View source for Force

{{Short description|Influence that can change motion of an object}}
{{hatnote group|
{{Other uses}}
{{redirect|Physical force}}
}}
{{Infobox physical quantity
| name = Force
| width =
| background =
| image = [[File:Force examples.svg|200px]]
| caption = Forces can be described as a push or pull on an object. They can be due to phenomena such as [[gravity]], [[magnetism]], or anything that might cause a mass to accelerate.
| unit = [[newton (unit)|newton]] (N)
| otherunits = [[dyne]], [[pound-force]], [[poundal]], [[kip (unit)|kip]], [[kilopond]]
| symbols = <math>\vec F</math>, {{mvar|F}}, {{math|'''F'''}}
| baseunits = [[kilogram|kg]]·[[metre|m]]·[[second|s]]<sup>−2</sup>
| dimension = wikidata
| derivations = {{math|1='''F''' = ''[[Mass|m]]''[[Acceleration|'''a''']]}}
}}

{{Classical mechanics|expanded=Fundamental concepts}}
In [[physics]], a '''force''' is an influence that can cause an [[Physical object|object]] to change its [[velocity]], i.e., to [[accelerate]], meaning a change in speed or direction, unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the [[Euclidean vector#Length|magnitude]] and [[Direction (geometry, geography)|direction]] of a force are both important, force is a [[Vector (geometric)|vector]] quantity. The [[SI unit]] of force is the [[newton (unit)|newton (N)]], and force is often represented by the symbol {{math|'''F'''}}.

Force plays a central role in classical mechanics, figuring in all three of [[Newton's laws of motion]], which specify that the force on an object with an unchanging [[mass]] is equal to the [[Product (mathematics)|product]] of the object's mass and the [[acceleration]] that it undergoes. 
Types of forces often encountered in [[classical mechanics]] include [[Elasticity (physics)|elastic]], [[friction]]al, [[Normal force|contact or "normal" forces]], and [[gravity|gravitational]]. The rotational version of force is [[torque]], which produces [[angular acceleration|changes in the rotational speed]] of an object. In an extended body, each part often applies forces on the adjacent parts; the distribution of such forces through the body is the internal [[stress (mechanics)|mechanical stress]]. In equilibrium these stresses cause no acceleration of the body as the forces balance one another. If these are not in equilibrium they can cause [[deformation (engineering)|deformation]] of solid materials, or flow in [[fluid]]s.

In [[modern physics]], which includes [[Theory of relativity|relativity]] and [[quantum mechanics]], the laws governing motion are revised to rely on [[fundamental interactions]] as the ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes.<ref>{{Cite web |last=Cohen |first=Michael |title=Classical Mechanics: a Critical Introduction |url=https://www.physics.upenn.edu/sites/default/files/Classical_Mechanics_a_Critical_Introduction_0_0.pdf |url-status=live |archive-url=https://web.archive.org/web/20220703194458/https://www.physics.upenn.edu/sites/default/files/Classical_Mechanics_a_Critical_Introduction_0_0.pdf |archive-date=July 3, 2022 |access-date=January 9, 2024 |website=[[University of Pennsylvania]]}}</ref>

==Development of the concept==
Philosophers in [[Classical antiquity|antiquity]] used the concept of force in the study of [[statics|stationary]] and [[dynamics (physics)|moving]] objects and [[simple machine]]s, but thinkers such as [[Aristotle]] and [[Archimedes]] retained fundamental errors in understanding force. In part, this was due to an incomplete understanding of the sometimes non-obvious force of [[friction]] and a consequently inadequate view of the nature of natural motion.<ref name="Archimedes">{{cite book |last=Heath |first=Thomas L. |author-link=Thomas Heath (classicist) |url=https://archive.org/details/worksofarchimede029517mbp |title=The Works of Archimedes |via=[[Internet Archive]] |access-date=2007-10-14 |year=1897 }}</ref> A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by [[Galileo Galilei]] and [[Sir Isaac Newton]]. With his mathematical insight, Newton formulated [[Newton's laws of motion|laws of motion]] that were not improved for over two hundred years.<ref name=uniphysics_ch2/>

