Definitions

# Coefficient of friction

The coefficient of friction is a dimensionless quantity symbolized by the Greek letter μ and is used to approximate the force of friction. Friction can be viewed, again as an approximation, as being of two primary types, static or kinetic.

The coefficient of static friction is defined as the ratio of the maximum static friction force (F) between the surfaces in contact to the normal (N) force. The coefficient of kinetic friction is defined as the ratio of the kinetic friction force (F) between the surfaces in contact to the normal force:

$mu = frac\left\{F\right\}\left\{N\right\}$

Both static and kinetic coefficients of friction depend on the pair of surfaces in contact; their values are usually approximately determined experimentally. For a given pair of surfaces, the coefficient of static friction is usually larger than that of kinetic friction; in some sets the two coefficients are equal, such as teflon-on-teflon.

The friction force is directed in the opposite direction of the resultant force acting on a body. In the case of kinetic friction, the direction of the friction force may or may not match the direction of motion: a block sliding atop a table with rectilinear motion is subject to friction directed along the line of motion; an automobile making a turn is subject to friction acting perpendicular to the line of motion (in which case it is said to be 'normal' to it). A motionless body is subject to static friction. The direction of the static friction force can be visualized as directly opposed to the force that would otherwise cause motion, were it not for the static friction preventing motion. In this case, the friction force exactly cancels the applied force, so the net force given by the Vector sum, equals zero. It is important to note that in all cases, Newton's first law of motion holds.

While it is often stated that the coefficient of friction (COF) is a "material property," it is better categorized as a "system property." Unlike true material properties (such as conductivity, dielectric constant, yield strength), the COF for any two materials depends on system variables like temperature, velocity, atmosphere and also what are now popularly described as aging and deaging times; as well as on geometric properties of the interface between the materials. For example, a copper pin sliding against a thick copper plate can have a COF that varies from 0.6 at low speeds (metal sliding against metal) to below 0.2 at high speeds when the copper surface begins to melt due to frictional heating. The latter speed, of course, does not determine the COF uniquely; if the pin diameter is increased so that the frictional heating is removed rapidly, the temperature drops, the pin remains solid and the COF rises to that of a 'low speed' test.

## The normal force

The normal force is defined as the net force compressing two parallel surfaces together; and its direction is perpendicular to the surfaces. In the simple case of a mass resting on a horizontal surface, the only component of the normal force is the force due to gravity, where $N=mg$. In this case, the magnitude of the friction force is the product of the mass of the object, the acceleration due to gravity, and the coefficient of friction. However, the coefficient of friction is not a function of mass or volume; it depends only on the material. For instance, a large aluminum block has the same coefficient of friction as a small aluminum block. However, the magnitude of the friction force itself depends on the normal force, and hence the mass of the block.

If an object is on a level surface and the force tending to cause it to slide is horizontal, the normal force $N$ between the object and the surface is just its weight, which is equal to its mass multiplied by the acceleration due to earth's gravity, g. If the object is on a tilted surface such as an inclined plane, the normal force is less, because less of the force of gravity is perpendicular to the face of the plane. Therefore, the normal force, and ultimately the frictional force, is determined using vector analysis, usually via a free body diagram. Depending on the situation, the calculation of the normal force may include forces other than gravity.

## Static friction and kinetic friction

Friction forces are categorized as either static or kinetic. The coefficient of static friction $mu_s$ characterizes friction when no movement exists between the two surfaces in question, and the kinetic coefficient $mu_k$ characterizes friction where motion occurs. While static and kinetic friction differ in value (the coefficient of static friction typically being greater than that of kinetic friction), both result from the electric force acting on microscopic irregularities in two adjacent surfaces. The friction force depends mostly on the resultant force acting on the body, while the motion of the body only influences the distinction between static or kinetic. While it is easier to visualize friction in terms of motion, its dependence is on the resultant force acting on a body.

The static friction force must be overcome by an applied force before an object can move. The maximum possible friction force between two surfaces before sliding begins is the product of the coefficient of static friction and the normal force: $F_\left\{max\right\} = mu_s N$. When there is no sliding occurring, the friction force can have any value from zero up to $F_\left\{max\right\}$. Any force smaller than $F_\left\{max\right\}$ attempting to slide one surface over the other is opposed by a frictional force of equal magnitude and opposite direction. Any force larger than $F_\left\{max\right\}$ overcomes the force of static friction and causes sliding to occur. The instant that sliding occurs, kinetic friction is applicable and static friction is no longer relevant.

When the resultant force acting along the plane of a surface increases in value, a proportional increase in static friction force results up to a maximum value $F_\left\{max\right\}$. Since this is the maximum value that static friction can take for any particular material, a further increase in the resultant force will produce motion. A lower value of friction, kinetic friction, replaces static friction for the duration of the movement.

When one surface is sliding over the other, the friction force between them is always the same and is given by the product of the coefficient of kinetic friction and the normal force: $F = mu_k N$. The coefficient of static friction is larger than the coefficient of kinetic friction, since it takes more force to make surfaces start sliding over each other than it does to keep them sliding once started.

These empirical relationships are only approximations. For example, the friction between surfaces sliding over each other may depend to some extent on the contact area, or on the sliding velocity. The friction force is electromagnetic in origin, as atoms of one surface stick to atoms of the other briefly before snapping apart, causing atomic vibrations and thus transforming into heat the work needed to maintain the sliding. The coefficient also changes slightly along a surface and is only an approximation. However, despite the complexity of the fundamental physics behind friction, the relationships are accurate enough to be useful in many applications.

## Frequent mistake

Occasionally it is maintained that µ is always < 1, but this is not true. While in most relevant applications µ < 1, a value above 1 merely implies that the force required to slide an object along the surface is greater than the normal force of the surface on the object. For example, silicone rubber or acrylic rubber-coated surfaces have a coefficient of friction that can be substantially larger than 1.

CRC Handbook of Chemistry & Physics - Values for Coefficient of Friction
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