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Malti⚖️ Atwood Machine — Pulley Tension & Acceleration
Simulate an Atwood machine: two masses connected by a string over a pulley. Adjust m1, m2 and pulley moment of inertia to see acceleration, tension asymmetry and motion in real time.
About the Atwood Machine
The Atwood machine — two masses connected by a string over a pulley — was invented in 1784 by the Reverend George Atwood, an English mathematician, as a way to slow down and directly verify free fall under Newton's second law. Because the net driving force is only the small difference between the two hanging weights, m1 − m2, while the inertia being accelerated is their sum, m1 + m2, the resulting acceleration a = (m1 − m2)g / (m1 + m2) can be made much smaller than g. That made it possible, with the crude clocks and rulers of the eighteenth century, to time the motion accurately enough to confirm that acceleration is constant and proportional to force, and even to obtain a usable estimate of g.
This simulator extends the classic setup with a pulley that can carry its own moment of inertia, I_p. An ideal, massless pulley simply redirects the string, so the tension is the same on both sides: T1 = T2 = 2m1m2g/(m1+m2). A real pulley has mass and must itself be angularly accelerated by the net torque from the two string tensions, which requires T1 ≠ T2 whenever the pulley resists spinning up. The full result, a = (m1 − m2)g / (m1 + m2 + I_p/R²), shows the extra inertia term I_p/R² acting exactly like additional mass in the denominator — it slows the system down without changing which side falls. Because it isolates a clean, adjustable relationship between force, mass and rotational inertia, the Atwood machine remains a staple of introductory mechanics labs for measuring g and demonstrating Newton's second law under controlled, low acceleration.
Frequently Asked Questions
What is an Atwood machine used for?
It is a classic teaching and measurement device: two masses hang from a string over a pulley, and the small difference between them produces a slow, easily measured acceleration. Physics labs use it to verify Newton's second law, to measure the local acceleration of gravity g, and to demonstrate how tension and acceleration depend on mass and, when the pulley itself has mass, on rotational inertia too.
Why is tension equal on both sides only for an ideal pulley?
An ideal pulley is treated as massless and frictionless, so it exerts no net torque of its own — it just redirects the string, and the same tension passes through unchanged, giving T1 = T2. A real pulley has mass and moment of inertia I_p, and the net torque from the two different tensions is what spins it up. That requires T1 and T2 to differ whenever the pulley is accelerating, which is exactly what this simulator lets you see by raising I_p above zero.
What happens if m1 equals m2?
The numerator of the acceleration formula, m1 − m2, becomes zero, so a = 0 regardless of the pulley's moment of inertia. Both tensions become equal to m·g (the weight of either mass), the system is in static equilibrium, and neither mass moves — the string sits balanced over the pulley just as a symmetric see-saw would remain level.
How does pulley moment of inertia change the result?
The moment of inertia enters the acceleration formula as an extra term, I_p/R², added to the combined mass m1 + m2 in the denominator: a = (m1 − m2)g / (m1 + m2 + I_p/R²). A larger I_p (for a fixed radius R) makes the denominator bigger and the acceleration smaller, because some of the gravitational energy that would otherwise speed up the masses is instead used to spin up the pulley. It also splits the tension: T1 = m1(g − a) and T2 = m2(g + a) are no longer equal once a is reduced by the extra inertia term.
What does this simulation ignore?
The model assumes a massless, perfectly inextensible string that never slips on the pulley, a frictionless axle, and no air resistance. It also assumes the string stays taut and the masses move purely vertically. Real experiments must contend with string stretch, bearing friction, air drag on larger masses, and finite string mass — all of which this idealised model sets aside to isolate the core Newton's-second-law relationship.
How is this used to measure g in real labs?
Because the acceleration a = (m1 − m2)g / (m1 + m2 + I_p/R²) is usually much smaller than free-fall acceleration g, it can be timed accurately with simple stopwatches or photogates over a measured drop distance. Rearranging the formula for g and plugging in the measured a, the known masses, and (if not negligible) the pulley's moment of inertia and radius gives an experimental value of g that a falling object alone would be far too fast to measure directly with basic equipment.
Who invented the Atwood machine and why?
The Reverend George Atwood described the device in 1784 in a paper to the Royal Society, specifically to provide a controlled, repeatable way to test and demonstrate the laws of uniformly accelerated motion that Newton had described roughly a century earlier. By trading free-fall's large, hard-to-measure acceleration for a small, adjustable one, Atwood turned an abstract law into a hands-on classroom demonstration — a role the machine still fills today.
What is the difference between T1 and T2 physically?
T1 is the tension in the string segment attached to m1, and T2 is the tension in the segment attached to m2. For an ideal, massless pulley these must be equal, because the pulley transmits tension without change. For a pulley with real rotational inertia, the two segments pull on the pulley's rim with different forces so that the net torque, (T1 − T2)·R, supplies the angular acceleration needed to spin the pulley up, which is why T1 and T2 diverge as I_p increases.
How do the masses' sizes on screen relate to their values?
Each block is drawn larger as its mass slider increases, purely as a visual cue — the simulation scales box size with the mass value so you can see at a glance which side is heavier without reading the numeric readout. The physics itself uses the exact slider values in the acceleration and tension formulas; the box size is cosmetic and does not feed back into the calculation.
Simulate an Atwood machine: two masses connected by a string over a pulley. Adjust m1, m2 and pulley moment of inertia to see acceleration, tension asymmetry and motion in real time.
2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install