Measuring Gravity with Nothing More Than a Pendulum

A simple school experiment can become a surprisingly sophisticated investigation into gravity, modelling, uncertainty and the limits of approximation.

A pendulum is one of those pieces of physics apparatus that can look almost too simple to be interesting.

A piece of string.
A small mass.
Something to hang it from.

Pull it to one side, let go, and it swings backwards and forwards.

At GCSE, the experiment is often reduced to:

Measure the time for ten oscillations, divide by ten, and calculate the period.

There is nothing wrong with that as a starting point. But the pendulum deserves much more attention than this.

With a ruler, a stopwatch, some string and a suitable bob, we can investigate gravitational acceleration, experimental uncertainty, mathematical models, damping, conservation of energy and even the point at which one of the standard approximations used in physics begins to break down.

In fact, I think the pendulum makes an excellent example of how the same experiment can grow with the student.

A GCSE student can measure a period.

An A-level student can determine g.

A more advanced student can ask whether the equation they are using is actually true.

And that last question is where experimental physics becomes particularly interesting.

The deceptively simple pendulum

For a simple pendulum, the familiar relationship is:

T = 2 x pi x sqrt(L/g)

where:

T = period of the pendulum in seconds
L = length of the pendulum in metres
g = gravitational field strength in m/s^2

Rearranging gives:

g = 4 x pi^2 x L / T^2

So, in principle, determining the acceleration due to gravity seems remarkably easy.

Measure L.

Measure T.

Put the values into the equation.

We should obtain something close to:

g = 9.81 m/s^2

But that immediately raises a much more interesting question.

How close can we actually get?

And what determines whether our result is good or poor?

Start at GCSE Level: Measuring the Period

The simplest version of the experiment is still a useful one.

Suspend a small dense mass from a piece of string.

Measure the pendulum length.

Displace the bob slightly.

Release it.

Measure the time taken for ten complete oscillations.

If ten oscillations take 18.2 seconds:

T = 18.2 / 10

T = 1.82 s

Already there is an important experimental lesson here.

Why time ten oscillations rather than one?

Suppose your reaction time introduces an uncertainty of approximately 0.2 seconds.

If you time one oscillation lasting about 2 seconds, that is a substantial proportion of the measurement.

But if you time ten oscillations lasting about 20 seconds, the same reaction-time uncertainty becomes a much smaller percentage of the total.

This is an excellent example of a general experimental principle:

When possible, measure a larger quantity and divide afterwards.

The same idea appears throughout practical science.

Rather than measuring the thickness of one sheet of paper, measure 100 sheets.

Rather than timing one oscillation, time ten or twenty.

But What Exactly Is the Length of a Pendulum?

This is one of the first places where students can introduce a systematic error.

The length is not simply the length of the string.

It is the distance from:

the pivot to the centre of mass of the pendulum bob.

If the string is 80.0 cm long and the spherical bob has a radius of 1.5 cm, the effective pendulum length is approximately:

80.0 + 1.5 = 81.5 cm

or:

L = 0.815 m

For a short pendulum, ignoring the radius of the bob can produce a significant error.

It is a small detail, but experiments are often won or lost through apparently small details.

Turning the Experiment into a Measurement of Gravity

We can now use:

g = 4 x pi^2 x L / T^2

Suppose:

L = 0.800 m

and:

T = 1.79 s

Then:

g = 4 x pi^2 x 0.800 / 1.79^2

which gives a value close to the expected gravitational acceleration.

But I would not particularly like students to stop there.

Putting two measurements into an equation gives a value for g, but it does not make full use of the experiment.

There is a much better method.

A Better A-Level Experiment: Vary the Pendulum Length

Instead of measuring one pendulum, measure several.

For example:

0.30 m
0.40 m
0.50 m
0.60 m
0.70 m
0.80 m
0.90 m
1.00 m

Measure the period for each.

The original equation is:

T = 2 x pi x sqrt(L/g)

Square both sides:

T^2 = 4 x pi^2 x L/g

Therefore:

T^2 = (4 x pi^2/g)L

This is now in the form:

y = mx

So if we plot:

T^2 against L

we should obtain a straight line.

Its gradient is:

gradient = 4 x pi^2/g

Therefore:

g = 4 x pi^2 / gradient

This is much more powerful experimentally.

