The Pendulum

The Small-Angle Approximation

why it's simple harmonic

For small swings the sine of the angle is nearly the angle itself, so the restoring force becomes proportional to the swing, the spring condition for SHM.

Example

For small swings the sine of the angle is nearly the angle itself, so the restoring force becomes proportional to the swing — exactly the spring condition for simple harmonic motion. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The restoring force is not quite proportional

A spring's restoring force is exactly proportional to displacement, which is what makes it simple harmonic. The pendulum's restoring force is the weight times the sine of the angle, and the sine is not exactly proportional to the angle.

Frestore=m gsin⁡θF_{\text{restore}} = m\,g\sin\theta

For small angles, sine of the angle is nearly the angle

But for a small angle, measured in radians, the sine of the angle is very close to the angle itself. So the restoring force becomes nearly the weight times the angle — proportional to the swing, just like a spring. That is why a gently swinging pendulum is simple harmonic. Swing it wide and the approximation breaks: the sine falls below the angle and the motion is no longer simple. The angle we used earlier to get a concrete restoring force is itself already too wide for this — it was just a convenient exact angle for a number; real simple harmonic motion needs much gentler swings.

sin⁡θ≈θ  ⇒  Frestore≈m g θ\sin\theta \approx \theta \;\Rightarrow\; F_{\text{restore}} \approx m\,g\,\theta
A small swing is nearly simple harmonicA bob hanging close to the bottom, displaced only a little from straight down.bob

Small swing: arc distance gives proportional force

For a small swing, the arc distance is length times angle. With length fixed, twice the arc distance means twice the angle and therefore twice the restoring force. This is the pendulum's spring-like region.

sθFapprox1 m1101 N2 m152 N3 m3103 N\begin{array}{c|c|c}s & \theta & F_{\text{approx}} \\ \hline 1\ \text{m} & \tfrac{1}{10} & 1\ \text{N} \\ 2\ \text{m} & \tfrac{1}{5} & 2\ \text{N} \\ 3\ \text{m} & \tfrac{3}{10} & 3\ \text{N}\end{array}

Same arc distance, longer string, gentler pull

Hold the sideways arc distance fixed. A longer string makes a smaller angle, so the restoring pull is gentler. That is why the length of the string changes the timing.

LθFapprox5 m152 N10 m1101 N20 m12012 N\begin{array}{c|c|c}L & \theta & F_{\text{approx}} \\ \hline 5\ \text{m} & \tfrac{1}{5} & 2\ \text{N} \\ 10\ \text{m} & \tfrac{1}{10} & 1\ \text{N} \\ 20\ \text{m} & \tfrac{1}{20} & \tfrac{1}{2}\ \text{N}\end{array}
mechanics sin(theta) is approximately theta for small angles, making the restoring force proportional to displacement; wide swings break it.