Editing Friction (section)

==Dry friction==
Dry friction resists relative lateral motion of two solid surfaces in contact. The two regimes of dry friction are 'static friction' ("[[stiction]]") between non-moving surfaces, and ''kinetic friction'' (sometimes called sliding friction or dynamic friction) between moving surfaces.

Coulomb friction, named after [[Charles-Augustin de Coulomb]], is an approximate model used to calculate the force of dry friction. It is governed by the model:
<math display="block">F_\mathrm{f} \leq \mu F_\mathrm{n},</math>
where
* <math>F_\mathrm{f}</math> is the force of friction exerted by each surface on the other.  It is parallel to the surface, in a direction opposite to the net applied force.
* <math>\mu</math> is the coefficient of friction, which is an empirical property of the contacting materials,
* <math>F_\mathrm{n}</math> is the [[normal force]] exerted by each surface on the other, directed perpendicular (normal) to the surface.

The Coulomb friction <math>F_\mathrm{f}</math> may take any value from zero up to <math>\mu F_\mathrm{n}</math>, and the direction of the frictional force against a surface is opposite to the motion that surface would experience in the absence of friction. Thus, in the static case, the frictional force is exactly what it must be in order to prevent motion between the surfaces; it balances the net force tending to cause such motion. In this case, rather than providing an estimate of the actual frictional force, the Coulomb approximation provides a threshold value for this force, above which motion would commence. This maximum force is known as [[Traction (engineering)|traction]].

The force of friction is always exerted in a direction that opposes movement (for kinetic friction) or potential movement (for static friction) between the two surfaces. For example, a [[curling]] stone sliding along the ice experiences a kinetic force slowing it down. For an example of potential movement, the drive wheels of an accelerating car experience a frictional force pointing forward; if they did not, the wheels would spin, and the rubber would slide backwards along the pavement. Note that it is not the direction of movement of the vehicle they oppose, it is the direction of (potential) sliding between tire and road.

===Normal force===
[[Image:Free body diagram2.svg|right|200px|thumb|[[Free-body diagram]] for a block on a ramp. Arrows are [[Euclidean vector|vectors]] indicating directions and magnitudes of forces. ''N'' is the normal force, ''mg'' is the force of [[gravity]], and ''F<sub>f</sub>'' is the force of friction.]]

{{Main|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 <math>N=mg\,</math>. In this case, conditions of equilibrium tell us that the magnitude of the friction force is ''zero'', <math>F_f = 0</math>. In fact, the friction force always satisfies <math>F_f\le \mu N</math>, with equality reached only at a critical ramp angle (given by <math>\tan^{-1}\mu</math>) 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 <math>N</math> might include forces other than gravity. If an object is on a {{em|level surface}} and subjected to an external force <math>P</math> tending to cause it to slide, then the normal force between the object and the surface is just <math>N = mg + P_y</math>, where <math>mg</math> is the block's weight and <math>P_y</math> is the downward component of the external force. Prior to sliding, this friction force is <math>F_f = -P_x</math>, where <math>P_x</math> is the horizontal component of the external force. Thus, <math>F_f \le \mu N</math> in general. Sliding commences only after this frictional force reaches the value <math>F_f = \mu N</math>. 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 {{em|tilted surface}} such as an inclined plane, the normal force from gravity is smaller than <math>mg</math>, 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 (geometric)|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 <math>F_f \le \mu N</math>, then the tentative assumption was correct, and it is the actual frictional force. Otherwise, the friction force must be set equal to <math>F_f = \mu N</math>, and then the resulting force imbalance would then determine the acceleration associated with slipping.

===Coefficient of friction===
{{see also|Coefficient of traction}}
{{expand section|explanation of why kinetic friction is always lower|date=August 2020}}
The '''coefficient of friction''' (COF), often symbolized by the Greek letter [[μ]], is a [[dimensionless]] [[scalar (physics)|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.<ref name="Air Brake Association-1921">{{cite book|author=Air Brake Association|title=The Principles and Design of Foundation Brake Rigging|url=https://books.google.com/books?id=DoNBAQAAMAAJ&pg=PA5|year=1921|publisher=Air brake association|page=5|access-date=2017-07-27|archive-date=2024-10-07|archive-url=https://web.archive.org/web/20241007090951/https://books.google.com/books?id=DoNBAQAAMAAJ&pg=PA5#v=onepage&q&f=false|url-status=live}}</ref>

