Friction

Free-body diagram for a block on a ramp. Arrows are vectors indicating directions and magnitudes of forces. N is the normal force, mg is the force of gravity, and F_f is the force of friction.

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 ${\displaystyle N=mg\,}$. In this case, conditions of equilibrium tell us that the magnitude of the friction force is zero, ${\displaystyle F_{f}=0}$. In fact, the friction force always satisfies ${\displaystyle F_{f}\leq \mu N}$, with equality reached only at a critical ramp angle (given by ${\displaystyle \tan ^{-1}\mu }$) that is steep enough to initiate sliding.

The friction coefficient is an empirical (experimentally measured) structural property that depends only on various aspects of the contacting materials, such as surface roughness. The coefficient of friction is not a function of mass or volume. 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 on the mass of the block.

Depending on the situation, the calculation of the normal force ${\displaystyle N}$ might include forces other than gravity. If an object is on a level surface and subjected to an external force ${\displaystyle P}$ tending to cause it to slide, then the normal force between the object and the surface is just ${\displaystyle N=mg+P_{y}}$, where ${\displaystyle mg}$ is the block's weight and ${\displaystyle P_{y}}$ is the downward component of the external force. Prior to sliding, this friction force is ${\displaystyle F_{f}=-P_{x}}$, where ${\displaystyle P_{x}}$ is the horizontal component of the external force. Thus, ${\displaystyle F_{f}\leq \mu N}$ in general. Sliding commences only after this frictional force reaches the value ${\displaystyle F_{f}=\mu N}$. Until then, friction is whatever it needs to be to provide equilibrium, so it can be treated as simply a reaction.

If the object is on a tilted surface such as an inclined plane, the normal force from gravity is smaller than ${\displaystyle mg}$, because less of the force of gravity is perpendicular to the face of the plane. The normal force and the frictional force are ultimately determined using vector analysis, usually via a free body diagram.

In general, process for solving any statics problem with friction is to treat contacting surfaces tentatively as immovable so that the corresponding tangential reaction force between them can be calculated. If this frictional reaction force satisfies ${\displaystyle F_{f}\leq \mu N}$, then the tentative assumption was correct, and it is the actual frictional force. Otherwise, the friction force must be set equal to ${\displaystyle F_{f}=\mu N}$, and then the resulting force imbalance would then determine the acceleration associated with slipping.

The coefficient of friction (COF), often symbolized by the Greek letter µ, is a dimensionless scalar value which equals the ratio of the force of friction between two bodies and the force pressing them together, either during or at the onset of slipping. The coefficient of friction depends on the materials used; for example, ice on steel has a low coefficient of friction, while rubber on pavement has a high coefficient of friction. Coefficients of friction range from near zero to greater than one. The coefficient of friction between two surfaces of similar metals is greater than that between two surfaces of different metals; for example, brass has a higher coefficient of friction when moved against brass, but less if moved against steel or aluminum.[28]

For surfaces at rest relative to each other, ${\displaystyle \mu =\mu _{\mathrm {s} }}$, where ${\displaystyle \mu _{\mathrm {s} }}$ is the coefficient of static friction. This is usually larger than its kinetic counterpart. The coefficient of static friction exhibited by a pair of contacting surfaces depends upon the combined effects of material deformation characteristics and surface roughness, both of which have their origins in the chemical bonding between atoms in each of the bulk materials and between the material surfaces and any adsorbed material. The fractality of surfaces, a parameter describing the scaling behavior of surface asperities, is known to play an important role in determining the magnitude of the static friction.[1]

For surfaces in relative motion ${\displaystyle \mu =\mu _{\mathrm {k} }}$, where ${\displaystyle \mu _{\mathrm {k} }}$ is the coefficient of kinetic friction. The Coulomb friction is equal to ${\displaystyle F_{\mathrm {f} }}$, and the frictional force on each surface is exerted in the direction opposite to its motion relative to the other surface.

Arthur Morin introduced the term and demonstrated the utility of the coefficient of friction.[12] The coefficient of friction is an empirical measurement — it has to be measured experimentally, and cannot be found through calculations.[29] Rougher surfaces tend to have higher effective values. Both static and kinetic coefficients of friction depend on the pair of surfaces in contact; 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.

