How to Solve an Atwood Machine: Acceleration & Tension Formulas

Solve an ideal Atwood machine with a = g|m₂−m₁|/(m₁+m₂) and T = 2m₁m₂g/(m₁+m₂). Worked examples and a free Atwood calculator for acceleration and tension.

What an Atwood Machine Asks

An Atwood machine is two masses connected by a string over a pulley. In intro physics the pulley is treated as massless and frictionless and the string does not stretch, so both masses have the same acceleration magnitude and the tension is the same throughout the string.

The heavier mass descends. Acceleration is less than g because the two masses must accelerate together. Tension sits between the two weights — it is not equal to either mg.

If you already have the two masses and only need checked values of a and T, open the free Atwood machine calculator first, then return here for free-body signs and traps.

The Two Formulas

  • a = g |m₂ − m₁| / (m₁ + m₂) — acceleration magnitude in m/s².
  • T = 2 m₁ m₂ g / (m₁ + m₂) — string tension in newtons.

    • Equivalently, after you know a: T = m_light(g + a) = m_heavy(g − a). Use the g given on the paper (often 9.8 or 10).

    Free-Body Diagrams

    For the descending (heavier) mass: m_h g − T = m_h a.

    For the rising (lighter) mass: T − m_l g = m_l a.

    Add the two equations and the tensions cancel, leaving a = g(m_h − m_l)/(m_h + m_l). Substitute back to get T.

    Each hanging mass is an F = ma problem. Refresh that with the force calculator.

    Units

  • Mass → kilograms (kg). Convert grams by dividing by 1000.
  • g → m/s² (usually 9.8 or 9.81; some papers use 10).
  • a → m/s². T → newtons (N).

    • Trap: leaving masses in grams while g is 9.8 m/s² makes T look thousands of times too small.

    Worked Examples

    Example 1: 1.0 kg and 3.0 kg

    g = 9.8 m/s². a = 9.8 × |3 − 1| / 4 = 4.9 m/s². T = 2(1)(3)(9.8)/4 = 14.7 N. The 3.0 kg mass descends. Check: T = 1(9.8 + 4.9) = 14.7 N.

    Example 2: equal masses

    m₁ = m₂ = 2.0 kg. Then a = 0 and T = mg = 19.6 N. The system does not accelerate; tension equals each weight.

    Example 3: find the heavier mass

    m₁ = 2.0 kg, a = 2.45 m/s², g = 9.8, m₂ heavier. m₂ = m₁(g + a)/(g − a) = 2.0(12.25)/7.35 ≈ 3.333 kg.

    To treat a as a standalone Δv/t check, use the acceleration calculator.

    Common Mistakes

  • Writing a = g as if one mass were in free fall.
  • Setting T equal to the heavier weight or the lighter weight.
  • Giving the two masses different acceleration magnitudes.
  • Mixing grams and kilograms.

    • Sanity check: a is between 0 and g, and T is between the two weights.

    Check With the Free Calculator

    The Atwood machine calculator returns acceleration, tension, and which mass descends, with the substituted working.

    Atwood machine calculator — a and T.

    Force calculator — F = ma on each mass.

    Acceleration calculator — a = Δv/t.

    Inclined plane calculator — another connected-mass F = ma setup.

    Learn a = g|Δm|/Σm and T = 2m₁m₂g/Σm, then confirm every answer on the calculator.

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