← Young Researchers ScienceSimple pendulum40 minutesStd 9–12

Measure Gravity With a String

✨ Six lengths of string, a stopwatch and one graph hand you the strength of Earth's gravity to within a few per cent.

A pendulum with its length and small swing angle marked, beside a graph of period squared in seconds squared against length in metres where six points lie on a straight line through the origin of gradient four pi squared over g. — How it works

🧰 You need

about 1.5 metres of thin string
a heavy nut or a bunch of keys as the bob
tape and a doorframe or shelf edge
a measuring tape
a phone stopwatch

👀 What you will see

Short pendulums swing noticeably faster than long ones. The plot of period squared against length is a straight line through the origin, not a curve. The gradient gives a value for g near 9.8 metres per second squared, usually a little off because of timing error.

🧪 Do it

Tape the string to a fixed edge and tie the bob on, then measure the length from the pivot to the centre of the bob.
Pull the bob aside by less than ten degrees and release it without a push.
Time twenty complete swings, divide by twenty to get the period, and repeat three times.
Shorten the string and repeat for six lengths between about 20 and 120 centimetres.
Tabulate length, mean time for twenty swings, period and period squared.
Plot period squared on the vertical axis against length, draw the best straight line and find its gradient.
Calculate gravity from the gradient using g equals four pi squared divided by the gradient.

💡 Why it happens

For small swings the period of a pendulum is T equals two pi times the square root of length divided by g, where g is the gravitational field strength. Squaring both sides gives T squared equals four pi squared over g, multiplied by length, so plotting T squared against length must give a straight line whose gradient is four pi squared divided by g. The period does not depend on the mass of the bob, nor much on the size of a small swing, so those are the variables you keep controlled. Timing twenty swings instead of one divides your reaction-time error by twenty, which is the single biggest improvement you can make.

🔁 Now try this

Hold the length fixed at one metre and change the bob mass in three steps, to confirm that period really is independent of mass.

🤔 Think about it

A pendulum clock runs slightly slow on a hot day, so what must be happening to its rod?

🧘 Ask the Acharya

“A pendulum clock runs slightly slow on a hot day, so what must be happening to its rod?” — ask him and he will explain it from your own chapters.

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