For surfaces at rest relative to each other, <math>\mu = \mu_\mathrm{s}</math>, where <math>\mu_\mathrm{s}</math> 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 [[adsorption|adsorbed material]]. The [[fractal]]ity 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.<ref name="Hanaor-2016">{{cite journal |last1=Hanaor |first1=D. |last2=Gan |first2=Y. |last3=Einav |first3=I. |year=2016 |title=Static friction at fractal interfaces |journal=Tribology International |volume=93 |pages=229–238 |arxiv=2106.01473 |doi=10.1016/j.triboint.2015.09.016 |s2cid=51900923}}</ref>

For surfaces in relative motion <math>\mu = \mu_\mathrm{k}</math>, where <math>\mu_\mathrm{k}</math> is the ''coefficient of kinetic friction''. The Coulomb friction is equal to <math>F_\mathrm{f}</math>, 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.<ref name="Dowson-1997" /> The coefficient of friction is an [[empirical]] [[measurement]]{{mdash}}it has to be measured [[experiment]]ally, and cannot be found through calculations.<ref name="Valentin L. Popov-2014">{{cite journal |author1=Valentin L. Popov |title=Generalized law of friction between elastomers and differently shaped rough bodies |journal=Sci. Rep. |date=17 Jan 2014 |volume=4 |page=3750 |doi=10.1038/srep03750 |pmid=24435002 |pmc=3894559 |bibcode= 2014NatSR...4.3750P}}</ref> 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 roughness|surface structure]].<ref name="Hanaor-2016" /> 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.{{Citation needed|date=December 2008}}

In systems with significant non-uniform stress fields, because local slip occurs before the system slides, the macroscopic coefficient of static friction depends on the applied load, system size, or shape; [[Friction#Laws of dry friction|Amontons' law]] is not satisfied macroscopically.<ref>{{Cite journal |last1=Otsuki |first1=M. |last2=Matsukawa |first2=H. |date=2013-04-02 |title=Systematic breakdown of Amontons' law of friction for an elastic object locally obeying Amontons' law |journal=Scientific Reports |volume=3 |pages=1586 |doi=10.1038/srep01586|pmid=23545778 |pmc=3613807 |arxiv=1202.1716 |bibcode=2013NatSR...3.1586O }}</ref>

====Approximate coefficients of friction====
{{Disputed section|date=November 2021}}
{| class="wikitable"
|-
! colspan="2" rowspan="2" data-sort-type="text"|Materials !! colspan="2"|Static Friction, <math>\mu_\mathrm{s}</math> !! colspan="2"|Kinetic/Sliding Friction, <math>\mu_\mathrm{k}\,</math>
|-
|-
!data-sort-type="number"| Dry and clean !!data-sort-type="number"| Lubricated
!data-sort-type="number| Dry and clean !!data-sort-type="number"| Lubricated
|-
! Aluminium
! Steel
| 0.61<ref name="Friction Factors">{{cite web
| url = http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm#method
| title = Friction Factors – Coefficients of Friction
| access-date = 2015-04-27
| archive-url = https://web.archive.org/web/20190201171526/http://www.roymech.co.uk/Useful_Tables/Tribology/co_of_frict.htm#method
| archive-date = 2019-02-01

