The Nonlinear Pendulum — When the Textbook Formula Breaks

Source

Simulation: Nonlinear pendulum "ghost race": the true sin θ dynamics (RK4-integrated) swings alongside a ghost obeying the small-angle formula, plus a T(θ₀)/T₀ elliptic curve with a slider-bound static marker.
Concept: T = 2π√(L/g) is an approximation that ignores amplitude. The exact period grows with release angle — only ~0.05% high at 5°, +7.3% at 60°, +18% at 90° — following the complete elliptic integral K.
Serves: Limits of the SHM idealization: reading the textbook period as the θ₀ → 0 limit; judging where linearization is trustworthy.
Inquiry: Predict whether a wider swing takes more, less, or the same time; release at 10° then 90° and watch the ghost separate; count laps until the ghost leads by a full swing.
Misconception: "The period is independent of amplitude" — true only as θ₀ → 0. Encoded as prediction choice B; rebuttal shown only when chosen.
Verified anchors: L = 1 m, g = 9.81 m/s²: T₀ = 2.0061 s; 5° → 2.0071 s (×1.0005); 60° → 2.1529 s (×1.0732); 90° → 2.3678 s (×1.1803). Computed here by the same AGM iteration used for every displayed number.

Pedagogy principles applied

ID	Principle	How it shows up here
G4b	Predict first, optional & non-blocking	Sim loads held at release (one click to run); prediction offered in the sidebar, never gates Play, sliders, or views.
H1/H2	Misconception structural, not standing text	"Exactly the same time" is prediction choice B; the targeted rebuttal appears only if selected. The naive model exists on screen only as the labeled ⚠ ghost, visibly falling out of step.
CF	Counterfactual mode	Both models implemented exactly — RK4 of θ̈ = −(g/L)sin θ vs T₀ = 2π√(L/g) — behind ⚠/✓ mode buttons; teaching by contrast.
C1/C2	Readout discipline	Two badges only (period; lap + phase gap). The T(θ₀) curve and the numeric table open on demand.
E1/J2	Animation honesty	True nonlinear integration in real time; view scales to current L (1 m bar = px/m); chart axes fixed over θ₀ ∈ [5°, 90°].
D6/D7	Complexity revealed by explicit controls	📈 curve, "why it breaks" panel, and the elliptic-integral formalism each behind an always-available toggle.
A3/A4	Formal tools off by default	K(k), the AGM identity, and the power series live behind the ⚛︎ The Physics button.
INV	Invariant made visible	The plotted T/T₀ curve is independent of L; step ⑤ has students verify √L scaling leaves the % lag per lap unchanged.
AUD	Machine-readable audit hook	window.__audit.at(θ₀) returns periodRatio via the identical AGM function that drives all displays.

The Nonlinear Pendulum

Teach me Hide Text

Speed

Held at θ₀ — press ▶ Play to release both pendulums from rest

Guided Inquiry

1 · Meet the ghost race

Let’s meet the two racers. The solid bob follows the true dynamics, θ̈ = −(g/L)sin θ. The dashed ⚠ ghost follows the textbook formula T₀ = 2π√(L/g), which is amplitude-blind. Both release from rest at the same θ₀ (L = 2 m), and they’re held at release until you press ▶ Play.

2 · Predict — wide vs gentle

Here’s your first prediction. Same pendulum, two releases from rest: a gentle 10° and a wide 90°. What do you think — one complete swing (out and back) at 90° takes…

3 · Watch at θ₀ = 10°

Let’s start gently. Press ▶ Play if needed. The ⚠ ghost hugs the real bob — it falls behind by only ~0.2% per lap. Near θ₀ → 0, the textbook formula is trustworthy.

4 · Race at θ₀ = 90°

Now for the big race. Both bobs are re-released at 90°. Count the laps until the gap badge shows the ghost leading by a full swing (it turns red). Each real lap costs +18%, so the formula’s lead keeps compounding.