Instead of depending upon one measurement, we are using an entire set of results.

Why the Graph Is So Important

There is an important difference between:

using an equation to obtain an answer

and:

testing whether the equation actually describes reality.

If T^2 is proportional to L, the graph should be a straight line through the origin.

That allows us to ask:

Is the graph actually straight?
Does it pass through the origin?
Are there anomalous results?
Is there evidence of a systematic error?
Does the gradient give a sensible value of g?

This is where the experiment starts becoming much more like real physics.

Investigation 1: Does the Mass of the Bob Matter?

This is an excellent prediction exercise.

Ask the student first:

If I replace a 50 g pendulum bob with a 200 g bob, what do you think will happen to the period?

Many people instinctively expect the heavier mass to swing differently.

Try it.

Keep the pendulum length constant.

Use several different masses.

Measure their periods.

Within experimental uncertainty, the period should remain essentially unchanged.

Notice that mass does not appear in:

T = 2 x pi x sqrt(L/g)

That is not an accident.

The restoring force becomes greater for a heavier bob, but its inertia also increases.

The mass cancels from the mathematics.

This connects beautifully with a much larger idea in physics: gravitational acceleration is independent of the mass of a falling object, provided effects such as air resistance can be ignored.

Investigation 2: Does Pendulum Length Matter?

Here the effect is very obvious.

A long pendulum swings slowly.

A short pendulum swings quickly.

But the relationship is not simply:

T proportional to L

Instead:

T proportional to sqrt(L)

This gives students an opportunity to distinguish between different mathematical relationships.

If the length is increased by a factor of four:

L becomes 4L

then:

T becomes 2T

It does not become 4T.

That distinction between linear, square and square-root relationships is enormously important in A-level science.

Investigation 3: Does Amplitude Matter?

This is where the experiment becomes especially interesting.

The standard pendulum equation comes with an assumption that is sometimes forgotten.

It assumes that the angle of oscillation is relatively small.

For small angles:

sin(theta) is approximately equal to theta

provided theta is measured in radians.

This is known as the small-angle approximation.

It allows the pendulum's motion to be treated approximately as simple harmonic motion.

But approximations are not laws of nature.

Eventually they fail.

And we can actually watch that happen.

Testing the Small-Angle Approximation

Set up a pendulum of fixed length.

Now measure its period using different starting angles.

For example:

5 degrees
10 degrees
15 degrees
20 degrees
30 degrees
45 degrees
60 degrees

Keep everything else as constant as possible.

At small angles, there will be very little noticeable difference.

But as the amplitude increases, the period becomes longer than predicted by the simple pendulum equation.

Approximately:

at 5 degrees, the difference is tiny;
at 10 degrees, the correction is around 0.2%;
at 20 degrees, it is approaching 1%;
at 30 degrees, it is around 2%;
by 45 degrees, the effect is several percent.

Suddenly the phrase "small-angle approximation" becomes something the student has actually observed rather than simply memorised.

What Counts as a "Small" Angle?

This leads to a wonderful scientific question.

Students sometimes want a definite answer:

Is 10 degrees a small angle?

But physics does not really work like that.

The better question is:

Small enough for what?

If your measurements have an uncertainty of several percent, the error introduced by using a 10-degree amplitude is probably insignificant.

If you are performing a very high-precision measurement, it may no longer be insignificant.

This is a central idea in modelling.

An approximation does not have to be perfectly true to be useful.

It simply has to be sufficiently accurate for the problem being considered.

Investigation 4: What Happens to the Amplitude?

Release the pendulum and leave it swinging.

The amplitude gradually decreases.

Eventually it stops.

Why?

Because the real pendulum is losing mechanical energy.

Some energy is transferred through:

air resistance;
friction at the pivot;
movement within the string;
sound;
vibrations transferred into the support.

The oscillation is therefore damped.

This gives us another possible experiment.

Measuring Damping

Start the pendulum from a known angle.

Record the maximum amplitude after:

10 oscillations
20 oscillations
30 oscillations
40 oscillations

The amplitude should gradually decrease.

With sufficiently careful measurements, students can investigate how rapidly energy is being removed from the system.

A phone camera can make this particularly interesting because the motion can be recorded and examined frame by frame.

You could even place a large protractor or angular scale behind the pendulum.

