Newton's laws and connected particles

Mathématiques A-level Updated Wed 7 Oct 2026

Newton's laws connect forces with motion. In mechanics questions you apply F = ma to a whole system, or to one part of it, to find an acceleration and then the force in a rope or tow bar.

Part 1 of 3: Learn it

In short

Resultant force = mass × acceleration, with the force in the direction of the acceleration.
Objects joined by a light, inextensible string have the same acceleration (and the same tension throughout).
Treat the system as one object to find a; then one part on its own to find the tension.

Where this is in your specification

Spec points: DfE R1 to R6 (AQA 7357 R1 to R6, Edexcel 9MA0 Mechanics topic 8, OCR A H240 3.03)

Board	Topic: Forces and Newton's laws
DfE content	R1-R6
AQA 7357	R1-R6
Edexcel 9MA0	Mechanics topic 8
OCR H240	3.03

Newton's three laws

First law: an object stays at rest or keeps a constant velocity unless a resultant force acts on it.
Second law: F = ma, where F is the resultant force in newtons, m the mass in kg and a the acceleration in m/s².
Third law: if A exerts a force on B, B exerts an equal and opposite force on A.

Forces to include

Force	Acts
Weight, mg	vertically downwards
Normal reaction, R	perpendicular to the surface
Tension or thrust	along a string (tension pulls) or rod (thrust can push)
Resistance or friction	against the direction of motion

Connected particles

For a car towing a trailer, both have the same acceleration, and the tow bar pulls the trailer forwards with the same force as it pulls the car backwards (third law). Over a smooth pulley, a string has the same tension on both sides, and the heavier particle accelerates down while the lighter one accelerates up at the same rate.

Quick check

A resultant force of 20 N acts on a 4 kg mass. Find its acceleration.

Show the answer

a = 20 ÷ 4 = 5 m/s².

Part 2 of 3: See it worked

Worked examples

Example 1

A 1200 kg car tows a 400 kg trailer. The driving force is 2400 N, with resistances of 300 N on the car and 100 N on the trailer. Find the acceleration and the tension in the tow bar.

Whole system: 2400 − 300 − 100 = 1600a, so a = 1.25 m/s²
Trailer alone: T − 100 = 400 × 1.25 = 500
T = 600 N

Answer: 1.25 m/s² and 600 N.

Example 2

Particles of 7 kg and 3 kg hang from a light string over a smooth pulley and are released from rest. Find the acceleration and the tension. (g = 9.8 m/s²)

7 kg: 7g − T = 7a. 3 kg: T − 3g = 3a
Adding the two equations: 4g = 10a, giving a = 3.92 m/s²
T = 3g + 3a = 3 × 13.72 = 41.16 N

Answer: 3.92 m/s² and 41.2 N (3 s.f.).

Example 3

A 600 kg lift accelerates upwards at 0.5 m/s². Find the tension in its cable.

Up is positive: T − 600g = 600 × 0.5
T = 600 × (9.8 + 0.5)

Answer: 6180 N.

Common mistakes

Putting the tension into the whole-system equation. Internal forces cancel out.
Using weight in kg instead of newtons.
Writing F = ma with F as one force instead of the resultant force.
Giving the two particles over a pulley different accelerations.

Quick check

What does "inextensible" tell you about a string?

Show the answer

It does not stretch, so the particles it joins have the same acceleration.

Part 3 of 3: Test yourself

Check yourself

Answer each one in your head or on paper first, then open it to check.

A resultant force of 20 N acts on a 4 kg mass. Find its acceleration.

a = 20 ÷ 4 = 5 m/s².

What does "inextensible" tell you about a string?

It does not stretch, so the particles it joins have the same acceleration.

State Newton's third law.

When one object exerts a force on another, the second exerts a force of the same size in the opposite direction on the first.

Jobs that use this

Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).

Ces fiches sont en anglais car elles suivent les programmes d'examen britanniques.

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