Most dry materials in combination have friction coefficient values between 0.3 and 0.6. Values outside this range are rarer, but teflon, for example, can have a coefficient as low as 0.04. A value of zero would mean no friction at all, an elusive property. Rubber in contact with other surfaces can yield friction coefficients from 1 to 2. 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.

While it is often stated that the 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, namely surface structure.[1] 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.

Materials		Static Friction, ${\displaystyle \mu _{\mathrm {s} }}$		Kinetic/Sliding Friction, ${\displaystyle \mu _{\mathrm {k} }\,}$
Dry and clean	Lubricated	Dry and clean	Lubricated
Aluminium	Steel	0.61[30]		0.47[30]
Aluminium	Aluminium	1.05–1.35[30]	0.3[30]	1.4[30]–1.5[31]
Gold	Gold			2.5[31]
Platinum	Platinum	1.2[30]	0.25[30]	3.0[31]
Silver	Silver	1.4[30]	0.55[30]	1.5[31]
Alumina ceramic	Silicon nitride ceramic				0.004 (wet)[32]
BAM (Ceramic alloy AlMgB14)	Titanium boride (TiB2)	0.04–0.05[33]	0.02[34][35]
Brass	Steel	0.35–0.51[30]	0.19[30]	0.44[30]
Cast iron	Copper	1.05[30]		0.29[30]
Cast iron	Zinc	0.85[30]		0.21[30]
Concrete	Rubber	1.0	0.30 (wet)	0.6–0.85[30]	0.45–0.75 (wet)[30]
Concrete	Wood	0.62[30][36]
Copper	Glass	0.68[37]		0.53[37]
Copper	Steel	0.53[37]		0.36[30][37]	0.18[37]
Glass	Glass	0.9–1.0[30][37]	0.005–0.01[37]	0.4[30][37]	0.09–0.116[37]
Human synovial fluid	Human cartilage		0.01[38]		0.003[38]
Ice	Ice	0.02–0.09[39]
Polyethene	Steel	0.2[30][39]	0.2[30][39]
PTFE (Teflon)	PTFE (Teflon)	0.04[30][39]	0.04[30][39]		0.04[30]
Steel	Ice	0.03[39]
Steel	PTFE (Teflon)	0.04[30]−0.2[39]	0.04[30]		0.04[30]
Steel	Steel	0.74[30]−0.80[39]	0.005–0.23[37][39]	0.42–0.62[30][37]	0.029–0.19[37]
Wood	Metal	0.2–0.6[30][36]	0.2 (wet)[30][36]	0.49[37]	0.075[37]
Wood	Wood	0.25–0.62[30][36][37]	0.2 (wet)[30][36]	0.32–0.48[37]	0.067–0.167[37]

Under certain conditions some materials have very low friction coefficients. An example is (highly ordered pyrolytic) graphite which can have a friction coefficient below 0.01.[40] This ultralow-friction regime is called superlubricity.

When the mass is not moving, the object experiences static friction. The friction increases as the applied force increases until the block moves. After the block moves, it experiences kinetic friction, which is less than the maximum static friction.

Static friction is friction between two or more solid objects that are not moving relative to each other. For example, static friction can prevent an object from sliding down a sloped surface. The coefficient of static friction, typically denoted as μ_s, is usually higher than the coefficient of kinetic friction. Static friction is considered to arise as the result of surface roughness features across multiple length scales at solid surfaces. These features, known as asperities are present down to nano-scale dimensions and result in true solid to solid contact existing only at a limited number of points accounting for only a fraction of the apparent or nominal contact area.[41] The linearity between applied load and true contact area, arising from asperity deformation, gives rise to the linearity between static frictional force and normal force, found for typical Amonton–Coulomb type friction.[42]

