Forces and Newton's Laws

A-Level Maths · Mechanics

Forces

A force is a push or pull (a vector), measured in newtons. Common forces:

Weight W = mg (downwards).
Normal reaction R (perpendicular to a surface).
Tension (in strings), thrust (in rods).
Friction F (opposes motion, along the surface).

Newton's laws

First law: an object stays at rest or moves at constant velocity unless a resultant force acts.
Second law: F = ma — the resultant force = mass × acceleration (in the direction of the resultant).
Third law: forces come in equal and opposite pairs acting on different objects.

Resolving forces

Split forces into perpendicular components (often horizontal/vertical, or along/perpendicular to a slope):

horizontal = F cos θ,   vertical = F sin θ

For an object in equilibrium, the components in each direction balance (resultant = 0). For accelerating objects, apply F = ma in each direction.

Friction

Friction opposes (attempted) motion, up to a maximum: F ≤ μR, where μ is the coefficient of friction and R the normal reaction.
At the point of slipping (limiting equilibrium), F = μR.

Connected particles

For objects connected by strings (e.g. over a pulley, or a car towing a trailer):

Treat the system together to find the acceleration (using the total mass and net driving force), then look at one object to find the tension.
The tension is the same throughout a light, inextensible string over a smooth pulley.

Motion on a slope

Resolve weight into components along the slope (mg sin θ, down the slope) and perpendicular (mg cos θ, giving the normal reaction R = mg cos θ). Then apply F = ma along the slope.

Worked example

A 2 kg block on a smooth horizontal surface is pulled by a horizontal force of 10 N. Find its acceleration.

F = ma → 10 = 2a → a = 5 m/s². ✓

Common mistakes

Forgetting to resolve forces into components before applying F = ma.
Using weight = m instead of mg.
On a slope, mixing up sin θ (along) and cos θ (perpendicular) components.

Exam tips

Draw a clear force diagram and choose sensible directions to resolve.
Apply F = ma in each direction; use equilibrium (resultant = 0) when not accelerating.
For connected particles, find acceleration from the whole system, then tension from one part.

Key facts to remember

F = ma (Newton's second law); weight W = mg; equilibrium ⇒ resultant force = 0.
Friction F ≤ μR (F = μR at limiting equilibrium).
Resolve forces (F cos θ / F sin θ); on a slope, weight components are mg sin θ (along) and mg cos θ (perpendicular).