Atwood Machine الصيغة (a & T): مجاني Calculator & الشغلed Examples

Solve Atwood a = g|m₂−m₁|/(m₁+m₂) and T = 2m₁m₂g/(m₁+m₂). مجاني-body signs, unit traps, examples, and a free Atwood calculator.

كيفية الحل an Atwood Machine (Quick Answer)

An ideal Atwood machine is two masses on a string over a massless, frictionless pulley. Both masses share the same acceleration magnitude a, and the string tension T is the same throughout.

  • a = g |m₂ − m₁| / (m₁ + m₂) — acceleration in m/s² (always less than g).
  • T = 2 m₁ m₂ g / (m₁ + m₂) — tension in newtons.
  • The heavier mass descends. Equivalently T = m_light(g + a) = m_heavy(g − a).
  • Use g from the paper (often 9.8 or 10) and masses in kilograms.

    • Need checked a and T now? Open the free Atwood machine calculator — then come back for free-body signs and traps.

    Atwood Machine الصيغةs (a and T)

    Students often search “atwood machine formula” or “atwood acceleration” for these two results. Once both masses and g are known, a and T are fixed — that is why a rearranging calculator covers most homework checks.

  • a = g |Δm| / Σm.
  • T = 2 m₁ m₂ g / Σm.

    • Solve both on the free Atwood machine calculator.

    مجاني-Body Diagrams

    Descending (heavier) mass: m_h g − T = m_h a.

    Rising (lighter) mass: T − m_l g = m_l a.

    Add the equations and the tensions cancel: a = g(m_h − m_l)/(m_h + m_l). Substitute back for T.

    Each hanging mass is an F = ma problem — refresh with the force calculator.

    Units

  • الكتلة → kilograms (convert grams ÷ 1000).
  • g → m/s² (9.8, 9.81, or 10 as given).
  • 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.

    الشغلed 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 → a = 0 and T = mg = 19.6 N. No acceleration; 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.

    Check Examples 1–3 on the Atwood machine calculator.

    أخطاء شائعة

  • 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: 0 ≤ a < g, and T sits between the two weights.

    مجاني Atwood Machine Calculator

    Use the free Atwood machine calculator for acceleration, tension, and which mass descends, with substituted steps.

    القوة calculator — F = ma on each mass.

    التسارع calculator — a = Δv/t.

    Inclined plane calculator — another connected-mass setup.

    Master a = g|Δm|/Σm and T = 2m₁m₂g/Σm — then verify every homework answer.

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