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: ${\displaystyle F_{\text{max}}=\mu _{\mathrm {s} }F_{\text{n}}}$. When there is no sliding occurring, the friction force can have any value from zero up to ${\displaystyle F_{\text{max}}}$. Any force smaller than ${\displaystyle F_{\text{max}}}$ attempting to slide one surface over the other is opposed by a frictional force of equal magnitude and opposite direction. Any force larger than ${\displaystyle F_{\text{max}}}$ overcomes the force of static friction and causes sliding to occur. The instant sliding occurs, static friction is no longer applicable—the friction between the two surfaces is then called kinetic friction. However, an apparent static friction can be observed even in the case when the true static friction is zero.[43]

An example of static friction is the force that prevents a car wheel from slipping as it rolls on the ground. Even though the wheel is in motion, the patch of the tire in contact with the ground is stationary relative to the ground, so it is static rather than kinetic friction. Upon slipping, the wheel friction changes to kinetic friction. An anti-lock braking system operates on the principle of allowing a locked wheel to resume rotating so that the car maintains static friction.

The maximum value of static friction, when motion is impending, is sometimes referred to as limiting friction,[44] although this term is not used universally.[3]

Kinetic friction, also known as dynamic friction or sliding friction, occurs when two objects are moving relative to each other and rub together (like a sled on the ground). The coefficient of kinetic friction is typically denoted as μ_k, and is usually less than the coefficient of static friction for the same materials.[45][46] However, Richard Feynman comments that "with dry metals it is very hard to show any difference."[47] The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: ${\displaystyle F_{k}=\mu _{\mathrm {k} }F_{n}}$. This is responsible for the Coulomb damping of an oscillating or vibrating system.

New models are beginning to show how kinetic friction can be greater than static friction.[48] Kinetic friction is now understood, in many cases, to be primarily caused by chemical bonding between the surfaces, rather than interlocking asperities;[49] however, in many other cases roughness effects are dominant, for example in rubber to road friction.[48] Surface roughness and contact area affect kinetic friction for micro- and nano-scale objects where surface area forces dominate inertial forces.[50]

The origin of kinetic friction at nanoscale can be explained by thermodynamics.[51] Upon sliding, a new surface forms at the back of a sliding true contact, and existing surface disappears at the front of it. Since all surfaces involve the thermodynamic surface energy, work must be spent in creating the new surface, and energy is released as heat in removing the surface. Thus, a force is required to move the back of the contact, and frictional heat is released at the front.

Angle of friction, θ, when block just starts to slide.

For certain applications, it is more useful to define static friction in terms of the maximum angle before which one of the items will begin sliding. This is called the angle of friction or friction angle. It is defined as:

$${\displaystyle \tan {\theta }=\mu _{\mathrm {s} }}$$

and thus:

$${\displaystyle \theta =\arctan {\mu _{\mathrm {s} }}}$$

where ${\displaystyle \theta }$ is the angle from horizontal and μ_s is the static coefficient of friction between the objects.[52] This formula can also be used to calculate μ_s from empirical measurements of the friction angle.

Determining the forces required to move atoms past each other is a challenge in designing nanomachines. In 2008 scientists for the first time were able to move a single atom across a surface, and measure the forces required. Using ultrahigh vacuum and nearly zero temperature (5 K), a modified atomic force microscope was used to drag a cobalt atom, and a carbon monoxide molecule, across surfaces of copper and platinum.[53]

The Coulomb approximation follows from the assumptions that: surfaces are in atomically close contact only over a small fraction of their overall area; that this contact area is proportional to the normal force (until saturation, which takes place when all area is in atomic contact); and that the frictional force is proportional to the applied normal force, independently of the contact area. The Coulomb approximation is fundamentally an empirical construct. It is a rule-of-thumb describing the approximate outcome of an extremely complicated physical interaction. The strength of the approximation is its simplicity and versatility. Though the relationship between normal force and frictional force is not exactly linear (and so the frictional force is not entirely independent of the contact area of the surfaces), the Coulomb approximation is an adequate representation of friction for the analysis of many physical systems.

When the surfaces are conjoined, Coulomb friction becomes a very poor approximation (for example, adhesive tape resists sliding even when there is no normal force, or a negative normal force). In this case, the frictional force may depend strongly on the area of contact. Some drag racing tires are adhesive for this reason. However, despite the complexity of the fundamental physics behind friction, the relationships are accurate enough to be useful in many applications.