}}</ref>
|
| 0.47<ref name="Friction Factors"/>
|
|-
!Aluminium
!Aluminium
|1.05–1.35<ref name="Friction Factors"/>
|0.3<ref name="Friction Factors"/>
|1.4<ref name="Friction Factors"/>–1.5{{cn|date=March 2024}}
|-
!Gold
!Gold
|
|
|2.5{{cn|date=March 2024}}
|
|-
!Platinum
!Platinum
|1.2<ref name="Friction Factors"/>
|0.25<ref name="Friction Factors"/>
|3.0{{cn|date=March 2024}}
|
|-
!Silver
!Silver
|1.4<ref name="Friction Factors"/>
|0.55<ref name="Friction Factors"/>
|1.5{{cn|date=March 2024}}
|
|-
! Alumina ceramic
! Silicon nitride ceramic
|
|
|
| 0.004 (wet)<ref name="Ferreira-2012">{{cite journal
| title = Ultra-low friction coefficient in alumina–silicon nitride pair lubricated with water
| doi=10.1016/j.wear.2012.07.030
| volume=296
| issue = 1–2
| journal=Wear
| pages=656–659| date = 2012-08-30
| last1 = Ferreira
| first1 = Vanderlei
| last2 = Yoshimura
| first2 = Humberto Naoyuki
| last3 = Sinatora
| first3 = Amilton
}}</ref>
|-
! [[Aluminium magnesium boride|BAM (Ceramic alloy AlMgB<sub>14</sub>)]]
! [[Titanium boride]] (TiB<sub>2</sub>)
| 0.04–0.05<ref name="Tian-2003">{{cite journal|doi=10.1063/1.1615677|title=Superhard self-lubricating AlMgB[sub 14] films for microelectromechanical devices|year=2003|last1=Tian|first1=Y.|last2=Bastawros|first2=A. F.|last3=Lo|first3=C. C. H.|last4=Constant|first4=A. P.|last5=Russell|first5=A.M.|last6=Cook|first6=B. A.|journal=Applied Physics Letters|volume=83|issue=14|page=2781|bibcode=2003ApPhL..83.2781T|url=http://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1004&context=mse_pubs|access-date=2019-01-31|archive-date=2024-10-07|archive-url=https://web.archive.org/web/20241007090944/https://dr.lib.iastate.edu/handle/20.500.12876/55639/|url-status=live}}</ref>
| 0.02<ref name="Kleiner, Kurt-2008">{{cite web
| url = https://www.newscientist.com/article/dn16102-material-slicker-than-teflon-discovered-by-accident.html
| title = Material slicker than Teflon discovered by accident
| author = Kleiner, Kurt
| date = 2008-11-21
| access-date = 2008-12-25
| archive-date = 2008-12-20
| archive-url = https://web.archive.org/web/20081220162702/http://www.newscientist.com/article/dn16102-material-slicker-than-teflon-discovered-by-accident.html
| url-status = live
}}</ref><ref name="Higdon-2011">{{cite journal|doi=10.1016/j.wear.2010.11.044
| title=Friction and wear mechanisms in AlMgB14-TiB2 nanocoatings
| year=2011|last1=Higdon|first1=C.|last2=Cook|first2=B.|last3=Harringa|first3=J.
| last4=Russell|first4=A.|last5=Goldsmith|first5=J.|last6=Qu|first6=J.|last7=Blau|first7=P.|journal=Wear|volume=271|issue=9–10
| pages=2111–2115}}</ref>
|
|
|-
! Brass
! Steel
| 0.35–0.51<ref name="Friction Factors"/>
| 0.19<ref name="Friction Factors"/>
| 0.44<ref name="Friction Factors"/>
|
|-
! Cast iron
! Copper
| 1.05<ref name="Friction Factors"/>
|
| 0.29<ref name="Friction Factors"/>
|
|-
! Cast iron
! Zinc
| 0.85<ref name="Friction Factors"/>
|
| 0.21<ref name="Friction Factors"/>
|
|-
! Concrete
! Rubber
| 1.0
| 0.30 (wet)
| 0.6–0.85<ref name="Friction Factors"/>
| 0.45–0.75 (wet)<ref name="Friction Factors"/>
|-
! Concrete
! Wood
| 0.62<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
|
|
|
|-
! Copper
! Glass
| 0.68<ref name="Barrett-1990" />
|
| 0.53<ref name="Barrett-1990" />
|
|-
! Copper
! Steel
| 0.53<ref name="Barrett-1990" />
|
| 0.36<ref name="Friction Factors"/><ref name="Barrett-1990" />
| 0.18<ref name="Barrett-1990" />
|-
! Glass
! Glass
| 0.9–1.0<ref name="Friction Factors"/><ref name="Barrett-1990" />
| 0.005–0.01<ref name="Barrett-1990" />
| 0.4<ref name="Friction Factors"/><ref name="Barrett-1990" />
| 0.09–0.116<ref name="Barrett-1990" />
|-
! Human synovial fluid
! Human cartilage
|
| 0.01<ref name="Coefficients of Friction of Human Joints">{{cite web
| url = http://hypertextbook.com/facts/2007/ConnieQiu.shtml
| title = Coefficients of Friction of Human Joints
| access-date = 2015-04-27
| archive-date = 2024-10-07
| archive-url = https://web.archive.org/web/20241007090943/https://hypertextbook.com/facts/2007/ConnieQiu.shtml
| url-status = live
}}</ref>
|
| 0.003<ref name="Coefficients of Friction of Human Joints"/>
|-
! Ice
! Ice
| 0.02–0.09<ref name="The Engineering Toolbox"/>
|
|
|
|-
! [[Polyethene]]
! Steel
| 0.2<ref name="Friction Factors"/><ref name="The Engineering Toolbox"/>
| 0.2<ref name="Friction Factors"/><ref name="The Engineering Toolbox"/>
|
|
|-
! [[PTFE]] (Teflon)
! PTFE (Teflon)
| 0.04<ref name="Friction Factors"/><ref name="The Engineering Toolbox">{{cite web
| url = http://www.engineeringtoolbox.com/friction-coefficients-d_778.html
| title = The Engineering Toolbox: Friction and Coefficients of Friction
| access-date = 2008-11-23
| archive-date = 2013-12-03
| archive-url = https://web.archive.org/web/20131203000601/http://www.engineeringtoolbox.com/friction-coefficients-d_778.html
| url-status = live
}}</ref>
| 0.04<ref name="Friction Factors"/><ref name="The Engineering Toolbox"/>
|
| 0.04<ref name="Friction Factors"/>
|-
! Steel
! Ice
| 0.03<ref name="The Engineering Toolbox"/>
|
|
|
|-
! Steel
! PTFE (Teflon)
| 0.04<ref name="Friction Factors"/>−0.2<ref name="The Engineering Toolbox"/>
| 0.04<ref name="Friction Factors"/>
|
| 0.04<ref name="Friction Factors"/>
|-
! Steel
! Steel
| 0.74<ref name="Friction Factors"/>−0.80<ref name="The Engineering Toolbox"/>
| 0.005–0.23<ref name="Barrett-1990" /><ref name="The Engineering Toolbox"/>
| 0.42–0.62<ref name="Friction Factors"/><ref name="Barrett-1990" />
| 0.029–0.19<ref name="Barrett-1990" />
|-
! Wood
! Metal
| 0.2–0.6<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
| 0.2 (wet)<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
| 0.49<ref name="Barrett-1990" />
| 0.075<ref name="Barrett-1990" />
|-
! Wood
! Wood
| 0.25–0.62<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8">[http://www.engineershandbook.com/Tables/frictioncoefficients.htm Coefficient of Friction] {{webarchive |url=https://web.archive.org/web/20090308124246/http://www.engineershandbook.com/Tables/frictioncoefficients.htm |date=March 8, 2009 }}. EngineersHandbook.com</ref><ref name="Barrett-1990">{{cite journal |last1=Barrett |first1=Richard T. |title=(NASA-RP-1228) Fastener Design Manual |url=http://hdl.handle.net/2060/19900009424 |website=NASA Technical Reports Server |publisher=NASA Lewis Research Center |access-date=3 August 2020 |page=16 |date=1 March 1990 |hdl=2060/19900009424 |archive-date=7 October 2024 |archive-url=https://web.archive.org/web/20241007091010/https://ntrs.nasa.gov/citations/19900009424 |url-status=live }}</ref>
| 0.2 (wet)<ref name="Friction Factors"/><ref name="Coefficient of Friction Archived March 8"/>
| 0.32–0.48<ref name="Barrett-1990" />
| 0.067–0.167<ref name="Barrett-1990" />
|}

