Atwood Machine

Two masses share a massless, inextensible rope over a pulley of radius R and rotational inertia I. The axle is frictionless and the rope does not slip. The masses start at rest; g = 9.81 m/s². Positive displacement s₁, velocity v₁ and acceleration a mean m₁ moves downward and m₂ upward.

a = (m₁ − m₂)g / (m₁ + m₂ + I/R²). The separate rope tensions are T₁ = m₁(g − a) and T₂ = m₂(g + a). Their difference supplies the pulley torque. The ideal comparison uses the same masses and radius with I = 0, so both tensions equal 2m₁m₂g/(m₁ + m₂). Equal masses are a valid equilibrium for every allowed inertia.

Motion uses the same constant acceleration: s₁ = at²/2 and v₁ = at. No slip gives angle s₁/R and angular velocity v₁/R, positive for m₁ downward. The two kinetic energies are (m₁ + m₂)v₁²/2 for translation and I(v₁/R)²/2 for rotation. Their sum equals the work of gravity, (m₁ − m₂)gs₁.

The display ends after either mass travels 0.35 m. It retains the arrival speed and energy immediately before any possible impact; no collision or stopping force is modelled. At equilibrium it remains at time zero. The sketch is not to geometric scale; its fixed pulley size and block sizes do not depict the entered radius or mass. Force arrows use one common length scale within each sketch; the table gives their numerical values.

Run starts or resumes; after arrival it restarts from rest. One step pauses and advances by 0.05 s, stopping at the travel limit. Reset returns an accepted solution to rest. Editing physical inputs discards the old result. Playback changes only the speed of the display. Leaving the tool or hiding the page pauses motion. Only the last valid parameters and playback speed are saved. The optional velocity plot uses the full travel time; its marker follows the current state.

Source: MIT OpenCourseWare, 8.01SC Classical Mechanics (2016), lesson 31.4: Atwood Machine. The tool uses m₁ downward as its positive sign convention.