This Atwood machine calculator finds the acceleration and string tension for two masses hanging over a pulley — the 1784 apparatus that first let physicists measure gravity’s acceleration in slow motion.
Ide idealizations stated: massless, frictionless pulley; inextensible string. The tension lands strictly between the two weights — it must hold up the lighter mass’s fall while holding back the heavier one’s — which is the sanity check your intuition should run on any Atwood answer.
a = (m₂ − m₁) g ÷ (m₁ + m₂) — Gravity, Diluted
The system accelerates as if the net weight (the difference) were pulling the total mass: difference in the numerator, sum in the denominator. Near-equal masses crawl — the property George Atwood exploited to time gravity with 18th-century clocks. Tension T = 2m₁m₂g/(m₁ + m₂) is the same on both sides of an ideal pulley, always between the two weights. General motion tools: our Acceleration Calculator. Part of our Physics Calculators collection.
Worked Example (Verified)
- 3 kg vs 5 kg (defaults): a = (2/8) × 9.81 = 2.45 m/s² · T = (2 × 3 × 5 × 9.81)/8 = 36.8 N — between the weights 29.4 N and 49.1 N, as required ✔
- Equal masses (5 and 5): a = 0, T = 49.1 N = either weight — the pulley becomes an expensive way to hang still
- Extreme ratio (1 kg vs 100 kg): a = (99/101) g ≈ 9.62 m/s² — approaching free fall; tension = 2×1×100×9.81/101 ≈ 19.4 N, which checks out from the light side’s own equation: rising at 9.62 m/s² needs T = m₁(g + a) = 1 × 19.43 ✔
- On the Moon (g = 1.62): the 3-vs-5 system accelerates at 0.41 m/s² — Atwood as a planetary-gravity demonstrator
Why Physicists Loved This Machine
- Before photogates, timing a free-falling object over meters was hopeless; diluting g to 0.1–1 m/s² made acceleration measurable with pendulum clocks.
- The system is Newton’s second law in its cleanest dress: one equation for the whole system, one per mass, tension the internal negotiator.
- Real pulleys add rotational inertia (effective extra mass I/r²) and friction — the standard second-week lab is measuring exactly those deviations.
- Elevator-and-counterweight engineering is an Atwood machine with a business model: the counterweight makes the motor lift only the difference.
Related Tools
- Acceleration Calculator — v, u, a, t in general
- Force Calculator — F = ma everywhere else
- Physics Calculators — the full collection
Frequently Asked Questions
How do you calculate Atwood machine acceleration?
a = |m₂ − m₁| × g ÷ (m₁ + m₂): the unbalanced weight divided by the total mass being moved. 5 kg against 3 kg gives (2/8) × 9.81 = 2.45 m/s² toward the heavier side.
How do you find the tension in an Atwood machine?
T = 2 m₁ m₂ g ÷ (m₁ + m₂) — derivable from either mass’s own equation (for the rising mass, T = m(g + a); for the falling one, T = m(g − a); both give the same number, 36.8 N in the worked example). Tension always lies between the two weights.
What if the masses are equal?
No acceleration: the system balances (or continues at constant velocity if already moving — Newton’s first law in the apparatus). Tension equals each weight exactly. Near-equal is the lab configuration: small difference, stately motion, precise timing.
Does pulley mass matter?
In reality, yes: a massive pulley must be rotationally accelerated, adding an effective mass of I/r² to the denominator and lowering a. Friction lowers it further. This calculator is the ideal (massless, frictionless) reference that real measurements are compared against — as Atwood himself intended.
Can one mass be on a table instead of hanging?
That’s the modified Atwood (one mass sliding, possibly with friction) — a different formula (the hanging weight drives total mass, the table mass contributes inertia but not driving weight; friction subtracts). This page covers the classic two-hanging-mass machine.
Is this Atwood machine calculator free?
Yes — free, instant, no sign-up.