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.<ref>{{cite journal |last=Dienwiebel |first=Martin |title=Superlubricity of Graphite |display-authors=etal |journal=Phys. Rev. Lett. |volume=92 |page=126101 |year=2004 |doi=10.1103/PhysRevLett.92.126101 |url=http://www.physics.leidenuniv.nl/sections/cm/ip/group/PDF/Phys.rev.lett/2004/92(2004)12601.pdf |issue=12 |bibcode=2004PhRvL..92l6101D |pmid=15089689 |s2cid=26811802 |access-date=2011-09-01 |archive-date=2011-09-17 |archive-url=https://web.archive.org/web/20110917120623/http://www.physics.leidenuniv.nl/sections/cm/ip/group/PDF/Phys.rev.lett/2004/92(2004)12601.pdf |url-status=live }}</ref>
This ultralow-friction regime is called [[superlubricity]].<ref>{{Citation |last=Müser |first=Martin H. |title=Theoretical Studies of Superlubricity |date=2015 |work=Fundamentals of Friction and Wear on the Nanoscale |pages=209–232 |editor-last=Gnecco |editor-first=Enrico |url=https://link.springer.com/10.1007/978-3-319-10560-4_11 |access-date=2025-04-25 |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-319-10560-4_11 |isbn=978-3-319-10559-8 |editor2-last=Meyer |editor2-first=Ernst}}</ref>