As of 2012, a single study has demonstrated the potential for an effectively negative coefficient of friction in the low-load regime, meaning that a decrease in normal force leads to an increase in friction. This contradicts everyday experience in which an increase in normal force leads to an increase in friction.[54] This was reported in the journal Nature in October 2012 and involved the friction encountered by an atomic force microscope stylus when dragged across a graphene sheet in the presence of graphene-adsorbed oxygen.[54]

Despite being a simplified model of friction, the Coulomb model is useful in many numerical simulation applications such as multibody systems and granular material. Even its most simple expression encapsulates the fundamental effects of sticking and sliding which are required in many applied cases, although specific algorithms have to be designed in order to efficiently numerically integrate mechanical systems with Coulomb friction and bilateral or unilateral contact.[55][56][57][58][59] Some quite nonlinear effects, such as the so-called Painlevé paradoxes, may be encountered with Coulomb friction.[60]

Dry friction can induce several types of instabilities in mechanical systems which display a stable behaviour in the absence of friction.[61] These instabilities may be caused by the decrease of the friction force with an increasing velocity of sliding, by material expansion due to heat generation during friction (the thermo-elastic instabilities), or by pure dynamic effects of sliding of two elastic materials (the Adams–Martins instabilities). The latter were originally discovered in 1995 by George G. Adams and João Arménio Correia Martins for smooth surfaces[62][63] and were later found in periodic rough surfaces.[64] In particular, friction-related dynamical instabilities are thought to be responsible for brake squeal and the 'song' of a glass harp,[65][66] phenomena which involve stick and slip, modelled as a drop of friction coefficient with velocity.[67]

A practically important case is the self-oscillation of the strings of bowed instruments such as the violin, cello, hurdy-gurdy, erhu, etc.

A connection between dry friction and flutter instability in a simple mechanical system has been discovered,[68] watch the movie Archived 2015-01-10 at the Wayback Machine for more details.

Frictional instabilities can lead to the formation of new self-organized patterns (or "secondary structures") at the sliding interface, such as in-situ formed tribofilms which are utilized for the reduction of friction and wear in so-called self-lubricating materials.[69]

Rolling resistance is the force that resists the rolling of a wheel or other circular object along a surface caused by deformations in the object or surface. Generally the force of rolling resistance is less than that associated with kinetic friction.[71] Typical values for the coefficient of rolling resistance are 0.001.[72] One of the most common examples of rolling resistance is the movement of motor vehicle tires on a road, a process which generates heat and sound as by-products.[73]

Any wheel equipped with a brake is capable of generating a large retarding force, usually for the purpose of slowing and stopping a vehicle or piece of rotating machinery. Braking friction differs from rolling friction because the coefficient of friction for rolling friction is small whereas the coefficient of friction for braking friction is designed to be large by choice of materials for brake pads.

Rubbing two materials against each other can lead to charge transfer, either electrons or ions. The energy required for this contributes to the friction. In addition, sliding can cause a build-up of electrostatic charge, which can be hazardous if flammable gases or vapours are present. When the static build-up discharges, explosions can be caused by ignition of the flammable mixture.

Belt friction is a physical property observed from the forces acting on a belt wrapped around a pulley, when one end is being pulled. The resulting tension, which acts on both ends of the belt, can be modeled by the belt friction equation.

In practice, the theoretical tension acting on the belt or rope calculated by the belt friction equation can be compared to the maximum tension the belt can support. This helps a designer of such a rig to know how many times the belt or rope must be wrapped around the pulley to prevent it from slipping. Mountain climbers and sailing crews demonstrate a standard knowledge of belt friction when accomplishing basic tasks.

Devices such as wheels, ball bearings, roller bearings, and air cushion or other types of fluid bearings can change sliding friction into a much smaller type of rolling friction.

Many thermoplastic materials such as nylon, HDPE and PTFE are commonly used in low friction bearings. They are especially useful because the coefficient of friction falls with increasing imposed load.[74] For improved wear resistance, very high molecular weight grades are usually specified for heavy duty or critical bearings.