Now the humble pendulum has introduced:

oscillations;
damping;
energy transfer;
data analysis.

Does Damping Change the Period?

Another good question is:

As the pendulum loses energy and its amplitude decreases, does its period change?

At small amplitudes, the change may be extremely difficult to detect.

At larger amplitudes things become more interesting because reducing amplitude also moves the pendulum towards the small-angle regime.

This is exactly the sort of question for which students should make a prediction before taking measurements.

An Important Practical Problem: Reaction Time

A student with a stopwatch is probably the biggest source of uncertainty in a basic pendulum experiment.

There are two reaction-time events:

starting the timer;
stopping the timer.

Timing many oscillations reduces the percentage effect.

There are other ways of improving the measurement.

You could use:

slow-motion video;
a light gate;
a motion sensor;
computer video analysis.

But there is also something rather satisfying about seeing how far we can get using little more than a stopwatch and careful experimental technique.

Where Should You Start Timing?

This sounds trivial.

It isn't.

Imagine starting the stopwatch when the pendulum reaches the extreme left of its motion.

That is a difficult moment to judge accurately because the bob slows down before reversing direction.

Instead, I prefer timing the bob as it passes through its equilibrium position.

It is moving fastest there and the crossing point can often be identified much more consistently.

A marker placed behind the pendulum makes this easier.

The student can count:

zero, one, two, three...

as the pendulum repeatedly passes the reference point in the same direction.

Repeats Matter

A single measurement should rarely be trusted when repeats are practical.

Suppose the times for 10 oscillations are:

18.12 s
18.25 s
18.17 s

Calculate the mean rather than simply selecting one value.

Repeating measurements also tells us something that a single result cannot:

how reproducible the experiment is.

If repeated timings differ dramatically, something is wrong with either the method or the system being measured.

Random Error and Systematic Error

The pendulum provides a very nice distinction between two important kinds of experimental uncertainty.

Random uncertainty

Examples include:

human reaction time;
slightly different release positions;
difficulty deciding exactly when the bob crosses the reference line.

Repeating the measurement helps reduce their effect.

Systematic uncertainty

Examples might include:

measuring the string rather than pivot-to-centre distance;
a ruler with an incorrect zero;
consistently including an extra part of the support in the measured length.

Repeating the measurement does not necessarily remove systematic error.

You can repeat the wrong measurement ten times and simply become very confident in the wrong answer.

That is an extremely important lesson in experimental science.

Don't Push the Pendulum

Another surprisingly common problem occurs when the pendulum is released.

The student pulls the bob sideways and then gives it a tiny push as they release it.

That immediately changes the initial conditions.

Instead, the bob should simply be released.

One simple method is to hold the bob between two fingers and open them without pushing.

For high-quality measurements, small details like this matter.

Keep the Motion in One Plane

A real pendulum often starts swinging in an ellipse rather than cleanly backwards and forwards.

That makes measurement much more difficult.

Try to ensure that:

the string is attached securely;
the release is straight;
the support does not twist;
the bob is not rotating unnecessarily.

A dense, compact bob is generally preferable to a large light object because air resistance is less significant relative to its weight.

Could We Measure g at Home?

Yes.

That is one reason I particularly like this experiment.

You do not need an expensive laboratory instrument to measure one of the fundamental characteristics of our environment.

You need:

string;
a small dense mass;
a secure support;
a ruler or tape measure;
a stopwatch.

Using careful measurements and a graph of T^2 against L, it is quite possible to obtain a respectable value for g.

And that is rather remarkable.

We are measuring the gravitational acceleration of planet Earth using what is essentially a weight hanging from a piece of string.

Taking It Further: Measuring g at Different Locations

There is an even more intriguing extension.

The value of g is not absolutely identical everywhere on Earth.

It varies slightly because of factors including:

latitude;
altitude;
the Earth's rotation;
local geology.

A domestic pendulum experiment is unlikely to resolve very small differences easily, but the idea introduces an important point:

9.81 m/s^2 is not some magical universal number.

It is an approximate value for gravitational acceleration close to the Earth's surface.

Move to the Moon and the experiment would produce a completely different period.

What Would a Pendulum Do on the Moon?

Because:

T = 2 x pi x sqrt(L/g)

a smaller gravitational acceleration means a longer period.

