Friction Force Formula: f = μN with Static and Kinetic Cases

The friction force formula is $f = \mu N$ : the force of friction equals the coefficient of friction multiplied by the normal force pressing the two surfaces together. A box pressed on the floor with 400 N, on a surface with $\mu = 0.25$ , feels a friction force of 100 N when it slides.

Friction always acts along the surface and against the sliding, or against the sliding that would happen without it. Two versions of the formula exist, one for surfaces gripping each other and one for surfaces already slipping. This page covers both, and it belongs to the laws of motion section.

Quick facts

Formula
$f = \mu N$
Static friction
$f_s \le \mu_s N$ , it adjusts up to a maximum
Kinetic friction
$f_k = \mu_k N$ , constant while sliding
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
Direction
Along the surface, opposite to the motion or the attempted motion

Static and kinetic forms of the formula of friction

When nothing is sliding yet, static friction matches whatever push you apply, up to a ceiling. Push a heavy cabinet gently and friction pushes back just as hard, so nothing moves. The ceiling is $f_{s,\max} = \mu_s N$ . Once your push beats it, the cabinet slips and kinetic friction takes over at $f_k = \mu_k N$ .

Static and kinetic friction compared
Feature	Static friction	Kinetic friction
When it acts	Surfaces not slipping	Surfaces sliding
Formula	$f_s \le \mu_s N$	$f_k = \mu_k N$
Size	Any value from 0 up to the maximum	Fixed for a given N and pair of surfaces
Coefficient	$\mu_s$ , usually larger	$\mu_k$ , usually smaller

That is why a stuck object is hardest to start and then slides more easily. The coefficients themselves have their own pages: static and kinetic.

Formula for friction force: how to use it

Find the normal force first. On a level floor with no other vertical push, N = mg. On a slope, N = mg cos θ. See the normal force formula.
Decide whether the body slides. If it is moving, use the kinetic coefficient. If it is at rest, static friction equals the applied force until that force reaches the maximum.
Multiply. $f = \mu N$ gives the kinetic value or the static limit.
Point it the right way. Draw friction along the surface, opposing the slip.

Frictional force formula: worked examples

Example 1: sliding a crate

A 40 kg crate slides across a level floor. The coefficient of kinetic friction is 0.30 (take g = 9.8 m/s²). How large is the friction force?

N = mg = 40 × 9.8 = 392 N.

$f_k = \mu_k N = 0.30 \times 392 = 117.6\ \text{N} \approx 118\ \text{N}$

Example 2: a block on a ramp

A 5.0 kg block slides down a 30° ramp with μk = 0.20. Find the friction on it.

$N = mg\cos 30^\circ = 5.0 \times 9.8 \times 0.866 = 42.4\ \text{N}$

$f_k = 0.20 \times 42.4 = 8.5\ \text{N, acting up the slope}$

Example 3: does it move

You push a 20 kg box with 50 N. The coefficient of static friction is 0.40. Does the box move, and what is the friction?

Maximum static friction: 0.40 × 20 × 9.8 = 78.4 N.

Your 50 N is less than 78.4 N, so the box stays put. Static friction is exactly 50 N, not 78.4 N.

Only use μN for static friction when the question says the body is just about to move. Otherwise static friction equals the force it balances.

Laws of friction and what friction does not depend on

Friction is proportional to the normal force, which is the content of the formula itself.
For ordinary dry surfaces, it does not depend on the apparent area of contact. A brick slides with the same friction on its wide face or its narrow face.
Kinetic friction hardly changes with sliding speed at the speeds in school problems.
The coefficient depends on the pair of materials and how rough or clean they are.

Rolling friction, met by wheels and ball bearings, is far smaller than sliding friction for the same load. Fluid friction, the drag in air or water, follows different rules and does depend on speed; see the drag force and viscous force pages.

Friction force formula physics: energy and work

Kinetic friction turns kinetic energy into heat. Over a sliding distance d, the work done by friction is $W = -f_k d$ . A 118 N friction force over 5.0 m removes 590 J from the crate in Example 1. The work done by friction formula page covers the full energy method, and the force formula page shows how friction feeds into $F = ma$ as one of the forces in the net sum.

Errors that cost marks with friction

Writing N = mg on a slope. On an incline, N = mg cos θ, which is less than mg.
Forgetting an angled pull. Pulling a sledge upward at an angle lifts it slightly and lowers N, so friction drops.
Always using $\mu_s N$ . Static friction is only that big at the point of slipping.
Drawing friction along the motion. It opposes the slipping of the surfaces in contact. For a driven wheel, though, friction on the tyre points forward, which is what pushes the car.

Frequently asked questions

What is the formula of friction?

The friction force is f = μN, where μ is the coefficient of friction and N is the normal force. Use μk for sliding surfaces and μs for the maximum grip before slipping.

What is the SI unit of frictional force?

Like every force, friction is measured in newtons (N), while μ itself carries no unit.

Does friction depend on the area of contact?

For dry solid surfaces in school physics, no. The friction depends on the normal force and the coefficient, not on how large the contact face is.

Why is static friction greater than kinetic friction?

Surfaces at rest settle into closer contact and form more tiny bonds. Once sliding, they have less time to grip, so the coefficient of kinetic friction is usually lower.

Is friction always opposite to motion?

It is opposite to the relative slipping of the two surfaces. For a person walking or a car accelerating, friction on the foot or tyre points forward and causes the motion.