By the early 20th century, [[Albert Einstein|Einstein]] developed a [[theory of relativity]] that correctly predicted the action of forces on objects with increasing momenta near the speed of light and also provided insight into the forces produced by gravitation and [[inertia]]. With modern insights into [[quantum mechanics]] and technology that can accelerate particles close to the speed of light, [[particle physics]] has devised a [[Standard Model]] to describe forces between particles smaller than atoms. The [[Standard Model]] predicts that exchanged particles called [[gauge boson]]s are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: [[strong force|strong]], [[electromagnetic force|electromagnetic]], [[weak force|weak]], and [[gravitational force|gravitational]].<ref name=FeynmanVol1>{{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |title=The Feynman lectures on physics. Vol. I: Mainly mechanics, radiation and heat|year=2010|publisher=Basic Books|location=New York|isbn=978-0465024933|edition=New millennium |title-link=The Feynman Lectures on Physics |author-link1=Richard Feynman |author-link2=Robert B. Leighton |author-link3=Matthew Sands}}</ref>{{rp|((2–10))}}<ref name=Kleppner />{{rp|79}} [[High energy physics|High-energy particle physics]] [[observation]]s made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental [[electroweak]] interaction.<ref name="final theory"/>

== Pre-Newtonian concepts ==
{{see also|Aristotelian physics|Theory of impetus}}
[[File:Aristoteles Louvre2.jpg|thumb|upright|[[Aristotle]] famously described a force as anything that causes an object to undergo "unnatural motion"]]
Since antiquity the concept of force has been recognized as integral to the functioning of each of the [[simple machine]]s. The [[mechanical advantage]] given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of [[work (physics)|work]]. Analysis of the characteristics of forces ultimately culminated in the work of [[Archimedes]] who was especially famous for formulating a treatment of [[buoyant force]]s inherent in [[fluid]]s.<ref name="Archimedes"/>

[[Aristotle]] provided a [[philosophical]] discussion of the concept of a force as an integral part of [[Physics (Aristotle)|Aristotelian cosmology]]. In Aristotle's view, the terrestrial sphere contained four [[Classical element|elements]] that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, were in their natural place when on the ground, and that they stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force.<ref>{{cite book |last=Lang |first=Helen S. |title=The order of nature in Aristotle's physics : place and the elements |year=1998 |publisher=Cambridge Univ. Press |location=Cambridge |isbn=978-0521624534 |edition=}}</ref> This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of [[projectile]]s, such as the flight of arrows. An archer causes the arrow to move at the start of the flight, and it then sails through the air even though no discernible efficient cause acts upon it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation requires a continuous medium such as air to sustain the motion.<ref name="Hetherington">{{cite book |first=Norriss S. |last=Hetherington |title=Cosmology: Historical, Literary, Philosophical, Religious, and Scientific Perspectives |page=[https://archive.org/details/cosmologyhistori0000unse/page/100 100] |publisher=Garland Reference Library of the Humanities |year=1993 |isbn=978-0-8153-1085-3 |url=https://archive.org/details/cosmologyhistori0000unse/page/100 }}</ref>

Though [[Aristotelian physics]] was criticized as early as the 6th century,<ref>{{cite book|first=Richard |last=Sorabji |chapter=John Philoponus |title=Philoponus and the Rejection of Aristotelian Science |jstor=44216227 |year=2010 |edition=2nd |publisher=Institute of Classical Studies, University of London |isbn=978-1-905-67018-5 |oclc=878730683 |page=47}}</ref><ref>{{cite book|first=Anneliese |last=Maier |author-link=Anneliese Maier |title=On the Threshold of Exact Science |publisher=University of Pennsylvania Press |year=1982 |editor-first=Steven D. |editor-last=Sargent |isbn=978-0-812-27831-6 |oclc=495305340 |page=79}}</ref> its shortcomings would not be corrected until the 17th century work of [[Galileo Galilei]], who was influenced by the late medieval idea that objects in forced motion carried an innate force of [[impetus theory|impetus]]. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the [[Aristotelian theory of gravity|Aristotelian theory of motion]]. He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their [[velocity]] unless acted on by a force, for example [[friction]].<ref name="Galileo">{{cite book|last=Drake |first=Stillman |year=1978 |title=Galileo At Work |location=Chicago |publisher=University of Chicago Press |isbn=0-226-16226-5}}</ref> Galileo's idea that force is needed to change motion rather than to sustain it, further improved upon by [[Isaac Beeckman]], [[René Descartes]], and [[Pierre Gassendi]], became a key principle of Newtonian physics.<ref>{{Cite book |last=LoLordo |first=Antonia |url=https://www.worldcat.org/oclc/182818133 |title=Pierre Gassendi and the Birth of Early Modern Philosophy |date=2007 |publisher=Cambridge University Press |isbn=978-0-511-34982-9 |location=New York |pages=175–180 |oclc=182818133}}</ref>

