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, 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. Real pulleys have rotational inertia that slows the acceleration further — see the formula explanation for the correction term.

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. This 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.