Atwood machine

Intermediate

Two masses over a pulley accelerate by the difference in weights.

Key relationa = (m₁−m₂)g/(m₁+m₂)

Two masses on either side of a light pulley accelerate at a = (m₁ − m₂)g/(m₁ + m₂), the heavier going down. The tension is T = 2m₁m₂g/(m₁ + m₂).

Worked example

3 kg and 1 kg: a = 2·9.8/4 = 4.9 m/s², T = 14.7 N.

Common mistake

T is not equal to either weight when accelerating.

In more detail

An Atwood machine has two masses connected over a pulley. The difference in masses sets the acceleration: a = (m1 − m2)g / (m1 + m2).

Lift counterweights are designed on the same principle, balancing masses to reduce motor load.
Physics labs use Atwood machines to measure g and test Newton's second law.
Lift and counterweight systems use the Atwood principle, where a heavier load on one side accelerates the system.

What happens if the masses are equal?

The acceleration is zero, so the system stays at rest or moves at constant speed.

Does pulley mass matter?

For an ideal massless frictionless pulley, no. A real pulley with mass needs its rotational inertia included.

What happens if the masses are equal?

The acceleration is zero, because the weights balance and the system stays at rest or moves at constant velocity.