A one-metre pendulum would therefore swing much more slowly on the Moon than on Earth.

This provides a lovely thought experiment.

If astronauts constructed a pendulum inside a suitable lunar habitat, they could use its period to measure lunar gravity.

The same mathematics would apply.

Only g would change.

From Pendulums to Clocks

Pendulums were historically enormously important because their period can be very stable.

That made them suitable as timing devices.

Pendulum clocks transformed accurate timekeeping.

But now our investigations show something rather important.

If amplitude affects period, then a good pendulum clock must prevent excessive changes in amplitude from affecting its timing.

Once again, what initially appears to be a simple physics experiment has an engineering application.

A Suggested Investigation Sequence

If I were developing this experiment with a student over several sessions, I might progress through it like this:

Stage 1 — GCSE

Measure the time for ten oscillations.

Calculate:

T = total time / number of oscillations

Learn about:

period;
frequency;
repeats;
averages;
reaction time.

Stage 2 — GCSE to A-Level Transition

Investigate how period changes with:

length;
bob mass;
amplitude.

Ask the student to make predictions before taking measurements.

Stage 3 — A-Level

Measure T for several lengths.

Calculate T^2.

Plot:

T^2 against L

Find the gradient.

Calculate:

g = 4 x pi^2 / gradient

Compare the experimental value with the accepted value.

Calculate percentage difference.

Stage 4 — Experimental Analysis

Investigate:

uncertainty in L;
uncertainty in T;
repeat measurements;
anomalous data;
best-fit lines;
worst acceptable gradients;
systematic errors.

Stage 5 — Testing the Model

Increase the starting amplitude.

Determine when the period begins to depart measurably from the small-angle prediction.

Now the experiment has become an investigation into the limitations of the model itself.

That, to me, is where it becomes especially valuable.

The Pendulum as a Lesson in Scientific Models

School physics can sometimes unintentionally give the impression that equations are perfect descriptions of nature.

They are not.

They are models.

The simple pendulum equation assumes, amongst other things:

the string has negligible mass;
the string does not stretch;
the bob behaves approximately like a point mass;
there is negligible air resistance;
there is negligible friction at the pivot;
the oscillations are small;
the gravitational field is uniform;
the support does not move.

None of those statements is completely true for our real pendulum.

Yet the equation works remarkably well.

That is one of the most important ideas a student can take away from the experiment.

A model can be extremely useful without being perfectly true.

One Experiment, Many Levels of Physics

I like experiments like the pendulum because there is no obvious point at which you have "finished" them.

A younger student might ask:

How long does one swing take?

A GCSE student might ask:

How does length affect the period?

An A-level student might ask:

Can I determine g from the gradient?

A more advanced student might ask:

At what amplitude does the small-angle approximation become experimentally unacceptable?

And another might ask:

How does damping modify the motion?

The apparatus has not changed very much.

What has changed is the sophistication of the question.

A Personal Reflection: This Is What Practical Science Should Be

One of the things I particularly enjoy about teaching practical science is taking apparatus that looks completely familiar and discovering that there is much more hidden within it.

A pendulum is not impressive because it is technologically complicated.

It is impressive precisely because it isn't.

Students can see almost everything that is happening.

There is no mysterious black box producing numbers on a screen.

Pull the bob aside.

Release it.

Watch gravity accelerate it towards the centre.

Watch its momentum carry it onwards.

Watch gravitational potential energy become kinetic energy and then become gravitational potential energy again.

Then start asking questions.

That process — observe, predict, measure, analyse, question the model and investigate again — is what experimental science is really about.

Conclusion: A Piece of String Can Measure the Earth

The pendulum is sometimes treated as little more than a convenient way of teaching students how to use a stopwatch.

It deserves better.

With exactly the same piece of apparatus we can investigate:

period;
frequency;
gravitational acceleration;
square-root relationships;
simple harmonic motion;
energy transfers;
damping;
experimental uncertainty;
graphical analysis;
systematic errors;
mathematical approximations;
the limitations of physical models.

And perhaps the most satisfying result of all is that we can obtain a measurement of the Earth's gravitational acceleration from little more than a length of string, a mass, a ruler and a clock.

That is the beauty of classical physics.

Sometimes the simplest apparatus produces the deepest questions.

18 August 2026