===Static friction===<!-- Traction (engineering) links here -->
[[File:Static kinetic friction vs time.png|400px|right|thumb|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.]]
{{Main|Stiction}}

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''<sub>s</sub>, 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.<ref>[https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces multi-scale origins of static friction] {{Webarchive|url=https://web.archive.org/web/20210918100737/https://www.researchgate.net/publication/283675011_Static_friction_at_fractal_interfaces |date=2021-09-18 }} 2016</ref> 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.<ref>{{cite journal | author= Greenwood J.A. and JB Williamson| title= Contact of nominally flat surfaces | journal= Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences | volume=295 | issue=1442 | year=1966}}</ref>

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: <math>F_\text{max} = \mu_\mathrm{s} F_\text{n}</math>. When there is no sliding occurring, the friction force can have any value from zero up to <math>F_\text{max}</math>. Any force smaller than <math>F_\text{max}</math> attempting to slide one surface over the other is opposed by a frictional force of equal magnitude and opposite direction. Any force larger than <math>F_\text{max}</math> 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.<ref>{{Cite journal|last1=Nakano|first1=K.|last2=Popov|first2=V. L.|date=2020-12-10|title=Dynamic stiction without static friction: The role of friction vector rotation| url=https://link.aps.org/doi/10.1103/PhysRevE.102.063001| journal=Physical Review E|volume=102|issue=6|page=063001| doi=10.1103/PhysRevE.102.063001|pmid=33466084 |bibcode=2020PhRvE.102f3001N |hdl=10131/00013921 |s2cid=230599544 |hdl-access=free}}</ref>

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''',<ref name="Bhavikatti-1994">{{cite book
| url = https://books.google.com/books?id=4wkLl4NvmWAC&pg=PA112
| title = Engineering Mechanics
| last = Bhavikatti
| first = S. S.
| author2 = K. G. Rajashekarappa
| page = 112
| access-date = 2007-10-21
| publisher = New Age International
| isbn = 978-81-224-0617-7
| year = 1994
| archive-date = 2024-10-07
| archive-url = https://web.archive.org/web/20241007091508/https://books.google.com/books?id=4wkLl4NvmWAC&pg=PA112#v=onepage&q&f=false
| url-status = live
}}</ref>
although this term is not used universally.<ref name="Beer-1996"/>

===Kinetic friction===
'''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''<sub>k</sub>, and is usually less than the coefficient of static friction for the same materials.<ref>{{cite book|title=Statics: Analysis and Design of Systems in Equilibrium |publisher=Wiley and Sons|year=2005 |isbn=978-0-471-37299-8|page=618 |quote=In general, for given contacting surfaces, ''μ''<sub>k</sub> < ''μ''<sub>s</sub> |author1=Sheppard, Sheri|author2=Tongue, Benson H.|author3=Anagnos, Thalia|author1-link=Sheri D. Sheppard}}
</ref><ref>
{{cite book
| title = Engineering Mechanics: Statics
|author1=Meriam, James L. |author2=Kraige, L. Glenn |author3=Palm, William John | publisher = Wiley and Sons
| year = 2002
| page= 330
| quote = Kinetic friction force is usually somewhat less than the maximum static friction force.
| isbn = 978-0-471-40646-4}}</ref> However, [[Richard Feynman]] comments that "with dry metals it is very hard to show any difference."<ref>{{cite web
| url = http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
| title = The Feynman Lectures on Physics, Vol. I, p. 12–5
| publisher = Addison-Wesley
| year = 1964
| author1 = Feynman, Richard P.
| author2 = Leighton, Robert B.
| author3 = Sands, Matthew
| access-date = 2009-10-16
| archive-date = 2021-03-10
| archive-url = https://web.archive.org/web/20210310234726/http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html
| url-status = live
}}</ref>
The friction force between two surfaces after sliding begins is the product of the coefficient of kinetic friction and the normal force: <math>F_{k} = \mu_\mathrm{k} F_{n}</math>. This is responsible for the [[Coulomb damping]] of an [[Oscillation#Damped oscillations|oscillating]] or [[Vibration#Types|vibrating]] system.

