Preview
Pendulum Lab
Measure period and explore pendulum dynamics — exact nonlinear ODE
In 1583, a bored medical student named Galileo watched a chandelier swing in the Pisa cathedral and timed it against his own pulse. Big swings, small swings — same beat. That observation became the pendulum clock, and eventually a formula every physics student memorizes: T = 2π√(L/g). Length and gravity set the beat; the bob's mass cancels out of the equation entirely, which is why a chandelier and a wrecking ball on equal chains keep equal time. But here is what the mantra leaves out: Galileo's pulse was not a precision instrument, and the tidy formula is a small-angle approximation. It was born from the shortcut sinθ ≈ θ, and shortcuts have jurisdictions. This simulation doesn't take the shortcut — it integrates the true equation α = −(g/L)sinθ every frame — so it can show you exactly where the formula's jurisdiction ends. Swing the pendulum at a polite 15° and the measured period matches the formula to a few milliseconds. Haul it up to 80° and let go: the formula keeps promising 2.837 seconds while the actual pendulum bills you 3.22 — a fourteen-percent overrun the equation cannot see, because sinθ falls ever further behind θ on wide swings and the restoring pull goes soft. In this lab you'll find the border yourself, stretch the period by √2 with a doubled string, slow the same pendulum 2.46-fold by moving it to the Moon, and watch the energy books leak one percent a second through the default damping — physics honest enough to disagree with its own textbook formula.
What you'll be able to do
- Bound the small-angle approximation with the sim's two period rows: at θ₀ = 15° measured and theory agree (≈2.849 vs 2.837 s, a 0.4% gap), at θ₀ = 80° the exact integrator runs ≈3.22 s against the formula's pinned 2.837 s — a 14% miss, because sinθ < θ weakens the restoring pull on wide swings
- Apply T = 2π√(L/g) across both of its levers: doubling length 2 → 4 m stretches the period by √2 (2.837 → 4.012 s), while taking Earth's g = 9.81 to the Moon's 1.62 stretches it by √(9.81/1.62) ≈ 2.46 (2.837 → 6.981 s) and Mars lands between (4.606 s) — length up, slower; gravity up, faster; mass, never
- Track energy through the swing with the engine's own books: KE = ½mL²ω² and PE = mgL(1−cosθ) trade against the release total mgL(1−cosθ₀) = 0.669 J at defaults, with the h row (L(1−cosθ), 0.068 m at the 15° turnaround) tying the PE account to a visible height — and the default damping c = 0.01 bleeding ≈1% of the total per second into heat
Formulas
Make a prediction
Default Earth state: L = 2 m, θ₀ = 15°, g = 9.81 — T theory reads 2.837 s and T measured agrees within a few milliseconds. Now drag the Angle slider to 80° (the Large Angle preset) and let it swing. What do the two period rows do?
No grading here — commit to a guess, then scroll down and test it yourself.
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Check
Did the lab agree with you?
Your prediction
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Answer: T theory stays pinned at 2.837 s while T measured climbs to about 3.22 s — the formula assumed sinθ ≈ θ, and at 80° the real restoring pull is weaker than the shortcut claims, so the true pendulum runs ~14% slow
T = 2π√(L/g) contains no amplitude — so the theory row cannot move, and it doesn't: 2.837 s at any angle. But that formula was derived by replacing sinθ with θ, a substitution that is 1% wrong at 14° and 29% wrong at 80° ((θ−sinθ)/θ = 0.295 at 1.396 rad). The engine here never made the substitution: it integrates α = −(g/L)sinθ exactly, and on wide swings sinθ < θ means the restoring pull is systematically weaker than the small-angle story — the pendulum loiters at the extremes and the measured clock runs long, to (2/π)K(sin40°) = 1.137× the formula, about 3.22 seconds (easing down a hair each cycle as the gentle default damping trims the swing). Option B is the memorized mantra outrunning its jurisdiction: 'amplitude doesn't matter' is a small-angle theorem — OpenStax scopes it verbatim to 'θ less about 15°' — not a law of pendulums; the sim's quiz Q2 makes the same point. Option C has the speed right and the geometry wrong: the bob does move faster through the bottom, but it must also travel a disproportionately longer arc against a relatively weaker pull — the loitering wins.
Quiz (0/3)
A pendulum has period 2s. What is its length?
How does the period change on Mars (g = 3.72 m/s²)?
Why does the small-angle formula fail for large swings?
You can now
- Bound the small-angle approximation with the sim's two period rows: at θ₀ = 15° measured and theory agree (≈2.849 vs 2.837 s, a 0.4% gap), at θ₀ = 80° the exact integrator runs ≈3.22 s against the formula's pinned 2.837 s — a 14% miss, because sinθ < θ weakens the restoring pull on wide swings
- Apply T = 2π√(L/g) across both of its levers: doubling length 2 → 4 m stretches the period by √2 (2.837 → 4.012 s), while taking Earth's g = 9.81 to the Moon's 1.62 stretches it by √(9.81/1.62) ≈ 2.46 (2.837 → 6.981 s) and Mars lands between (4.606 s) — length up, slower; gravity up, faster; mass, never
- Track energy through the swing with the engine's own books: KE = ½mL²ω² and PE = mgL(1−cosθ) trade against the release total mgL(1−cosθ₀) = 0.669 J at defaults, with the h row (L(1−cosθ), 0.068 m at the 15° turnaround) tying the PE account to a visible height — and the default damping c = 0.01 bleeding ≈1% of the total per second into heat