Why Small Swings Keep Time

A pendulum clock keeps time because, for small swings, sin θ is almost exactly θ. Their ratio is trapped between cos θ and 1, and both of those head to 1 as the swing shrinks, so the ratio has nowhere to go but 1. That's the squeeze theorem.

How it works

Computing limits: the squeeze theorem

In one sentence

Squeeze theorem: if $g(x) \le f(x) \le h(x)$ near $a$ (except possibly at $a$ ) and $\lim_{x\to a}g(x) = \lim_{x\to a}h(x) = L$ , then $\lim_{x\to a}f(x) = L$ .

General formula

{\htmlData{sym=2}{\cos\theta}} \le {\htmlData{sym=1}{\frac{\sin\theta}{\theta}}} \le 1 \;\;\Rightarrow\;\; \lim_{{\htmlData{sym=0}{\theta}}\to0}{\htmlData{sym=1}{\frac{\sin\theta}{\theta}}} = 1, \qquad {\htmlData{sym=4}{T}} \approx 2\pi\sqrt{{\htmlData{sym=3}{L}}/g}

If a function is trapped between two others that meet at the same point, it has to go there too.

What the symbols mean

$\theta$
Swing angle
How far the pendulum is from straight down, in radians.
$\frac{\sin\theta}{\theta}$
The ratio
How close the small-angle approximation is: 1 means exact.
$\cos\theta$
Lower bound
Always below the ratio for 0 < |θ| < π/2.
$L$
Length
Pendulum length; the period depends only on L and g for small swings.
$T$
Period
Time for one full swing, the same for every small amplitude.

Gravity pulls the bob back with a force proportional to $\sin\theta$ , not $\theta$ , which makes the exact motion hard to solve.
Geometry on the unit circle shows $\cos\theta \le \frac{\sin\theta}{\theta} \le 1$ for $0 < |\theta| < \frac{\pi}{2}$ .
Both bounds approach 1 as $\theta \to 0$ , so the ratio is squeezed to 1: $\sin\theta \approx \theta$ for small swings.
With $\sin\theta$ replaced by $\theta$ , the period is $2\pi\sqrt{L/g}$ , the same for every small swing. At 10° the error is about half a percent.

Learn the math

Limit PropertiesCalc I · Lesson 7, Limits Computing LimitsCalc I · Lesson 8, Limits Derivatives of Trig FunctionsCalc I · Lesson 18, Derivatives