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.

Forces and the apparatus at rest update directly when physical inputs change. Run starts or resumes; after arrival it restarts from rest. One step pauses and advances by 0.05 s up to the travel limit. Reset returns the current inputs to rest; it does not replace them with defaults. Playback changes only display speed. Leaving the tool or hiding the page pauses motion. Raw input drafts, including incomplete or invalid values and decimal commas, and playback speed survive reload. Older valid numeric settings migrate automatically. Time, motion, plots and camera state are never stored; reopening recalculates at rest. The optional velocity plot spans the full travel time and marks 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.