Pendulum

2 min read#oscillations

A swinging bob that looks like a simple harmonic oscillator only when it barely moves — and reveals its nonlinear soul at large angles.

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Pendulum

A pendulum is a bob of mass m swinging on a rod of length L under gravity. Resolving the gravitational pull along the arc gives an equation of motion for the angle \theta from vertical:

\ddot\theta = -\frac{g}{L}\sin\theta.

That single \sin\theta is the whole story. It makes the pendulum nonlinear, and it is the difference between a textbook idealization and the real, slightly unruly thing on the wall.

The small-angle approximation

For small swings, \sin\theta \approx \theta, and the equation collapses to that of a Simple Harmonic Oscillator:

\ddot\theta \approx -\frac{g}{L}\theta, \qquad T \approx 2\pi\sqrt{\frac{L}{g}}.

In this regime the period is independent of amplitude — Galileo's famous observation. But the approximation quietly fails as the swing grows: a pendulum released from near-horizontal takes noticeably longer per swing than the formula predicts, because \sin\theta < \theta weakens the restoring pull at large angles.

Released from 2.4 rad — far outside the small-angle regime. Watch it linger near the top of each swing.

The full nonlinear swing

Across its whole range the pendulum has a far richer Phase Portrait than the SHO's tidy ellipses:

  • For low energy it oscillates, tracing closed loops around the stable hanging Fixed Point at \theta = 0.
  • The upright position \theta = \pi is an unstable fixed point — a saddle.
  • Give it enough energy and it stops swinging back and forth and instead rotates over the top, circulating forever.

The boundary between swinging and spinning is a special trajectory called the separatrix. Push the idea further — pin the pivot to a motor, or chain two together into a Double Pendulum — and the pendulum becomes a gateway to Chaos.

Why does a wide-swinging pendulum have a longer period than the small-angle formula T = 2π√(L/g) predicts?

See also