Atwood’s Machine — The Elegant Experiment That Slows Down Gravity
There are some pieces of physics apparatus that look almost too simple to be interesting.
Atwood’s machine is one of them.
Two masses hang on either side of a pulley. They are joined by a string. Make one mass slightly heavier than the other, release the system, and the heavier mass moves down while the lighter mass rises.
That is essentially the whole apparatus.
Yet from this simple arrangement we can investigate Newton’s laws, resultant force, acceleration, tension, conservation of energy, experimental uncertainty and even the rotational inertia of the pulley itself.
For me, that is what makes Atwood’s machine such an elegant classical physics experiment. It takes something that normally happens rather too quickly — acceleration under gravity — and slows it down enough for us to study it carefully.
And it raises a very useful question:
If gravity is pulling strongly on both masses, why can the resulting acceleration be surprisingly small?
A Machine Designed to Make Acceleration Measurable
The Atwood machine is named after the English mathematician and physicist George Atwood, who described the apparatus in the eighteenth century.
The problem Atwood faced was a practical one.
If you simply drop an object, it accelerates at approximately:
g = 9.81 m/s^2
That is quite a large acceleration.
Drop something through one metre and the whole event is over in less than half a second. That was particularly awkward in the days before electronic timers, light gates, motion sensors and high-speed video.
Atwood's clever idea was to allow gravity to act on two connected masses.
Most of the gravitational forces effectively oppose one another. Only the difference between them produces the acceleration of the system.
The result is a controllable acceleration that can be much smaller than g.
That makes Newton's second law much easier to investigate experimentally.
The Simplest Atwood Machine
Imagine two masses:
m1
and
m2
connected by a light string passing over a pulley.
Suppose:
m2 > m1
The second mass therefore travels downwards and the first mass travels upwards.
At first glance we might be tempted to say that the force accelerating the system is simply:
m2g
But that cannot be correct.
Gravity is also pulling downwards on m1.
Because the two masses are connected, the gravitational effects oppose each other as far as the motion of the complete system is concerned.
The driving force is therefore:
F = m2g - m1g
or:
F = (m2 - m1)g
But what mass is being accelerated?
Both masses are moving.
Therefore the total moving mass is:
m1 + m2
Newton's second law tells us:
F = ma
so:
(m2 - m1)g = (m1 + m2)a
and therefore:
a = (m2 - m1)g / (m1 + m2)
That is the central equation of the Atwood machine.
It is also a lovely example of why simply remembering F = ma is not enough. We have to decide which force and which mass belong in the equation.
A Small Imbalance Can Move a Large Mass
This is where the experiment becomes particularly interesting.
Suppose the two sides contain:
m1 = 0.245 kg
m2 = 0.255 kg
The total moving mass is:
0.245 + 0.255 = 0.500 kg
But the difference between the masses is only:
0.255 - 0.245 = 0.010 kg
The theoretical acceleration is:
a = 0.010 x 9.81 / 0.500
a = 0.1962 m/s^2
So although gravity itself produces an acceleration of about 9.81 m/s^2, our Atwood machine accelerates at only about:
0.20 m/s^2
That is around one-fiftieth of g.
Suddenly the motion is slow enough to watch.
That is the cleverness of the machine.
The Best Investigation: Keep the Total Mass Constant
One particularly satisfying experiment is to investigate how acceleration changes when the difference between the masses changes, while keeping the total mass constant.
This is important experimentally.
If we changed both the driving force and the total mass at the same time, interpreting the results would become more difficult.
Instead, imagine starting with:
250 g | 250 g
The machine is balanced.
Now transfer 5 g from one side to the other:
245 g | 255 g
The total mass is still:
500 g
but the mass difference is now:
10 g
Transfer another 5 g:
240 g | 260 g
The total remains 500 g, while the difference becomes 20 g.
We might continue with:
235 g | 265 g
230 g | 270 g
225 g | 275 g
The total mass remains unchanged throughout.
Only the imbalance changes.
What Are We Really Changing?
The driving force is:
F = (m2 - m1)g
So increasing the mass difference increases the resultant force.
At the same time:
m1 + m2
remains constant.
Newton tells us:
a = F / m
If mass remains constant, acceleration should therefore be directly proportional to force.
That gives us a prediction before we even perform the experiment:
Double the mass difference and the acceleration should approximately double.
That is excellent experimental physics.
We have a theory.
We make a prediction.
Then we collect measurements and see whether nature agrees.
A Possible Set of Results
Suppose the total moving mass is 0.500 kg.
We might obtain something resembling:
| Mass difference | Driving force | Ideal acceleration |
|---|---|---|
| 0.010 kg | 0.0981 N | 0.196 m/s^2 |
| 0.020 kg | 0.1962 N | 0.392 m/s^2 |
| 0.030 kg | 0.2943 N | 0.589 m/s^2 |
| 0.040 kg | 0.3924 N | 0.785 m/s^2 |
| 0.050 kg | 0.4905 N | 0.981 m/s^2 |
Plot:
acceleration against mass difference
and, for an ideal Atwood machine, we should obtain a straight line.
