Connected particles and pulleys

The idea

Particles joined by a light inextensible string share the same acceleration, and the tension is the same all along a string over a smooth pulley. Write $F = ma$ for each particle separately, then solve together.

Mark the direction of motion and the tension on each particle.
Write $F = ma$ for each particle.
Add the equations to remove $T$ , find $a$ , then substitute back for $T$ .

Watch out The particles share an acceleration, but the resultant force on each is different.

See it worked

Worked example

Particles of mass $5\text{ kg}$ and $3\text{ kg}$ hang over a smooth pulley. Find the acceleration and the tension.

$5\text{ kg}$ : $5g - T = 5a$ . $3\text{ kg}$ : $T - 3g = 3a$ .
Add: $2g = 8a$ , so $a = 2.45\text{ m s}^{-2}$ .
$T = 3(9.8 + 2.45) = 36.75\text{ N}$ .

Answer $a = 2.45\text{ m s}^{-2}$ , $T = 36.75\text{ N}$

Your turn

Question 1Non-calculator3 marks

Particles of mass $4\text{ kg}$ and $6\text{ kg}$ hang over a smooth pulley and are released from rest. Find the acceleration. Use $g = 9.8\text{ m s}^{-2}$ .

Question 2Calculator allowed3 marks

For the same particles ( $4\text{ kg}$ and $6\text{ kg}$ over a smooth pulley), find the tension in the string.

Question 3Non-calculator3 marks

A car of mass $1000\text{ kg}$ tows a trailer of mass $400\text{ kg}$ with a light rigid towbar. The driving force is $2100\text{ N}$ and there is no resistance. Find the tension in the towbar.

In past papers

20 official questions in 20 papers since 2018. The latest:

All A-level Maths past papers

A-level Maths Pro, Pearson Edexcel 9MA0

A-level Maths Pro

£129 once

Price rises to £199 on 1 January. Save £70 by joining before then.

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.

No VAT is charged.