The pendulum

A pendulum is a weight hanging on a string and swinging to and fro. One swing there and back is called a period. A one-metre string with a small weight takes about two seconds for it: thirty swings a minute.

A weight on a string. One swing there and back is the period $T$. For $L=1$ m: $T\approx 2$ s, so $60/2=30$ swings a minute.

Period $T$; $L=1\text{m}$ gives $T\approx 2\text{s}$ and $60/2=30$ a minute.

Hang a heavier weight on the same string: the time does not change. Make the swing a little wider (only small swings): the time does not change either. Only the length of the string matters.

Heavier weight: same $T$. Slightly wider swing: same $T$. Only $L$ matters: $T=f(L)$.

Mass $m$, small amplitude $a$: $T\ne f(m)$, $T\ne f(a)$; $T=f(L)$.

Make the string four times shorter, twenty-five centimetres, and the period halves to one second. Make it four times longer, four metres, and the period doubles to four seconds. The period grows like the square root of the length.

$L\to L/4$ gives $T\to T/2$; $L\to 4L$ gives $T\to 2T$. So $T\propto \sqrt{L}$: $0.25$ m gives $1$ s, and $4$ m gives $4$ s.

$T\propto \sqrt{L}$; $T(L/4)=T/2$, $T(4L)=2T$.

The exact rule is: the period equals two pi times the square root of the length over the pull of gravity. The surprise is that a string and a clock can measure gravity. Ten swings of a one-metre string took twenty seconds and a tenth of a second, so one swing took two point zero one seconds, and gravity came out at about nine point eight metres per second squared.

Exactly, $T=2\pi \sqrt{L/g}$, so $g=4{\pi }^{2}L/T^2$. Ten swings with $L=1$ m took $20.1$ s: $T=2.01$ and $g=39.48/4.04\approx 9.8$ m/s${}^2$. A string and a clock measure gravity.

$T=2\pi \sqrt{L/g}$, $T=20.1/10=2.01$, $g=4{\pi }^{2}L/T^2=39.48/4.04\approx 9.8$.