Alternatively plot:
acceleration against driving force
Again, Newton's second law predicts a straight line.
The gradient can even be connected to the total accelerated mass.
Now our very simple pulley has become an experimental test of:
F = ma
How Do We Measure the Acceleration?
There are several possibilities.
Method 1 — Measure Distance and Time
Release the masses from rest and measure the time taken to travel a known distance.
For motion starting from rest:
s = 1/2 at^2
Therefore:
a = 2s / t^2
If the mass travels 0.50 m in 1.80 seconds:
a = (2 x 0.50) / 1.80^2
a = 1.00 / 3.24
a = 0.309 m/s^2
The problem is timing.
Human reaction time becomes a significant source of uncertainty, particularly for short runs.
Method 2 — Video the Motion
A smartphone capable of slow-motion recording can make the experiment much better.
Place a scale beside one of the moving masses and record the motion.
Individual frames can then be used to determine:
position;
time;
velocity;
acceleration.
This is also an excellent opportunity to show students that physics experiments do not always require expensive specialist equipment.
Method 3 — Use Light Gates
Light gates make the experiment considerably more precise.
A card attached to one of the masses can interrupt one or more beams, allowing velocity and acceleration to be calculated electronically.
Method 4 — Use a Motion Sensor
With a suitable motion sensor or data-logging system, position can be recorded continuously.
This produces graphs of:
position against time
velocity against time
and possibly:
acceleration against time.
The velocity-time graph is particularly satisfying.
If acceleration is approximately constant, velocity increases linearly with time.
The gradient gives the acceleration directly.
Resolving the Forces on Each Mass
There is another way of analysing the machine, and this is particularly useful for A-level Physics.
Instead of treating the two masses as one system, consider each mass separately.
For the heavier mass m2:
Weight acts downward:
m2g
Tension acts upward:
T
Therefore:
m2g - T = m2a
For the lighter mass m1:
Tension acts upward.
Weight acts downward.
Therefore:
T - m1g = m1a
Now add the two equations:
m2g - T + T - m1g = m2a + m1a
The tensions cancel:
(m2 - m1)g = (m1 + m2)a
Giving us once again:
a = (m2 - m1)g / (m1 + m2)
This is a beautiful piece of mechanics because it demonstrates why choosing a complete system can simplify a problem.
The tension is very important to each individual mass.
But when we consider the complete two-mass system, tension becomes an internal force and disappears from the final equation.
That is a powerful idea that appears repeatedly in mechanics.
Tension Is Not Simply Equal to the Weight
Another useful misconception can be explored using Atwood's machine.
Students often become accustomed to situations where:
T = mg
But that is only true when the mass has zero acceleration.
In an Atwood machine the masses accelerate.
For the lighter mass:
T - m1g = m1a
Therefore:
T = m1(g + a)
So the tension is greater than the weight of the lighter mass.
For the heavier mass:
m2g - T = m2a
Therefore:
T = m2(g - a)
The tension is less than the weight of the heavier mass.
It must, of course, be the same tension throughout an ideal light string.
Again, a very simple experiment exposes some quite subtle mechanics.
The Real Machine Is More Interesting Than the Ideal One
Classroom calculations usually make several assumptions:
the pulley is frictionless;
the pulley has negligible mass;
the string has negligible mass;
the string does not stretch;
the string does not slip over the pulley;
air resistance is negligible.
Real apparatus politely refuses to obey all of these assumptions.
And that is where the experiment becomes even more interesting.
Friction in the Pulley
A real pulley has bearings.
Those bearings have friction.
When the mass difference is very small, you may find that the machine does not move at all.
Perhaps:
249 g | 251 g
remains stationary.
Increase the imbalance slightly and suddenly it begins to move.
That immediately tells us something important.
The driving force must first overcome friction.
If we plot acceleration against driving force, our experimental line may therefore fail to pass exactly through the origin.
Far from being a failed experiment, this is useful evidence about the real apparatus.
The Pulley Has to Accelerate Too
There is another wonderfully subtle effect.
The pulley rotates.
That means some of the driving force is being used not merely to accelerate the hanging masses but also to accelerate the rotation of the pulley.
In the ideal equation we wrote:
a = (m2 - m1)g / (m1 + m2)
But for a real pulley with rotational inertia, the acceleration will usually be slightly smaller.
At a more advanced level the pulley can effectively contribute an additional inertial term.
For a pulley of moment of inertia I and radius r:
a = (m2 - m1)g / (m1 + m2 + I/r^2)
We have not lost Newton's laws.
We have simply discovered that there was another part of the system whose acceleration we originally ignored.
