Atwood Machine with a Massive Pulley

Two weights on a rope over a pulley: a classic system for seeing how one constraint couples two motions. The same gravity pulls on both masses, yet it is the rope tension that keeps the lighter one from falling freely. Vary the mass ratio and the acceleration runs from zero (balanced masses) toward g (one mass dominating). Because the pulley here has real mass, spinning it up takes a net torque, so the two rope tensions are not equal: T₁ on the heavy side exceeds T₂ on the light side by exactly I a / R². The panel beside the rig plugs the live numbers into a = (m₁ − m₂) g / (m₁ + m₂ + I/R²), and the diagnostic below tracks the acceleration and both tensions against the mass ratio.

Figure 1. Atwood machine. Top: the two masses, their weight and tension force vectors, live acceleration and tension. Bottom: how acceleration and tension vary with the mass ratio m₁/m₂, with a cursor at the current system state.

WHAT TO TRY

Set m₁ = m₂: the system freezes at zero acceleration, and the two tensions stay equal even with a heavy pulley (no spin-up means no net torque).
Raise m₁ with m₂ fixed: the acceleration climbs toward g, and on the diagnostic both tension curves rise while T₁ pulls ahead of T₂.
Slide the pulley mass M down to 0 to recover the ideal massless pulley, where T₁ = T₂; raise it and the system grows sluggish while the tension gap T₁ − T₂ widens.
Read the equation panel beside the rig: it plugs the current masses into a, and shows the pulley's I/R² adding to the effective mass.