A common way to reduce friction is by using a lubricant, such as oil, water, or grease, which is placed between the two surfaces, often dramatically lessening the coefficient of friction. The science of friction and lubrication is called tribology. Lubricant technology is when lubricants are mixed with the application of science, especially to industrial or commercial objectives.

Superlubricity, a recently discovered effect, has been observed in graphite: it is the substantial decrease of friction between two sliding objects, approaching zero levels. A very small amount of frictional energy would still be dissipated.

Lubricants to overcome friction need not always be thin, turbulent fluids or powdery solids such as graphite and talc; acoustic lubrication actually uses sound as a lubricant.

Another way to reduce friction between two parts is to superimpose micro-scale vibration to one of the parts. This can be sinusoidal vibration as used in ultrasound-assisted cutting or vibration noise, known as dither.

The work done by friction can translate into deformation, wear, and heat that can affect the contact surface properties (even the coefficient of friction between the surfaces). This can be beneficial as in polishing. The work of friction is used to mix and join materials such as in the process of friction welding. Excessive erosion or wear of mating sliding surfaces occurs when work due to frictional forces rise to unacceptable levels. Harder corrosion particles caught between mating surfaces in relative motion (fretting) exacerbates wear of frictional forces. As surfaces are worn by work due to friction, fit and surface finish of an object may degrade until it no longer functions properly.[75] For example, bearing seizure or failure may result from excessive wear due to work of friction.

In the reference frame of the interface between two surfaces, static friction does no work, because there is never displacement between the surfaces. In the same reference frame, kinetic friction is always in the direction opposite the motion, and does negative work.[76] However, friction can do positive work in certain frames of reference. One can see this by placing a heavy box on a rug, then pulling on the rug quickly. In this case, the box slides backwards relative to the rug, but moves forward relative to the frame of reference in which the floor is stationary. Thus, the kinetic friction between the box and rug accelerates the box in the same direction that the box moves, doing positive work.[77]

When sliding takes place between two rough bodies in contact, the algebraic sum of the works done is different from zero, and the algebraic sum of the quantities of heat gained by the two bodies is equal to the quantity of work lost by friction, and the total quantity of heat gained is positive.[78][79] In a natural thermodynamic process, the work done by an agency in the surroundings of a thermodynamic system or working body is greater than the work received by the body, because of friction. Thermodynamic work is measured by changes in a body's state variables, sometimes called work-like variables, other than temperature and entropy. Examples of work-like variables, which are ordinary macroscopic physical variables and which occur in conjugate pairs, are pressure – volume, and electric field – electric polarization. Temperature and entropy are a specifically thermodynamic conjugate pair of state variables. They can be affected microscopically at an atomic level, by mechanisms such as friction, thermal conduction, and radiation. The part of the work done by an agency in the surroundings that does not change the volume of the working body but is dissipated in friction, is called isochoric work. It is received as heat, by the working body and sometimes partly by a body in the surroundings. It is not counted as thermodynamic work received by the working body.

• Automobile brakes inherently rely on friction, slowing a vehicle by converting its kinetic energy into heat. Incidentally, dispersing this large amount of heat safely is one technical challenge in designing brake systems. Disk brakes rely on friction between a disc and brake pads that are squeezed transversely against the rotating disc. In drum brakes, brake shoes or pads are pressed outwards against a rotating cylinder (brake drum) to create friction. Since braking discs can be more efficiently cooled than drums, disc brakes have better stopping performance.[80]
• Rail adhesion refers to the grip wheels of a train have on the rails, see Frictional contact mechanics.
• Road slipperiness is an important design and safety factor for automobiles[81]
  • Split friction is a particularly dangerous condition arising due to varying friction on either side of a car.
  • Road texture affects the interaction of tires and the driving surface.

• A tribometer is an instrument that measures friction on a surface.
• A profilograph is a device used to measure pavement surface roughness.

• Friction is used to heat and ignite matchsticks (friction between the head of a matchstick and the rubbing surface of the match box).[82]
• Sticky pads are used to prevent object from slipping off smooth surfaces by effectively increasing the friction coefficient between the surface and the object.