New models are beginning to show how kinetic friction can be greater than static friction.<ref name="Volokitin, A. I-2002" >{{cite journal |title=Theory of rubber friction: Nonstationary sliding |journal=Physical Review B |volume=65 |page=134106 |doi=10.1103/PhysRevB.65.134106 |author1=Persson, B. N. |author2=Volokitin, A. I |year=2002 |bibcode=2002PhRvB..65m4106P |issue=13 |url=http://juser.fz-juelich.de/record/25870/files/17249.pdf |access-date=2019-01-31 |archive-date=2021-09-18 |archive-url=https://web.archive.org/web/20210918100743/https://juser.fz-juelich.de/record/25870/files/17249.pdf |url-status=live }}</ref> In many other cases roughness effects are dominant, for example in rubber to road friction.<ref name="Volokitin, A. I-2002" /> Surface roughness and contact area affect kinetic friction for micro- and nano-scale objects where surface area forces dominate inertial forces.<ref>{{cite book |last=Persson |first=B. N. J. |title=Sliding friction: physical principles and applications |url=https://books.google.com/books?id=1jb-nZMnRGYC&q=kinetic+friction |access-date=2016-01-23 |year=2000 |publisher=Springer |isbn=978-3-540-67192-3 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007091447/https://books.google.com/books?id=1jb-nZMnRGYC&q=kinetic+friction#v=snippet&q=kinetic%20friction&f=false |url-status=live }}</ref>

The origin of kinetic friction at nanoscale can be rationalized by an energy model.<ref>{{cite journal |last1=Makkonen |first1=L |year=2012 |title=A thermodynamic model of sliding friction |doi=10.1063/1.3699027 |journal=AIP Advances |volume=2 |issue= 1|page=012179 |bibcode=2012AIPA....2a2179M |doi-access=free }}</ref> During 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.

[[File:Free body.svg|thumb|Angle of friction, ''θ'', when block just starts to slide]]

===Angle of friction===
{{For|the maximum angle of static friction between granular materials|Angle of repose}}
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:
<math display="block">\tan{\theta} = \mu_\mathrm{s}</math>
and thus:
<math display="block">\theta = \arctan{\mu_\mathrm{s}}</math>
where <math>\theta</math> is the angle from horizontal and ''μ<sub>s</sub>'' is the static coefficient of friction between the objects.<ref>{{cite book |last1=Nichols |first1=Edward Leamington |last2=Franklin |first2=William Suddards |title=The Elements of Physics |publisher=Macmillan |page=101 |year=1898 |volume=1 |url=https://books.google.com/books?id=8IlCAAAAIAAJ |access-date=2020-06-07 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083034/https://books.google.com/books?id=8IlCAAAAIAAJ |url-status=live }}</ref> This formula can also be used to calculate ''μ<sub>s</sub>'' from empirical measurements of the friction angle.

===Friction at the atomic level===
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]].<ref>{{Cite journal |last1=Ternes |first1=Markus |last2=Lutz |first2=Christopher P. |last3=Hirjibehedin |first3=Cyrus F. |last4=Giessibl |first4=Franz J. |author-link5= Andreas J. Heinrich |last5=Heinrich |first5=Andreas J. |title=The Force Needed to Move an Atom on a Surface |journal=[[Science (journal)|Science]] |volume=319 |issue=5866 |pages=1066–1069 |date=2008-02-22 |doi=10.1126/science.1150288 |pmid=18292336 |bibcode=2008Sci...319.1066T |s2cid=451375 |url=https://epub.uni-regensburg.de/25284/1/The%20Force%20Needed%20to%20Move%20an%20Atom%20on%20.pdf |archive-url=https://web.archive.org/web/20180720063201/https://epub.uni-regensburg.de/25284/1/The%20Force%20Needed%20to%20Move%20an%20Atom%20on%20.pdf |archive-date=2018-07-20 |url-status=live }}</ref>

===Limitations of the Coulomb model===
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.

