Learn9709Paper 4 · MechanicsNewton's laws of motionConnected particles and pulleys

Newton's laws of motion · Subtopic 3 of 4

Connected Particles and Pulleys

Two objects joined by a taut string move together with the same acceleration. Write $F=ma$ for each one (or for the whole system) and solve for the acceleration and the tension.

Cambridge 9709 Paper 4Edexcel 9MA0 Paper 3AQA 7357 Paper 2≈ 40 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Model strings as light and inextensible, and pulleys as smooth.
Write $F=ma$ for each particle and solve for $a$ and $T$ .
Use the whole-system shortcut to find $a$ quickly.
Handle towing problems with tension or thrust in a tow bar, and what happens when a string goes slack.

Pause and think

A $3$ kg mass and a $2$ kg mass hang over a smooth pulley. The $3$ kg mass is heavier, so it falls. Is the tension in the string $30$ N, $20$ N, or something in between?

Section 2 of 14

Core idea

Start hereA string over a pulley

A block sits on a table, tied by a string over a pulley to a hanging weight. Let go: both move together. The string pulls the block forward and holds the weight back with the same tension. Two equations, one for each object, solve it.

The big idea

Particles joined by a light inextensible string share the same acceleration; over a smooth pulley the tension is the same on both sides. Write $F=ma$ for each; add the equations to eliminate $T$ .

1. Table and hanging mass

A block on a table pulled by a hanging mass

The string pulls A forwards with tension

T

and holds B up with the same

T

. Both move at the same rate

a

Worked exampleA (3 kg) on a smooth table, B (2 kg) hanging

Find $a$ and $T$ .

Goal: $a$ , $T$ .
Strategy: One equation each.

$A:\ T=3a$
Only $T$ horizontally.
$B:\ 20-T=2a$
Weight down, tension up.
$20=5a\Rightarrow a=4,\ T=12$
Add.

Answer

4

m s

^{-2}

;

12

✓ Check

T<20

: B must accelerate down ✓.

Worked exampleRough table, $\mu=0.2$

Repeat.

Goal: $a$ , $T$ .
Strategy: Friction on A: $0.2\times30=6$ N.

$T-6=3a,\quad20-T=2a$
Equations.
$14=5a\Rightarrow a=2.8,\ T=14.4$
Add; substitute.

Answer

2.8

m s

^{-2}

;

14.4

✓ Check

20-14.4=5.6=2(2.8)

✓.

2. Two hanging masses

Worked example5 kg and 3 kg over a pulley

Find $a$ , $T$ , and distance moved in 2 s from rest.

Goal: Values.
Strategy: Heavier goes down.

$50-T=5a,\quad T-30=3a$
Each mass.
$20=8a\Rightarrow a=2.5,\ T=37.5$
Add.
$s=\tfrac12(2.5)(4)=5\text{ m}$
suvat.

Answer

2.5

m s

^{-2}

;

37.5

5

✓ Check

T

between the two weights (

30<37.5<50

) ✓.

Edge caseThe string goes slack If the falling mass hits the floor,

T

becomes zero and the other particle continues with only friction (or nothing) acting.

Pause and think

Why does $T$ cancel when you add the equations?

Summary

Same $a$ , same $T$ .
One equation per particle; add.

Inextensible: Does not stretch.

HistoryAtwood's machine. In 1784 the Cambridge tutor George Atwood described a machine of two nearly equal masses over a pulley. Because the acceleration

g\dfrac{m_1-m_2}{m_1+m_2}

is small, he could time the motion accurately and check Newton's laws. Versions of it are still used in school laboratories.

Fun factLifts cheat with a counterweight. A lift car hangs on one side of a pulley and a counterweight on the other, roughly the mass of the car plus about half its full load. The motor only has to supply the difference, which is why a lift can carry a dozen people using about as much power as a kettle.

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