Pendulum lab
Set the length, release angle and gravity, and compare the exact period with the small-angle formula from class.
Result
L = 1 m · θ₀ = 30° · g = 9.81
The period doesn't care about the mass — a lead bob and a wooden one swing in step. Only length, gravity, and (a little) amplitude matter.
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θ = 21.3° · speed 1.14 m/s · KE 0.65 · PE 0.67 J/kg
How is this calculated?
Using the tool
The exact period comes from the elliptic integral and sits side by side with the small-angle formula — along with exactly how much that formula understates at your amplitude. Scrub the time slider to ride one full swing; the bob and its velocity vector show where the pendulum is and how fast it's moving at every instant.
The physics
A pendulum trades height for speed and back, and gravity is the banker: the bob released at θ₀ falls through a height L(1 − cos θ₀), so its speed at the bottom is √(2gh) — pure energy conservation, exact at any angle. The famous T = 2π√(L/g) is a different kind of statement: it assumes the restoring force is proportional to the angle (sin θ ≈ θ), which is only true for small swings. The honest period multiplies that formula by an elliptic-integral factor that grows with amplitude — about 1.7% long at 30°, 18% at 90°, and unboundedly as the release point approaches straight up, where the pendulum hovers forever. The mass appears in the force and in the inertia and cancels — Galileo's point: heavy and light swing together.
New to the math? Learn the concept: Physics of motion, made visual