Time a pendulum

The question

Does a pendulum's swing time depend on how heavy it is, how far it swings, or how long it is? And can I use it to measure the strength of gravity, $g$?

Learn just enough first

Look up the pendulum formula, $T = 2\pi\sqrt{L/g}$, and what "small swings" means for it
Read how Galileo noticed that a swinging lamp kept time

Kit

About 1.2 m of string or thread that doesn't stretch
A small heavy weight: a big nut, a bunch of washers or a padlock
Tape to fix the string to the edge of a table or door frame
Tape measure and your phone's stopwatch

My prediction

Which will change the swing time: a heavier weight, a bigger swing, or a longer string? Write down your guess for each before you start.

Method

Hang the weight so it swings freely. Measure the length $L$ from the fixing point to the middle of the weight.
Pull it aside a little — less than about 15° — and let go without pushing.
Time 10 full swings (there and back is one swing) and divide by 10. Timing many swings makes your reaction time matter much less.
Repeat each timing three times and take the average.
Try five lengths, from about 0.2 m to 1 m. Then, at one length, try a heavier weight and a bigger swing.

Data

Length $L$ (m)	Time for 10 swings (s)	Period $T$ (s)	$T^2$ (s²)
0.20
0.40
0.60
0.80
1.00

Result

Squaring the formula gives a straight line:

$$ T^2 = \frac{4\pi^2}{g}\,L $$

Plot $T^2$ against $L$ (the notebook draws the graph from your table). The slope is $4\pi^2/g$, so

$$ g = \frac{4\pi^2}{\text{slope}} $$

The accepted value in the UK is about 9.81 m/s².

Explain it to a friend

A pendulum's swing time depends only on its length (and on gravity), not on how heavy it is or — for small swings — how far it goes. That's why pendulums made good clocks, and why a longer pendulum ticks more slowly.

New questions

What happens to the period for really big swings, say 60°?
How would this pendulum behave on the Moon, where $g$ is about 1.6 m/s²?