Atwood Machine Acceleration

Hang two masses over a light, frictionless pulley and only the difference in weight drives the system, while the total mass has to be accelerated — giving a = (m₁ − m₂)g ⁄ (m₁ + m₂). With 3 kg against 2 kg the acceleration is (1 ⁄ 5) × 9.80665 ≈ 1.96 m/s², a fifth of free fall. Make the masses nearly equal and the acceleration becomes as gentle as you like, which is precisely the point: George Atwood built his machine in 1784 to slow gravity down enough to verify Newton's laws with the crude clocks of the day.

Two sanity checks fall out immediately. Equal masses give a = 0, the system balances. Let m₂ → 0 and a → g, plain free fall. The idealisation to watch is the pulley: a real one has rotational inertia and bearing friction, so measured accelerations run a few percent low, and the rope's own mass matters once the masses are small. Elevator counterweights are the industrial version of the same trick — balancing most of the car's weight so the motor only has to handle the difference.