===="Negative" coefficient of friction====
{{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.<ref name="Deng-2012"/>  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.<ref name="Deng-2012">{{cite journal |last1=Deng |first1=Zhao |display-authors=etal |date=October 14, 2012 |title=Adhesion-dependent negative friction coefficient on chemically modified graphite at the nanoscale |journal=[[Nature (journal)|Nature]] |bibcode=2012NatMa..11.1032D |doi=10.1038/nmat3452 |pmid=23064494 |volume=11 |issue=12 |pages=1032–7}}
* {{cite magazine |date=2012-10-17 |title=At the nanoscale, graphite can turn friction upside down |magazine=R&D Magazine |url=http://www.rdmag.com/news/2012/10/nanoscale-graphite-can-turn-friction-upside-down |archive-url=https://web.archive.org/web/20130731031301/http://www.rdmag.com/news/2012/10/nanoscale-graphite-can-turn-friction-upside-down |archive-date=2013-07-31}}</ref>

===Numerical simulation of the Coulomb model===
Despite being a simplified model of friction, the Coulomb model is useful in many [[numerical simulation]] applications such as [[multibody system]]s 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 [[Numerical integration|numerically integrate]] mechanical systems with Coulomb friction and bilateral or unilateral contact.<ref>{{cite journal |last1=Haslinger |first1=J. |title=Approximation of the Signorini problem with friction, obeying the Coulomb law |journal=Mathematical Methods in the Applied Sciences |volume=5 |issue=1 |pages=422–437 |year=1983 |bibcode=1983MMAS....5..422H |doi=10.1002/mma.1670050127 |last2=Nedlec |first2=J.C. |hdl=10338.dmlcz/104086 |url=http://dml.cz/bitstream/handle/10338.dmlcz/104086/AplMat_29-1984-3_6.pdf |access-date=2019-09-19 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083034/http://dml.cz/bitstream/handle/10338.dmlcz/104086/AplMat_29-1984-3_6.pdf |url-status=live }}</ref><ref>{{cite journal |last1=Alart |first1=P. |last2=Curnier |first2=A. |title=A mixed formulation for frictional contact problems prone to Newton like solution method |journal=Computer Methods in Applied Mechanics and Engineering |volume=92 |pages=353–375 |year=1991 |bibcode=1991CMAME..92..353A |doi=10.1016/0045-7825(91)90022-X |issue=3 |url=https://hal.science/hal-04264964/file/Alart1991.pdf |access-date=2024-03-29 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007091441/https://hal.science/hal-04264964/file/Alart1991.pdf |url-status=live }}</ref><ref>{{cite journal |last1=Acary |first1=V. |last2=Cadoux |first2=F. |last3=Lemaréchal |first3=C. |last4=Malick |first4=J. |title=A formulation of the linear discrete Coulomb friction problem via convex optimization |journal=Journal of Applied Mathematics and Mechanics |volume=91 |issue=2 |pages=155–175 |year=2011 |doi=10.1002/zamm.201000073 |bibcode=2011ZaMM...91..155A |s2cid=17280625 |url=https://hal.inria.fr/inria-00495734/document |access-date=2018-04-20 |archive-date=2024-10-07 |archive-url=https://web.archive.org/web/20241007083036/https://hal.inria.fr/inria-00495734/document |url-status=live }}</ref><ref>{{cite journal |last1=De Saxcé |first1=G. |last2=Feng |first2=Z.-Q. |title=The bipotential method: A constructive approach to design the complete contact law with friction and improved numerical algorithms |journal=Mathematical and Computer Modelling |volume=28 |issue=4 |pages=225–245 |year=1998 |doi=10.1016/S0895-7177(98)00119-8|doi-access=free }}</ref><ref>{{cite journal |last1=Simo |first1=J.C. |last2=Laursen |first2=T.A. |title=An augmented lagrangian treatment of contact problems involving friction |journal=Computers and Structures |volume=42 |issue=2 |pages=97–116 |year=1992 |doi=10.1016/0045-7949(92)90540-G|doi-access=free }}</ref> Some quite [[Nonlinear system#Types of nonlinear dynamic behaviors|nonlinear effects]], such as the so-called [[Painlevé paradox]]es, may be encountered with Coulomb friction.<ref>{{cite book |last1=Acary |first1=V. |last2=Brogliato |first2=B. |title=Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics |publisher=[[Springer Science+Business Media|Springer Verlag Heidelberg]] |volume=35 |year=2008}}</ref>

