Pumping

Every pendulum has a natural frequency — its isochronous period. If you push it in time with its natural frequency, the amplitude grows rapidly. This is resonance.

This is why you pump a swing at the right moment: push during the forward swing, release at the top. Small pushes timed to the natural frequency add energy efficiently. Push at the wrong time and you slow the swing.

Resonance can be destructive: the Tacoma Narrows Bridge collapsed in 1940 when wind vortices drove oscillations at the bridge's natural frequency.

The Double Pendulum and Chaos

Add a second pendulum hanging below the first and everything changes. The double pendulum is one of the simplest examples of a chaotic dynamical system.

Two double pendulums started with positions that differ by less than a millimetre will behave completely differently within a few seconds. This is called sensitive dependence on initial conditions — or the butterfly effect.

The equations of motion are still deterministic (no randomness — the same starting state always produces the same result). But the motion is so sensitive that any tiny uncertainty in the initial state leads to completely different long-term behaviour.

Chaos ≠ randomness: A chaotic system follows exact physical laws. If you knew the initial conditions with perfect precision, you could predict the future perfectly. The problem is that in practice perfect precision is impossible, and any error, however small, grows exponentially.

Try It Yourself

Double Pendulum Simulation — Release up to 120 pendulums with slightly different initial angles and watch chaos unfold in slow motion.

Home experiment: Tie a weight to a string. Try different string lengths and measure the period with a stopwatch. Verify that doubling the length increases the period by √2 ≈ 1.41. Does the mass affect the period? Try it!

🕰️ Open Pendulum →