How it works
An Atwood machine is two masses connected by a rope over a single pulley — the classic teaching rig for Newton's laws. The heavier mass falls, the lighter one rises, and both share the same magnitude of acceleration because the rope is inextensible.
Solving the two Newton's-second-law equations together gives a = (m₁ − m₂)g / (m₁ + m₂) and T = 2·m₁·m₂·g / (m₁ + m₂). The acceleration is only the weight difference spread over the total mass, so nearly equal masses barely move — which is exactly why the machine lets you measure g with a slow, easily timed fall.
This calculator assumes an ideal setup: a massless inextensible rope, a massless frictionless pulley, and g ≈ 9.81 m/s². Real pulleys have inertia and friction that reduce the acceleration slightly.
Use it in real life
Teaching g: because the acceleration is a small, adjustable fraction of g, an Atwood machine turns free fall into a fall slow enough to time by hand — historically one of the first accurate ways to measure gravity.
Elevators with counterweights: a lift and its counterweight are a powered Atwood machine. Matching the counterweight to a typical load means the motor only has to move the small difference, saving energy.
Sanity check: set both masses equal and the acceleration is zero while the tension equals each weight — the system just hangs balanced, exactly as intuition demands.
Frequently asked questions
What is the formula for an Atwood machine?
Acceleration a = (m₁ − m₂)g / (m₁ + m₂) and rope tension T = 2·m₁·m₂·g / (m₁ + m₂), for an ideal massless rope and frictionless pulley.
Why is the tension not just equal to the heavier weight?
Because the heavier mass is accelerating downward, the rope supports less than its full weight; and the lighter mass is accelerating upward, so the rope pulls harder than its weight. The single tension settles between the two weights.
What happens if the two masses are equal?
The acceleration is zero and the tension equals the weight of either mass (T = mg). The system is balanced and stays at rest or moves at constant speed.