PHYSICSLAB

How a pendulum works

A simple pendulum is a mass — the bob — swinging from a fixed pivot. Gravity pulls the bob straight down, but the string constrains it to an arc, so only the component of gravity along that arc acts to restore the bob toward the lowest point. That restoring force is what makes the pendulum oscillate back and forth.

For small swings (below about 15°), the restoring force is very nearly proportional to the displacement, which is the defining condition for simple harmonic motion — and why a small-angle pendulum keeps almost perfect time. As you increase the starting angle the motion stays periodic but is no longer perfectly simple-harmonic, and the period grows slightly. This simulator captures that effect because it integrates the full equation of motion with RK4, rather than assuming the small-angle approximation.

Experiments to try in this simulation

1. Length versus period: run a single pendulum and time ten full swings at 1 m, then at 4 m. The longer pendulum is slower — but quadrupling the length only doubles the period, because the period depends on the square root of length.

2. Mass makes no difference: change the bob's mass while holding length and angle fixed. The period does not change. Gravity accelerates every mass equally, so a heavy bob and a light bob keep the same time.

3. Gravity across the solar system: switch gravity from Earth to Moon to Jupiter. On the Moon the same pendulum swings far more slowly; on Jupiter, faster — a direct look at the inverse-square-root relationship with g.

4. Coupled energy transfer: add a second pendulum, raise the coupling slider, and start only one of them moving. Watch the energy flow back and forth between them — the hallmark of coupled oscillators and resonance.

Key equations

For small angles, the period of a simple pendulum is T = 2π√(L/g), where L is the length and g is the gravitational acceleration. Notice what is absent from that formula: the mass of the bob, and (to first approximation) the amplitude. The angular frequency is ω = √(g/L) = 2π/T.

The bob's energy trades between gravitational potential energy, U = mgL(1 − cos θ), at the top of each swing and kinetic energy at the bottom. In the ideal frictionless case that total energy is conserved; adding damping removes a little energy each cycle, shrinking the amplitude over time — which you can watch directly on the energy graph.

Real-world applications

Pendulums were the heart of accurate timekeeping for nearly 300 years, from Huygens's pendulum clock in 1656 until the quartz era. The same physics drives a metronome and once powered the seismometers that recorded earthquakes. A Foucault pendulum slowly rotates its plane of swing, giving direct visible proof that the Earth turns. Coupled and damped pendulums model resonance and vibration control — the 660-tonne tuned mass damper that steadies the Taipei 101 skyscraper in high winds is, in essence, an enormous pendulum.