# Coefficient of friction

The coefficient of friction , also called the coefficient of friction (symbol µ or  f ), is a quantity in the dimension number for the ratio of the frictional force to the contact force between two bodies. The term belongs to the field of tribology .

## Physical meaning

When specifying a coefficient of friction, a distinction is made between sliding friction and static friction : With sliding friction, the friction surfaces move relative to one another, while with static friction they do not. In the case of Coulomb friction, the sliding coefficient is constant. In practice, a corresponding temperature , speed and pressure dependency can be seen, which indicates an influence of the surface change and the nature of the never ideally flat surface (but not on the coefficient of friction itself) and thus apparently influences the material properties.

The coefficient of friction in metals is measured on polished surfaces in order to largely exclude mechanical interlocking (form fit). The decisive factors are the adhesion and cohesion forces between the materials. Depending on the material, van der Waals forces or, in polarized materials, hydrogen bonds, similar forces are formed between the surfaces. The material adhesion is highest with ionic materials such as B. table salt .

## Calculation of the frictional force

With the help of the coefficient of friction, the maximum static or sliding friction force between two bodies can be calculated.

Static friction: ${\ displaystyle F _ {\ mathrm {R, H}} \ leq \ mu _ {\ mathrm {H}} \ cdot F _ {\ mathrm {N}}}$
maximum static friction: ${\ displaystyle F _ {\ mathrm {R, max}} = \ mu _ {\ mathrm {H}} \ cdot F _ {\ mathrm {N}}}$
Sliding friction: ${\ displaystyle F _ {\ mathrm {R, G}} = \ mu _ {\ mathrm {G}} \ cdot F _ {\ mathrm {N}}}$

Here is the frictional force or the coefficient of friction and the normal force (force perpendicular to the surface). The coefficient of friction determines how large the friction force is in relation to the normal force; a higher coefficient of friction means a greater frictional force. ${\ displaystyle F _ {\ mathrm {R}}}$${\ displaystyle \ mu _ {\ mathrm {H}}}$${\ displaystyle \ mu _ {\ mathrm {G}}}$${\ displaystyle F _ {\ mathrm {N}}}$

For example, in order to push a metal block, one must first apply a force that is higher than the static friction force. If the block then slides over the ground, the smaller sliding friction force is sufficient. Because the coefficient of friction depends on the surface (dry, wet, ...), the friction forces also depend on it to the same extent.

To change the adhesion, you can also change the normal force, which in turn can be seen from the formula. On the plane the normal force corresponds to the weight force , in steep curves the component of the vector sum of weight force and centrifugal force perpendicular to the roadway. In motorsport, the normal force is increased by wings ( English spoilers ), which use the airstream to press the vehicle against the ground.

## Examples

The coefficients of friction from tables are only approximate. The friction depends on many different factors (material pairing, surface, lubrication, temperature, humidity, wear, normal force, etc.), so that the "correct" values ​​cannot be found in a table.

The most accurate results are obtained from a test under real conditions. Here, too, however, it should be noted that the relationships between the test and real use can change.

It always applies:

${\ displaystyle \ mu \ leq \ mu _ {\ mathrm {H}}}$
Static friction and sliding friction
(guide values, dry)
Material pairing Static friction Sliding friction
Steel on steel 0.2 0.1
Steel on wood 0.5 0.4
Steel on stone 0.8 0.7
Stone on wood 0.9 0.7
Leather on metal 0.6 0.4
Wood on wood 0.5 0.4
stone on stone 1.0 0.9
Steel on ice 0.03 0.01
Steel on concrete 0.35 0.20

### Static friction coefficients

Static friction coefficients µ H (guide values)
Material pairing dry little greasy lubricated with water
Bronze on bronze 0.18 0.11
Gray cast iron 0.56 0.73
steel 0.19 0.18
Cast iron on Oak 0.98
Gray cast iron 0.2 0.21
Oak on oak 0.58 0.71
Leather straps on Oak 0.49
Gray cast iron 0.48 0.28 0.12 0.38
Brass on oak 0.62 0.15
Steel on bronze 0.19
Oak 0.11 0.65
ice 0.027
Gray cast iron 0.19
steel 0.15 0.13
aluminum 0.19
Hemp rope on wood 0.5

A driven or braked tire always has a slip with respect to the surface on which it rolls. With small transmitted tangential forces, this slip is so small that it can be neglected for many applications. With a higher tangential force, the slip initially increases slightly, then increases more and more. This means that a maximum tangential force can be transmitted with a given pressure. This is similar to the transition from static friction to sliding friction. The quotient between the tangential force and the normal force is called the adhesion coefficient. Its maximum indicates the maximum force a tire can transmit as drive or braking force for a given normal force.

