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7.6MechanicsStretch

Connected Particles

Two particles connected by a string over a pulley move together with the same acceleration. Writing Newton’s second law for each particle gives simultaneous equations for the acceleration and tension.

28 min Applications of Forces
Edexcel A Level Maths: 7.6 Pulleys and Connected ParticlesCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Set up equations for connected particles
  • Use the same acceleration for both
  • Apply F = ma to each particle
  • Solve for acceleration and tension
1

The shared acceleration

An inextensible string means both particles have the same magnitude of acceleration aa and the tension TT is the same throughout (for a light, smooth pulley). Apply F=maF = ma to each particle separately.

heavier: mg−T=ma;lighter: T−m′g=m′a\text{heavier: } mg - T = ma; \quad \text{lighter: } T - m'g = m'a
One equation per particle.
Example 1Masses 3 kg and 2 kg over a smooth pulley (g = 9.8). Find a.
13g−T=3a3g - T = 3a and T−2g=2aT - 2g = 2a.
2Add: g=5a⇒a=9.85=1.96g = 5a \Rightarrow a = \tfrac{9.8}{5} = 1.96 m/s².
Answer
a=1.96a = 1.96 m/s²

Tip — Adding the two equations eliminates T and gives the acceleration directly.

Formula recap

mg−T=mamg - T = ma
Heavier particle (descending).
T−m′g=m′aT - m'g = m'a
Lighter particle (ascending).

Common mistakes to avoid

Using different accelerations for the two particles.
A taut inextensible string gives them the same acceleration.
Letting the tension differ on each side of a smooth pulley.
For a light, smooth pulley the tension is the same throughout.

Key takeaways

  • Connected particles share the same acceleration.
  • Tension is equal throughout a light, smooth pulley.
  • Apply F = ma to each, then add to eliminate T.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. Connected particles on a taut string have:

    • the same acceleration
    • different accelerations
    • zero acceleration
    • opposite speeds
    Show answer and explanation

    Answer: the same acceleration

    Inextensible string.

  2. Over a light, smooth pulley the tension is:

    • the same throughout
    • larger on the heavy side
    • zero
    • equal to the weight
    Show answer and explanation

    Answer: the same throughout

    Smooth pulley.

  3. For the descending mass, Newton’s law is:

    • mg−T=mamg - T = ma
    • T−mg=maT - mg = ma
    • mg+T=mamg + T = ma
    • T=maT = ma
    Show answer and explanation

    Answer: mg−T=mamg - T = ma

    Net downward force.

  4. Masses 3 kg and 2 kg over a smooth pulley, a ≈ (g=9.8):

    • 1.961.96 m/s²
    • 9.89.8 m/s²
    • 0.980.98 m/s²
    • 4.94.9 m/s²
    Show answer and explanation

    Answer: 1.961.96 m/s²

    g/5.

  5. Adding the two equations eliminates:

    • the tension T
    • the acceleration
    • the mass
    • gravity
    Show answer and explanation

    Answer: the tension T

    T cancels.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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