How Does a Pendulum Work?

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.

The Period Formula

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

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.

The Isochronism Myth: Big Swings Run Slow

Galileo's isochronism is only approximately true. The formula T = 2π√(L/g) is derived using the small-angle approximation sin(θ) ≈ θ, which works well for swings under about 15-20°. Push the amplitude wider and the real period grows longer than the formula predicts — a real pendulum clock loses time if you let it swing too wide.

The exact period is given by an infinite series:

where θ₀ is the maximum swing angle in radians. Plug in a few numbers to see how small the effect really is:

• At θ₀ = 10° (0.175 rad), the correction factor is about 1.0019 — the period is only 0.19% longer than the small-angle formula. Practically unnoticeable.
• At θ₀ = 45° the factor grows to roughly 1.04 — a 4% longer period. A clock ticking a 2-second period would drift by nearly a minute every hour.
• At θ₀ = 90° the factor is about 1.18 — the pendulum takes almost a fifth longer per swing than the "ideal" formula suggests.

This is exactly why Huygens' and later precision pendulum clocks were built with narrow swing arcs of just a few degrees: keeping θ₀ small isn't a stylistic choice, it's what makes the isochronism approximation good enough to keep accurate time. It's also why a pendulum clock that has been bumped or over-wound (swinging wider than designed) will visibly run slow.

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.

Measuring g With a Pendulum

Because the period formula rearranges so cleanly — g = 4π²L/T² — a simple pendulum is still one of the most accessible ways to measure the local strength of gravity, and it was the standard method for well over two centuries. Kater's pendulum, invented by British physicist Henry Kater in 1817, refined the idea into a "reversible" pendulum that could swing from either of two knife-edge pivots. By adjusting a sliding weight until the period was identical from both pivots, Kater could cancel out most sources of measurement error and determine g to better than 1 part in 10,000 — without ever needing to precisely measure the pendulum's centre of mass, which is very hard to locate exactly in a physical object.

Kater-style pendulums were used to map variations in g across the Earth's surface throughout the 19th and early 20th centuries — g is not perfectly constant; it varies by about 0.5% between the equator (where it is weaker, partly due to the Earth's bulge and partly due to centrifugal effect from rotation) and the poles (where it is strongest). These gravity surveys helped geologists infer the shape of the Earth and locate dense underground structures such as ore deposits, long before satellites made gravimetry routine.

You can reproduce a rough version of this at home: time 20 full swings of a string pendulum rather than just one, divide the total time by 20 to get an accurate T, then solve g = 4π²L/T². With a 1-metre string and careful timing, a stopwatch measurement can get within a percent or two of 9.81 m/s² — remarkably close for classroom equipment measuring something that once required Kater's precision reversible pendulum.

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.

Try It Yourself

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