5 · Read the T(θ₀) curve

Let’s look at every amplitude at once. The 📈 T(θ₀) curve is open: amber is the formula’s period, and cyan is the exact one. Where do you think the two agree?

Read it: the flat amber line is the formula’s claim (same T for every amplitude). The cyan curve is the exact period — they touch only as θ₀ → 0, and by 90° the real period is already +18%.

6 · What the formula ignores

One last question. Now change the length L while θ₀ stays fixed. What do you think happens to the ghost’s lead per lap?

Nothing changes at fixed θ₀: both periods stretch by √L together — the ghost’s lead per lap doesn’t change. Amplitude, not length, is what T₀ forgets.

Listen

Controls

▾

release θ₀5°

5°90°

length L2.00 m

1 m3 m

Second pendulum

length L₂1.00 m

1 m3 m

Overlays

Readouts

▾

Period

T = ?

release, then ▶ Play

Pendulum 1

T₁ = —

L₁ = —

Pendulum 2

T₂ = —

L₂ = —

Race

Exact laps—

Ghost leadpredict · then ▶

Formal tools — where T₀ comes from, and what replaces it

1 · The approximation the textbook makes

Dropping everything after the first term of sin θ removes θ₀ from the answer entirely.

2 · The true equation of motion (integrated live here, RK4)

3 · Exact period — complete elliptic integral

4 · How this page computes K — AGM iteration

K(0) = π/2 recovers T = T₀; K grows with θ₀ (scope here: up to 90°). The AGM converges to machine precision in a handful of steps.

The Physics

The nonlinear pendulum · what this sim shows, the physics behind it, and every control

At a glance

Textbooks give a pendulum's period as T = 2π√(L/g), and that formula never mentions how far the pendulum swings. This sim races a real pendulum against the formula, so you can see where the formula holds, where it fails, and by how much.

What you'll learn

Where T₀ = 2π√(L/g) comes from, and the approximation it rests on.
How the period of a real pendulum depends on its release angle θ₀, and why.
How to read the correction factor T/T₀ off the curve, and turn it into a period.
Why the length sets the time scale, while only the release angle sets the size of the formula's error.

Both bobs start from rest at the same angle θ₀. Solid: the real pendulum. Dashed: the formula.

What's on screen

✓ Exact bob (solid cyan): the real pendulum.
⚠ Small-angle ghost (dashed amber): swings as θ₀ cos(2πt/T₀), the motion the approximation sin θ ≈ θ predicts.
Readouts → Race card: laps completed, and how far ahead the ghost is, in swings.
📈 T(θ₀) curve (opens next to the stage): the ratio T/T₀ against release angle. Amber sits at 1 (the formula); cyan is the real pendulum.

See itRelease both bobs from 30°. Watch for: the dashed ghost slowly pulling ahead of the real bob.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
release θ₀	The angle both bobs start from, measured from the vertical.	5°–90° opens at 5°
length L	The string length. It sets the time scale of every swing.	1–3 m opens at 2 m
Second pendulum	Adds a green pendulum with its own length L₂, so two lengths race side by side. The ghost is hidden while it is on.	L₂: 1–3 m
Small angle (Overlays)	Shows or hides the dashed formula ghost.	on / off
📈 T(θ₀) curve (Overlays)	Opens the plot of T/T₀ against release angle, with a marker at your θ₀.	on / off
Readouts	The Period card (the real bob's T) and the Race card (laps, and the ghost's lead in swings).	panel
▶ Play / ⏸ Pause	Releases both bobs from rest, or freezes them so you can compare positions.	top bar
Speed	Plays the motion faster or slower on screen. The physics and the period stay the same.	0.25×–4×
↻ Reset	Restarts the mode you are in from its beginning.	top bar

Fixed: g = 9.81 m/s², no friction or air drag.

The textbook formula

Gravity's pull along the swing is proportional to sin θ. The solid bob follows this equation:

The true equation of motion.

For small angles, with θ in radians, sin θ ≈ θ (at 10° they differ by 0.5%). With that swap the motion becomes simple harmonic, and its period no longer depends on θ₀:

The textbook shortcut.

