Coefficient of Friction Formula: μ = f / N, Values, Examples

Q: How is the coefficient of friction calculated?

μ = f / N, the friction force divided by the normal force. Use limiting friction for the static coefficient and sliding friction for the kinetic one.

Q: What is the unit of coefficient of friction?

None at all: friction and normal force are both in newtons, so the units cancel.

Q: What is the difference between static and kinetic coefficients of friction?

The static coefficient describes the maximum grip before slipping starts. The kinetic coefficient describes friction once sliding is under way and is usually smaller.

Q: How is the coefficient of friction related to the angle of repose?

At the angle of repose, the block is just about to slide and μs = tan θ. A steeper angle of repose means a larger coefficient.

The coefficient of friction formula is $\mu = \frac{f}{N}$ : divide the friction force by the normal force pressing the surfaces together. If it takes a steady 15 N to drag a 50 N block across a bench, then $\mu = \frac{15}{50} = 0.30$ .

μ (the Greek letter mu) is a pure number that describes a pair of surfaces, such as rubber on dry road or steel on ice. It has two versions: the static coefficient for surfaces about to slip and the kinetic coefficient for surfaces already sliding. Both sit inside the friction force formula $f = \mu N$ and belong to the laws of motion section.

Quick facts

Formula
$\mu = \frac{f}{N}$
Static version
$\mu_s = \frac{f_{s,\max}}{N}$
Kinetic version
$\mu_k = \frac{f_k}{N}$
Unit
None, it is a ratio of two forces
Angle of repose
$\mu_s = \tan\theta$
Usual range
About 0.05 to 1, and it can be above 1

Formula for coefficient of friction, term by term

Terms in μ = f / N
Symbol	Meaning	Unit
$\mu$	Coefficient of friction for the two surfaces	none
$f$	Friction force: limiting value for μs, sliding value for μk	N
$N$	Normal force between the surfaces	N

Both forces are in newtons, so the units cancel and μ has no unit and no dimensions, [M⁰ L⁰ T⁰]. It depends on what the surfaces are made of, how smooth they are and whether they are dry, wet or oily. For school problems it does not depend on the area of contact or on the sliding speed.

Static and kinetic coefficients

The static coefficient $\mu_s$ uses the largest friction force the surfaces can supply before slipping starts, called limiting friction. The kinetic coefficient $\mu_k$ uses the friction during sliding. For almost every pair of materials $\mu_k < \mu_s$ . Separate pages cover the coefficient of static friction formula and the coefficient of kinetic friction formula, including how to measure each one.

Approximate coefficients for common pairs
Surfaces	Static μs	Kinetic μk
Rubber on dry concrete	about 1.0	about 0.8
Rubber on wet concrete	about 0.5 to 0.7	about 0.3 to 0.5
Wood on wood	0.25 to 0.5	about 0.2
Steel on steel, dry	about 0.7	about 0.6
Ice on ice	about 0.1	about 0.03

Real values vary a lot with surface finish, so treat the table as a guide and always use the number a question gives you.

The coefficient of friction formula on a tilted plane

Tilt a board with a block on it until the block just begins to slide. At that angle θ, the force down the slope $mg\sin\theta$ equals limiting friction $\mu_s mg\cos\theta$ . Cancel mg:

$\mu_s = \frac{\sin\theta}{\cos\theta} = \tan\theta$

The angle is called the angle of repose. It gives a quick lab method needing only a protractor, with no force meter at all.

Worked examples on the formula of coefficient of friction

Example 1: dragging at steady speed

A 50 N block is pulled at constant speed across a level bench by a 15 N horizontal force. Find μk.

Steady speed means friction balances the pull, so f = 15 N. On a level bench N = 50 N.

$\mu_k = \frac{15}{50} = 0.30$

Example 2: from an angle

A coin on a book cover starts to slide when the cover is tilted to 25°. Find μs.

$\mu_s = \tan 25^\circ = 0.466 \approx 0.47$

Example 3: mass in grams

An 800 g wooden block slides steadily when a spring balance pulls it horizontally with 2.4 N. With g = 9.8 m/s², find μk.

Convert: 800 g = 0.800 kg, so $N = 0.800 \times 9.8 = 7.84\ \text{N}$ .

$\mu_k = \frac{2.4}{7.84} = 0.306 \approx 0.31$

Can the coefficient be greater than 1

Yes. A coefficient above 1 just means friction can be larger than the normal force. Soft rubber on clean rough surfaces and some sticky materials manage this. Racing tyres on a warm dry track are a familiar case. What μ cannot be is negative, since friction always opposes the slip.

The angle form helps here: $\mu_s = 1$ means a block stays put on a 45° slope, and values above 1 mean steeper slopes still hold it.

Mistakes when finding μ

Using the mass in kg as N. You need the force mg in newtons.
Taking N = mg on a slope; use mg cos θ unless the angle of repose method applies.
Using an applied force that is not balanced by friction. Only at steady speed or at the point of slipping does the pull equal friction.
Giving μ a unit such as N. It has none.

If your μ comes out above about 1.5 for ordinary materials, check whether you divided by mass instead of weight.

The normal force formula page shows how to get N in every common set up.

Frequently asked questions

How is the coefficient of friction calculated?

μ = f / N, the friction force divided by the normal force. Use limiting friction for the static coefficient and sliding friction for the kinetic one.

What is the unit of coefficient of friction?

None at all: friction and normal force are both in newtons, so the units cancel.

What is the difference between static and kinetic coefficients of friction?

The static coefficient describes the maximum grip before slipping starts. The kinetic coefficient describes friction once sliding is under way and is usually smaller.

How is the coefficient of friction related to the angle of repose?

At the angle of repose, the block is just about to slide and μs = tan θ. A steeper angle of repose means a larger coefficient.