This is one of the reasons I particularly like classical experiments: increasing the precision does not merely improve the same answer. It often reveals another layer of physics.
An Interesting Extension: Keep the Difference Constant Instead
We can reverse the experiment.
Suppose we maintain the same difference between the masses but increase the total mass.
For example:
95 g | 105 g
195 g | 205 g
295 g | 305 g
In each case the mass difference is:
10 g
Therefore the driving force is approximately constant.
But the total mass being accelerated becomes progressively larger.
Since:
a = F / m
we expect the acceleration to decrease.
This allows students to investigate the other half of Newton's second law.
The first investigation asks:
What happens when force changes but mass remains constant?
The second asks:
What happens when mass changes but force remains approximately constant?
Together, they provide a remarkably complete investigation of F = ma.
Could We Measure g Using an Atwood Machine?
Yes.
Rearrange:
a = (m2 - m1)g / (m1 + m2)
to give:
g = a(m1 + m2) / (m2 - m1)
Measure the masses accurately and determine the acceleration experimentally.
You can then calculate an experimental value of gravitational field strength.
It probably will not produce 9.81000 m/s^2.
That is not the point.
The interesting question becomes:
Why doesn't it?
Possible explanations include:
pulley friction;
pulley rotational inertia;
inaccurate masses;
timing uncertainty;
string mass;
air resistance;
the release method;
the pulley not being perfectly aligned.
That turns a demonstration into a genuine experimental investigation.
Don't Push the Masses
One seemingly trivial experimental detail is actually extremely important.
The masses must be released, not pushed.
Giving the system even a small initial velocity affects measurements based on:
s = 1/2 at^2
because that equation assumes:
u = 0
A good release mechanism makes the experiment significantly more repeatable.
This is exactly the sort of small practical detail that students begin to appreciate when they actually perform experiments rather than merely reading descriptions of them.
Repeat the Measurements
One timing result tells us surprisingly little.
Repeat each mass configuration several times.
For example:
5 trials for each mass difference.
Calculate the mean acceleration and investigate the spread of the measurements.
If one measurement is very different, do not simply delete it because it is inconvenient.
Ask why.
Did the string catch?
Did the mass swing?
Was the release poor?
Did the pulley hesitate before moving?
Experimental physics is partly about collecting numbers.
It is much more about deciding which numbers we should trust.
Why This Experiment Is Such Good Teaching Physics
What I particularly like about Atwood's machine is that almost everyone can understand the apparatus immediately.
There is no mysterious black box.
You can see the masses.
You can see the string.
You can see the pulley.
You can see which side is heavier.
Yet from that simple arrangement we can explore:
Newton's first and second laws;
resultant force;
acceleration;
tension;
free-body diagrams;
simultaneous equations;
experimental uncertainty;
friction;
rotational motion;
energy;
modelling assumptions.
It is an excellent example of how sophisticated physics does not necessarily require sophisticated-looking equipment.
The Physics of a Nearly Balanced System
There is also something slightly counter-intuitive about watching an Atwood machine.
Imagine 500 g of mass being accelerated by an imbalance of only 10 g.
Gravity is pulling on both sides with forces of several newtons.
Yet the resultant force may be less than one tenth of a newton.
Large forces can therefore exist within a system while producing only a small overall acceleration.
That idea matters far beyond pulley experiments.
It appears whenever opposing forces nearly balance:
vehicles moving at constant speed;
aircraft in level flight;
terminal velocity;
objects floating;
structures under load;
orbital mechanics;
mechanical control systems.
The motion depends not on how large the individual forces are, but on their resultant.
From an Eighteenth-Century Pulley to Modern Physics
Atwood designed his machine at a time when accurately measuring rapidly changing motion was extremely difficult.
Today we can attach sensors, use computer data logging, film the experiment at hundreds of frames per second and fit mathematical models to thousands of measurements.
Yet the underlying apparatus hardly needs to change.
Two masses.
One string.
One pulley.
That longevity tells us something.
A great experiment does not merely demonstrate an effect.
It isolates an idea.
Atwood's machine isolates Newton's second law beautifully.
A Simple Apparatus With a Lot to Say
Atwood's machine may never compete visually with exploding chemicals, giant electrical discharges or dramatic vacuum demonstrations.
But that is part of its charm.
Transfer a tiny mass from one side to the other and the entire system begins to accelerate.
Increase that imbalance and the acceleration increases predictably.
Keep the force constant but increase the mass and the acceleration falls.
Measure carefully enough and even the imperfections — friction, pulley inertia and experimental uncertainty — become physics worth investigating.
The experiment begins with one of the most familiar equations in science:
F = ma
But actually performing it reveals what that equation really means.
For me, that is classical experimental physics at its best: simple apparatus, a clear question, a prediction that can be tested, and enough hidden complexity to reward anyone who decides to look a little more closely.
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