From simple harmonic motion to the butterfly effect — explore pendulums from first principles using PHYSIX's spring-mass system.
A simple pendulum oscillates with a period that depends only on its length and local gravity — not on the mass of the bob or the amplitude (for small angles). This is one of the most important results in classical mechanics.
PHYSIX implements pendulums using a spring force with a very high stiffness (k ≈ 40–80 N/m) connecting the bob to a pinned anchor. For small angles, this is equivalent to a rigid rod pendulum.
Click 🗑 CLEAR, then click 🏔 Surface in the topbar to move to a Earth-surface viewport.
Drag a 🔵 Sphere to the top of the canvas. In Properties, set mass = 1×10³⁰ and check Pinned. This is your pivot. Rename it "Pivot".
Drag another 🔵 Sphere below the pivot. Set mass = 1 kg, radius = 0.16 m. Position it exactly 2 m below the pivot on the y-axis.
From the GRV force tab, drag Uniform g onto the bob. Set gy = −9.80665 m/s².
From the MCH force tab, drag Spring onto the bob. Set target = Pivot, k = 40 N/m, L₀ = 2 m, damping = 0.1.
Drag the bob 0.3 m to the right. Press ▶ PLAY. Measure the period with the simulation clock — compare it to T = 2π√(2/9.8) ≈ 2.84 s.
A double pendulum is two pendulums connected end-to-end. For small angles it behaves predictably, but at larger amplitudes it becomes one of the simplest examples of deterministic chaos in nature.
In Presets → Oscillations, click 🌀 Double Pendulum. PHYSIX loads two spring-connected bobs with a shared pivot.
At small angles the motion looks nearly periodic. Let it run for 10–20 seconds and watch the trail pattern build up.
Reset and drag the outer bob to a near-horizontal position. Press Play. The motion quickly becomes irregular — this is chaos.
Chaos means that tiny differences in starting position lead to wildly different outcomes over time. This is not randomness — the system is fully deterministic — but it is unpredictable in practice.
The rate at which nearby trajectories diverge is measured by the Lyapunov exponent λ. For a double pendulum at large amplitude, λ > 0, confirming chaotic behaviour.