The Pendulum

Pendulum Period

timing the swing

In the small-angle approximation the period depends only on length and gravity, amplitude-independent, but only approximately, unlike the spring.

Example

In the small-angle approximation the period depends only on the length and gravity, T = 2 pi root L over g — amplitude-independent, but only approximately, unlike the spring. 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 small-swing period

In the small-angle approximation the period — the time for one full swing over and back — depends only on the string length and gravity, not on the mass or (to this approximation) the amplitude.

T=2πLgT = 2\pi\sqrt{\frac{L}{g}}
A pendulum of length LA bob on a string of length L from a fixed support.bob

A worked value

With a length of 10 metres and gravity 10, the length over gravity is 1 second squared, whose square root is 1 second, so the period is two pi seconds — the same clean value as our spring.

T=2π10 m10 m/s2=2π1 s2=2π sT = 2\pi\sqrt{\frac{10\ \text{m}}{10\ \text{m}/\text{s}^{2}}} = 2\pi\sqrt{1\ \text{s}^{2}} = \hlmath{2\pi}\ \text{s}

Same gravity, longer string, longer period

Hold gravity fixed. The length appears inside the square root, so increasing length makes the period longer, but not in a one-to-one way.

LL/gT10 m1 s22π s40 m4 s24π s90 m9 s26π s\begin{array}{c|c|c}L & L/g & T \\ \hline 10\ \text{m} & 1\ \text{s}^{2} & 2\pi\ \text{s} \\ 40\ \text{m} & 4\ \text{s}^{2} & 4\pi\ \text{s} \\ 90\ \text{m} & 9\ \text{s}^{2} & 6\pi\ \text{s}\end{array}

Same length, stronger gravity, shorter period

Hold length fixed. Stronger gravity makes length over gravity smaller, so the square root is smaller and the swing returns sooner. These are training gravities for exact comparison.

gL/gT10 m/s21 s22π s40 m/s214 s2π s90 m/s219 s223π s\begin{array}{c|c|c}g & L/g & T \\ \hline 10\ \text{m}/\text{s}^{2} & 1\ \text{s}^{2} & 2\pi\ \text{s} \\ 40\ \text{m}/\text{s}^{2} & \tfrac{1}{4}\ \text{s}^{2} & \pi\ \text{s} \\ 90\ \text{m}/\text{s}^{2} & \tfrac{1}{9}\ \text{s}^{2} & \tfrac{2}{3}\pi\ \text{s}\end{array}

Exact for the spring, approximate for the pendulum

One honest difference: the spring's period truly does not depend on amplitude — that is exact. The pendulum's amplitude-independence is only the small-angle approximation. Swing a real pendulum wide and each swing actually takes a little longer, because the sine falls below the angle. The clean formula is a small-swing idealization, like our clean numbers throughout.

spring: exact≠pendulum: small-angle only\text{spring: exact} \quad\ne\quad \text{pendulum: small-angle only}
mechanics With L = 10 m and g = 10 the period is a clean 2 pi seconds; the honest contrast is exact (spring) vs small-angle-only (pendulum).