Key idea. The formula says every swing takes the same time, whatever the release angle. That is only as true as sin θ ≈ θ.

See itRelease at a gentle 10°. Watch for: the ghost staying with the real bob lap after lap (at first it hides right behind it).

Why the real pendulum runs late

On either side of the vertical, |sin θ| < |θ|, so the real pull back is weaker than the formula assumes.
Energy conservation makes this exact: at every angle the real bob moves more slowly than the ghost, so every real swing takes longer.
The wider the release, the bigger the gap, and the longer the swing.

Real pull (cyan) against the formula's pull (amber).

release θ₀	real swing is longer by	ghost leads by one full swing after
10°	0.2%	≈ 524 laps
30°	1.7%	≈ 57 laps
90°	18%	≈ 5.5 laps

One lap, or swing, is out and back: one full period.

See itRace both bobs from 90°. Watch for: the Ghost lead on the Race card growing every lap.

The exact period

The real period has no formula in elementary functions. It equals T₀ times a correction factor that depends only on θ₀:

K is the complete elliptic integral of the first kind: a standard special function, like sin or log, that the sim evaluates for you.

θ₀ L

These sliders move the sim too.

Released close to 180° (almost upside down), the period grows without limit. The sim stops at 90°.

See itOpen the T/T₀ curve at your θ₀. Watch for: the cyan (real) curve rising above the flat amber (formula) line.

What the length changes

In T = T₀ × (factor set by θ₀), the length appears only inside T₀ = 2π√(L/g).

Length sets the time scale. T₀ grows as √L, so going from 1 m to 3 m stretches every period by √3 ≈ 1.73.
The release angle sets the error. T/T₀ has no L in it, so the ghost's lead, counted in swings, is the same at any length.

See itRace from 60° with L = 1 m, then L = 3 m. Watch for: each run stops after 2 laps, and the card shows both Ghost leads side by side.

See itOpen the T/T₀ curve, then drag the length slider. Watch for: the marker on the curve does not move, whatever the length.

Where it matters

Small-angle estimate of the extra time (θ₀ in radians).

Pendulum clocks. A clock is adjusted for its usual swing, but if the swing size drifts, its rate drifts too: from 2.0° to 2.2° is about 1.4 s a day. This is called circular error, and it is why clock swings are kept small and steady.
Huygens, who built the first pendulum clock, added curved guides (cycloidal cheeks) that shorten the string on wide swings. On a cycloid the period does not depend on the amplitude at all.
Measuring g. Timing a pendulum gives g, but a 10° swing already makes the answer about 0.4% too small unless you correct for it.

Go deeper: how the sim computes K(k)

The same factor three ways: integral, AGM, series.

The sim uses the arithmetic–geometric mean (AGM): repeatedly replace two numbers by their average and geometric mean. It converges in a few steps.
The series shows the first correction is θ₀²/16: double a small amplitude and the extra time grows four times.

Summary

In one line. T₀ = 2π√(L/g) is the small-angle limit of the true period. Every real swing, released from rest, takes longer than it, by a fraction set only by the release angle.

Key points

T₀ = 2π√(L/g) comes from replacing sin θ by θ.
Since sin θ < θ, every real swing is longer than T₀, and more so at wider angles.
The real period is T₀ times a factor that depends only on θ₀ (read T/T₀ off the curve).
Length sets the time scale; the release angle alone sets the error.

Common mix-ups

"The period doesn't depend on amplitude." Only for very small swings.
"Wide swings are faster, because the bob moves faster." It does move faster. With a pull proportional to θ, the extra speed would exactly cover the longer path. The real pull, sin θ, grows more slowly, so the speed falls short and the swing takes longer.
"A longer pendulum makes the error worse." The fractional error does not depend on L.

Things to try on your own

Set θ₀ to 45° and count the laps until the Ghost lead reaches one full swing. Compare with the table in section 04.
Turn on Second pendulum, give it L₂ = 1 m against L = 3 m, and compare the two periods.