Max. Adhesion coefficient µ H (guide values)
pairing dry wet, clean wet, lubricated icy
Pneumatic tires on arable soil 0.45 0.2 <0.2
Pneumatic tires on dirt road 0.45 0.2 <0.2
Pneumatic tires on wooden pavement 0.55 0.3 0.2 <0.2
Pneumatic tires on small pavement 0.55 0.3 0.2 <0.2
Pneumatic tires on cobblestones 0.6 0.4 0.3 <0.2
Pneumatic tires on gravel, rolled 0.7 0.5 0.4 <0.2
Pneumatic tires on gravel, rolled, asphalted 0.6 0.4 0.3 <0.2
Grapple wheels on arable soil 0.5
Tracked vehicles on arable land 0.8

### Coefficients of sliding friction

Coefficients of sliding friction µ G (reference values)
Material pairing dry little greasy lubricated with water
Bronze on bronze 0.20 0.06
Bronze on gray cast iron 0.21 0.08
Bronze on steel 0.18 0.16 0.07
Gray cast iron on bronze 0.20 0.15 0.08
Gray cast iron on oak 0.49 0.19 0.22
Gray cast iron on gray cast iron 0.28 0.15 0.08 0.31
Oak on oak 0.34 0.1 0.25
Leather straps on oak 0.27 0.29
Leather straps on gray cast iron 0.56 0.27 0.12 0.36
Brass on oak 0.60 0.44 0.24
Steel on bronze 0.18 0.16 0.07
Steel on oak 0.5 0.08 0.26
Steel on ice 0.014
Steel on gray cast iron 0.18 0.01
Steel on steel 0.12 0.01
Steel on brass 0.2
Steel on white metal 0.2 0.1 0.04
blocked car wheel on pavement 0.5 0.2

## Geometric interpretation

Figure 1: Resulting force within the friction cone

It can also be viewed as the tangent of the angle of friction . This is the smallest angle at which a body would slide down on an inclined plane. It applies ${\ displaystyle \ mu}$ ${\ displaystyle \ varphi}$

${\ displaystyle \ mu = \ tan (\ varphi)}$.

Take a car, for example: The tangent is known from everyday life as the incline of ascending roads and inclines, which is indicated on traffic signs (for example: 12% means an incline, over a length of 100 m the distance increases by 12 m). With a coefficient of static friction of one, slopes of a maximum of 100% (45 °) can be overcome. In reality, the climbing ability of vehicles is usually limited by the installed engine power and the overall transmission ratio of the gearbox - the exceptions are poor road conditions. On black ice or on a snow-covered road, the coefficient of static friction is very low, so that even slight inclines cannot be overcome or braking downhill is no longer possible.

Friction cone: Within the friction cone (Figure 1), systems are stable even under load (e.g. ladder on the ground) and are referred to as self-locking ; outside the friction cone, the friction force is no longer sufficient to keep the system at rest, it kicks a movement. Relevant technical systems are e.g. B. worm gears that are self-locking or not depending on screw pitch, material pairing and lubrication conditions.

## Limits

Reach caused by the occurring forces stresses the yield stress , the scope ends Coulomb model. The friction factor model takes its place .

## Common mistakes

### "Μ is always less than one"

It is sometimes claimed that it should apply. only means that normal and frictional force are equal. With several material pairings, for example surfaces coated with silicone rubber or acrylic rubber, the coefficient of friction is significantly greater than one. ${\ displaystyle \ mu \ leq 1}$${\ displaystyle \ mu = 1}$

### "Static friction is the coefficient of static friction times normal force"

The formula is often used for static friction

${\ displaystyle F _ {\ mathrm {R, H}} = \ mu _ {\ mathrm {H}} \ cdot F _ {\ mathrm {N}}}$

specified. However, the value calculated in this way only designates the limit case of the maximum possible push or pull force that counteracts the frictional force and at which the object can still be stationary. If this is exceeded, the mostly smaller sliding friction force acts immediately : ${\ displaystyle F _ {\ mathrm {R}}}$

${\ displaystyle F _ {\ mathrm {R, G}} = \ mu _ {\ mathrm {G}} \ cdot F _ {\ mathrm {N}}}$

Apparently this is z. B. in avalanches or landslides . Here the masses are close to the adhesive force. Small vibrations can locally exceed the static friction.

## literature

• Valentin L. Popov: Contact Mechanics and Friction. A text and application book from nanotribology to numerical simulation. Springer-Verlag, Berlin a. a. 2009, ISBN 978-3-540-88836-9 .

## swell

1. ^ Rainer Müller: Classic mechanics: from long jump to Mars flight . Walter de Gruyter, 2009, p. 115 ( limited preview in Google Book search).
2. a b c Horst Kuchling: Pocket book of physics. VEB Fachbuchverlag, Leipzig 1986, ISBN 3-87144-097-3 .