i About this experiment — click to learn the physics ▼
What you're looking at
A simple pendulum is a bob of mass m on a rigid, massless rod of length L, free to swing about a fixed pivot. Pull it to an initial angle θ₀ and release: gravity pulls it back toward the bottom, it overshoots, and it oscillates back and forth. Press Swing to animate it, and add damping to model friction and air resistance gradually stealing its energy.
The restoring force
At an angle θ, the component of gravity along the arc pulls the bob back toward equilibrium. Newton's second law for rotation gives the equation of motion:
θ̈ = −(g / L)·sin θ − b·θ̇ with damping b
The simulation solves this with a 4th-order Runge–Kutta integrator, so it stays accurate even at large swing angles where a simple formula would fail.
Small angles: simple harmonic motion
For small swings, sin θ ≈ θ, and the equation becomes that of a simple harmonic oscillator. Its period is famously independent of amplitude and mass, depending only on length and gravity:
This is why a longer pendulum swings more slowly, and why the same pendulum on the Moon (weaker g) takes much longer to complete a swing.
Large angles: the period grows
Once the amplitude is large, the sin θ term matters and the true period is longer than T₀. The exact value involves a complete elliptic integral, which the Period T card computes. At 90° the period is about 18% longer than the small-angle value; near 170° it more than doubles.
Energy exchange
Without damping, total mechanical energy is conserved — it simply shuttles between two forms, shown by the bar above the swing:
- Kinetic energy ½·m·L²·θ̇² is greatest at the bottom, where the bob moves fastest.
- Potential energy m·g·L·(1 − cos θ) is greatest at the turning points, where the bob is highest and momentarily still.
The maximum speed, reached at the lowest point, follows from energy conservation: v_max = √(2·g·L·(1 − cos θ₀)).
What is damping?
Damping is any resistance that gradually drains energy from the motion — for a real pendulum it's mostly air resistance and friction at the pivot. Unlike gravity, which only redirects energy back and forth between kinetic and potential, a damping force removes mechanical energy from the system, turning it into heat. It always pushes against the direction of motion and grows with speed, modelled here as a force proportional to the bob's velocity:
The visible effect: each swing is a little smaller than the last, so the bob traces a shrinking arc inside a slowly decaying envelope, and the energy bar steadily empties until the pendulum settles at rest at the bottom. With the damping control at 0 there's no energy loss and it swings forever; turn it up and the higher the value, the faster the oscillations die away. (Push it very high and the bob barely swings at all — it just eases back to the bottom.) Real clocks fight this loss by feeding in a tiny push each swing to keep the amplitude steady.
Things to try
Change the mass and confirm the period doesn't move at all. Quadruple the length and see the period exactly double. Compare a small 10° swing (T ≈ T₀) with a wide 150° swing on the same pendulum to see the large-angle slowdown. Finally, nudge the damping up a little and watch the swings shrink lap by lap as energy leaks away — then set it high and see the bob barely swing before stopping.