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.

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, whose square root is 1, so the period is two pi seconds — the same clean value as our spring.

T=2π1010 m/s2=2π1=2π sT = 2\pi\sqrt{\frac{10}{10\ \text{m}/\text{s}^{2}}} = 2\pi\sqrt{1} = \hlmath{2\pi}\ \text{s}

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).