===Dry friction and instabilities===
Dry friction can induce several types of instabilities in mechanical systems which display a stable behaviour in the absence of friction.<ref>{{cite book |last=Bigoni |first=D. |title=Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability |publisher=Cambridge University Press, 2012 |isbn=978-1-107-02541-7|date=2012-07-30 }}</ref>
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 (engineer)|George G. Adams]] and [[João Arménio Correia Martins]] for smooth surfaces<ref>{{cite journal |last=Adams |first=G. G. |title=Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction |doi=10.1115/1.2896013 |journal=Journal of Applied Mechanics |year=1995 |volume=62 |issue=4 |pages=867–872 |bibcode=1995JAM....62..867A }}</ref><ref>{{cite journal |last=Martins |first=J.A., Faria, L.O. & Guimarães, J. |title=Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions |doi=10.1115/1.2874477 |journal=Journal of Vibration and Acoustics |year=1995 |volume=117 |issue=4 |pages=445–451}}</ref> and were later found in periodic rough surfaces.<ref>{{cite journal |last1=M |first1=Nosonovsky |last2=G. |first2=Adams G. |title=Vibration and stability of frictional sliding of two elastic bodies with a wavy contact interface |doi=10.1115/1.1653684 |journal=Journal of Applied Mechanics |year=2004 |volume=71 |issue=2 |pages=154–161 |bibcode=2004JAM....71..154N }}</ref> In particular, friction-related dynamical instabilities are thought to be responsible for [[Brake#Noise|brake squeal]] and the 'song' of a [[glass harp]],<ref>{{cite journal |last2=J. |first2=Hultén |last1=J. |first1=Flint |title=Lining-deformation-induced modal coupling as squeal generator in a distributed parameter disk brake model |doi=10.1006/jsvi.2001.4052 |bibcode=2002JSV...254....1F |journal= Journal of Sound and Vibration|year=2002 |volume=254 |issue=1 |pages=1–21}}</ref><ref>{{cite journal |last1=M. |first1=Kröger |last2=M. |first2=Neubauer |last3=K. |first3=Popp |s2cid=16395796 |title=Experimental investigation on the avoidance of self-excited vibrations |journal=Phil. Trans. R. Soc. A |doi=10.1098/rsta.2007.2127 |year=2008 |pages=785–810 |volume=366 |issue=1866 |pmid=17947204 |bibcode=2008RSPTA.366..785K }}</ref> phenomena which involve stick and slip, modelled as a drop of friction coefficient with velocity.<ref>{{cite journal |last2=L. |first2=Ruina, A. |last1=R. |first1=Rice, J. |title=Stability of Steady Frictional Slipping |journal=Journal of Applied Mechanics |volume=50 |year=1983 |pages=343–349 |url=http://ruina.tam.cornell.edu/research/topics/friction_and_fracture/stability_steady.pdf |archive-url=https://web.archive.org/web/20100622193459/http://ruina.tam.cornell.edu/research/topics/friction_and_fracture/stability_steady.pdf |archive-date=2010-06-22 |url-status=live |doi=10.1115/1.3167042 |issue=2 |bibcode=1983JAM....50..343R |citeseerx=10.1.1.161.5207 }}</ref>

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 [[Aeroelastic flutter#Flutter|flutter]] instability in a simple mechanical system has been discovered,<ref>{{cite journal
 |url          = http://www.ing.unitn.it/~bigoni
 |author1      = Bigoni, D.
 |author2      = Noselli, G.
 |title        = Experimental evidence of flutter and divergence instabilities induced by dry friction
 |journal      = Journal of the Mechanics and Physics of Solids
 |year         = 2011
 |volume       = 59
 |pages        = 2208–2226
 |doi          = 10.1016/j.jmps.2011.05.007
 |issue        = 10
 |bibcode      = 2011JMPSo..59.2208B
 |citeseerx    = 10.1.1.700.5291
 |access-date  = 2011-11-30
 |archive-date = 2020-08-18
 |archive-url  = https://web.archive.org/web/20200818112658/http://www.ing.unitn.it/~bigoni/

}}</ref> watch the [http://www.ing.unitn.it/~bigoni/flutter.html movie] {{Webarchive|url=https://web.archive.org/web/20150110065917/http://www.ing.unitn.it/~bigoni/flutter.html |date=2015-01-10 }} 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.<ref>{{cite book
| title = Friction-Induced Vibrations and Self-Organization: Mechanics and Non-Equilibrium Thermodynamics of Sliding Contact
| first = Michael
| last = Nosonovsky
| year = 2013
| publisher = CRC Press
| isbn = 978-1-4665-0401-1
| url = http://www.crcpress.com/product/isbn/9781466504011
| page = 333}}</ref>