The Real Pendulum

Small swings look like a perfect sine wave, but big swings run slower, and a hard enough push sends the pendulum over the top. The phase portrait shows every case at once.

θ 1.00 radω 0.00 rad/s

t = 0.0

How it works

Nonlinear ODEs and phase portraits

In one sentence

A pendulum of length $L$ with damping $b$ obeys $\theta'' + b\theta' + (g/L)\sin\theta = 0$ . For small angles $\sin\theta \approx \theta$ and the motion is simple harmonic, but large swings take longer, and above the separatrix it spins over the top.

General formula

{\htmlData{sym=0}{\theta}}'' + {\htmlData{sym=1}{b}}\,{\htmlData{sym=0}{\theta}}' + \frac{\htmlData{sym=2}{{g}}}{\htmlData{sym=3}{{L}}}{\htmlData{sym=4}{\sin\theta}} = 0

$\theta''$ is angular acceleration, $b\theta'$ is friction, and $(g/L)\sin\theta$ is gravity's restoring pull: strongest at $90^\circ$ , zero at the bottom and at the very top.

What the symbols mean

$\theta$
Angle
How far the pendulum is from hanging straight down, in radians.
$b$
Damping
Air resistance and friction at the pivot. It makes the swings shrink over time.
$g$
Gravity
Gravitational acceleration, about 9.8 m/s² on Earth.
$L$
Length
How long the pendulum is. Longer pendulums swing more slowly.
$\sin\theta$
Restoring pull
Gravity pulling the bob back to the bottom. It is sin θ, not θ, which makes wide swings slower than small ones.

Rewrite it as a system: $\theta' = \omega$ and $\omega' = -\tfrac{g}{L}\sin\theta - b\,\omega$ .
Near the bottom $\sin\theta \approx \theta$ , which gives simple harmonic motion with period $2\pi\sqrt{L/g}$ .
The separatrix divides rocking back and forth from spinning over the top.

Learn the math

Taylor SeriesCalc II · Lesson 39, Power Series and Taylor Series Applications of SeriesCalc II · Lesson 40, Power Series and Taylor Series The Phase PlaneDiff Eq · Lesson 32, Systems of Differential Equations