Atwood Machine Calculator

The Atwood machine was invented by George Atwood in 1784 to demonstrate Newton’s second law. Two masses hang from a string over a frictionless pulley. The heavier mass descends and the lighter one rises, but at an acceleration less than free fall, making the motion slow enough to measure by hand.

Deriving the equations

For masses m1 > m2, applying Newton’s second law to each mass:

For m1 (descending): m1 x g - T = m1 x a

For m2 (ascending): T - m2 x g = m2 x a

Solving simultaneously:

Acceleration: a = (m1 - m2) x g / (m1 + m2)

Tension: T = 2 x m1 x m2 x g / (m1 + m2)

These assume a massless, frictionless pulley and an inextensible string.

What a real pulley does to the answer. A pulley of moment of inertia I and radius R has to be spun up too, and the string tension is what spins it. That adds an I/R² term to the effective mass of the system:

a = (m1 − m2) × g / (m1 + m2 + I/R²)

For a solid disc pulley of mass M, I = ½MR², so I/R² = M/2 and the correction is simply half the pulley’s mass added to the denominator. A 200 g pulley on a 5 kg and 3 kg pair drops the acceleration from 2.453 to 2.423 m/s², about 1%, which is why Atwood’s original apparatus used a light pulley on fine bearings. Get the pulley mass up to the same order as the difference between the two weights and the correction stops being a rounding error.

The calculator below uses the idealised equations, so treat its answer as the ceiling and expect a real bench setup to come in a percent or two under.

Velocity after falling distance d

Starting from rest: v = sqrt(2 x a x d)

Checking edge cases

If m1 = m2: a = 0, T = m1 x g. The system is in equilibrium, with each mass supporting its own weight.

If m2 = 0: a = g, T = 0. The mass m1 falls freely, which makes sense: a free-falling mass has no tension in a rope that supports nothing.

Historical significance

Atwood used this device to verify that acceleration is proportional to net force. In 1784 this was still being established experimentally. The machine was used in physics teaching for over a century. It also appears in elevator design: a counterweight is essentially one side of an Atwood machine, reducing the motor load significantly.