Atwood Machine

Masses land hard. Put a book or a foam block on the floor under the descending side and keep hands off the string while it runs.

Clamp a low-friction pulley to a stand or a door frame so both strings hang free and vertical, with at least 1.5 m of drop below it. A ball-bearing pulley is worth using; a nail or a hook adds friction that swamps the effect.

Tie a light, non-stretching string over the pulley with an equal mass on each end — 200 g a side is a good start. Adjust until the system stays put wherever you place it. That balance point is your zero.

Give one side a gentle push. If it keeps drifting at roughly constant speed, friction is low. If it stops quickly, the pulley is binding — fix that now, because friction is the main source of error in everything that follows.

Mark a start height and a finish point on the floor or stand, and measure the vertical distance between them in metres. Use as long a drop as you have room for — timing error matters less over a longer run.

Transfer a small extra mass — 5 to 20 g — onto one hanger. This difference is the only unbalanced force acting; everything else still has to be accelerated by it.

Hold the heavy side at the start mark, release without pushing, and start the stopwatch at the instant of release. Stop it when the mass reaches the finish point.

Human reaction time is roughly 0.2 s and scatters both ways. Five runs and a mean gets most of it out; write down every individual time, not just the average.

The system starts from rest, so s = ½at². Rearranged: a = 2s ÷ t². Use your mean time and your measured drop in metres.

For masses m₁ and m₂ over an ideal pulley: a = (m₁ − m₂)g ÷ (m₁ + m₂), with g = 9.81 m/s². Note that only the difference drives it while the total resists it.

Measured acceleration should come out slightly below predicted. A shortfall of a few per cent is friction and pulley inertia. A shortfall of thirty per cent means the pulley is bad or the string is stretching.

Move a second rider across from one side to the other, keeping the total mass identical. Re-time. Doubling the mass difference should double the acceleration — force up, mass constant.

Now add the same extra mass to both sides, keeping the difference fixed. Re-time. Acceleration falls, because the same net force now has more to move.

With total mass held constant, plot your measured a on the vertical axis against (m₁ − m₂) on the horizontal. The points should fall on a straight line through the origin — that straight line is Newton's second law, drawn by hand.

The gradient of that line is g ÷ (m₁ + m₂). Multiply the gradient by your total mass and you have measured the acceleration due to gravity — expect a value a little under 9.81 m/s², for the same reasons as step 11.

The instrument exists because clocks were slow. George Atwood described this machine in his 1784 Treatise on the Rectilinear Motion and Rotation of Bodies at Cambridge. Newton's Principia was a century old and uncontested, but demonstrating uniform acceleration directly needed timing to a small fraction of a second, which pendulum clocks and human observers could not deliver on a two-metre free fall. Atwood's insight was not new physics — it was that you can dilute gravity. The factor (m₁ − m₂)/(m₁ + m₂) is a dial: set it to 1/100 and a fall that took 0.6 seconds now takes six.

Where the missing acceleration goes. The ideal formula assumes a massless, frictionless pulley and an inextensible massless string. A real pulley has to be spun up, and its rotational inertia acts like extra mass bolted to the system: the honest expression is a = (m₁ − m₂)g ÷ (m₁ + m₂ + I/r²), where I is the pulley's moment of inertia and r its radius. This is why your measured value sits below prediction, and why the discrepancy grows as you make the mass difference smaller — the pulley's fixed contribution becomes a larger share of the total. The error is not sloppiness; it is a real term you left out.

The tension is not the weight. A frequent misconception is that the string tension equals the weight of the lighter mass. It does not — if it did, nothing would accelerate. For the ideal case T = 2m₁m₂g ÷ (m₁ + m₂), which lies strictly between the two weights. The heavy side is held back and the light side is pulled up by the same string, and that only works if the tension is greater than one weight and less than the other.

Why it is still in laboratories. Photogates and light gates have replaced the stopwatch, but the machine itself is unchanged, because it isolates one relationship with almost nothing else in the way. Modern variants replace the hanging mass with a magnetic or motorised drive to study driven oscillation; the geometry is Atwood's.