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 Video by Zeeshan Zamurred Applications of Forces

Edexcel A Level Maths: 7.6 Pulleys and Connected ParticlesWatch the full walkthrough 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

The shared acceleration

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

heavier: m g - T = ma; lighter: T - m^{'} g = m^{'} a

One equation per particle.

3 g - T = 3 a

and

T - 2 g = 2 a

2Add:

g = 5 a \Rightarrow a = \frac{9.8}{5} = 1.96

m/s².

Answer

a = 1.96

m/s²

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

Formula recap

m g - T = ma

Heavier particle (descending).

T - 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

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