In the early 17th century, before Newton's ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'', the term "force" ({{lang-la|vis}}) was applied to many physical and non-physical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named {{lang|la|vis viva}} (live force) by [[Gottfried Wilhelm Leibniz|Leibniz]]. The modern concept of force corresponds to Newton's {{lang|la|vis motrix}} (accelerating force).<ref>{{Cite book |author-link1=Vladimir Arnold |first1=V. I. |last1=Arnold |first2=V. V. |last2=Kozlov |first3= A. I. |last3=Neĩshtadt |url=https://www.worldcat.org/oclc/16404140 |chapter=Mathematical aspects of classical and celestial mechanics |title=Encyclopaedia of Mathematical Sciences, Dynamical Systems III |volume=3 |date=1988 |publisher=Springer-Verlag |others=Anosov, D. V. |isbn=0-387-17002-2 |location=Berlin |oclc=16404140}}</ref>

== Newtonian mechanics ==
{{main|Newton's laws of motion}}
Sir Isaac Newton described the motion of all objects using the concepts of [[inertia]] and force. In 1687, Newton published his magnum opus, ''[[Philosophiæ Naturalis Principia Mathematica]]''.<ref name=uniphysics_ch2/><ref name="Principia">{{Cite book |last=Newton |first=Isaac |title=The Principia Mathematical Principles of Natural Philosophy |publisher=University of California Press |year=1999 |isbn=978-0-520-08817-7 |location=Berkeley |author-link=Isaac Newton}} This is a recent translation into English by [[I. Bernard Cohen]] and Anne Whitman, with help from Julia Budenz.</ref> In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day.<ref name="Principia"/> The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.<ref>{{cite book |last=Howland |first=R. A. |title=Intermediate dynamics a linear algebraic approach |date=2006 |publisher=Springer |location=New York |isbn=978-0387280592 |pages=255–256 |edition=Online-Ausg.}}</ref>

=== First law ===
{{main|Newton's first law}}
Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line.<ref name="Principia"/> The latter follows from the former because of the [[principle of relativity|principle that the laws of physics are the same]] for all [[inertial frame of reference|inertial observers]], i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.<ref name="mermin2005">{{cite book|first=N. David |last=Mermin |author-link=N. David Mermin |title=It's About Time: Understanding Einstein's Relativity |publisher=Princeton University Press |year=2005 |isbn=978-0-691-21877-9}}</ref>{{rp|1–7}}

<!-- This location for image in the text seems to line up with the next heading on desktop -->
[[File:GodfreyKneller-IsaacNewton-1689.jpg|upright|thumb|[[Sir Isaac Newton]] in 1689. His ''Principia'' presented his three laws of motion in geometrical language, whereas modern physics uses [[differential calculus]] and [[Vector (mathematics and physics)|vector]]s.]]

=== Second law ===
{{main|Newton's second law}}
According to the first law, motion at constant speed in a straight line does not need a cause. It is ''change'' in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion. 
[[Newton's second law]] states that the net force acting upon an object is equal to the [[time derivative|rate]] at which its [[momentum]] changes with [[time]]. If the mass of the object is constant, this law implies that the [[acceleration]] of an object is directly [[Proportionality (mathematics)|proportional]] to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the [[mass]] of the object.<ref name="openstax-university-physics" />{{rp|pages=204–207}}

A modern statement of Newton's second law is a vector equation:
<math display="block" qid=Q104212301>\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t},</math>
where <math> \vec{p}</math> is the [[momentum]] of the system, and <math> \vec{F}</math> is the net ([[Vector (geometric)#Addition and subtraction|vector sum]]) force.<ref name="openstax-university-physics" />{{rp|page=399}} If a body is in equilibrium, there is zero ''net'' force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an ''unbalanced'' force acting on an object it will result in the object's momentum changing over time.<ref name="Principia"/>

In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of [[Momentum#Linear momentum of a particle|momentum]],
<math display="block">\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t} = \frac{\mathrm{d}\left(m\vec{v}\right)}{\mathrm{d}t},</math>
where ''m'' is the [[mass]] and <math> \vec{v}</math> is the [[velocity]].<ref name=FeynmanVol1/>{{rp|((9-1,9-2))}} If Newton's second law is applied to a system of [[Newton's Laws of Motion#Open systems|constant mass]], ''m'' may be moved outside the derivative operator. The equation then becomes
<math display="block">\vec{F} = m\frac{\mathrm{d}\vec{v}}{\mathrm{d}t}.</math>
By substituting the definition of [[acceleration]], the algebraic version of [[Newton's second law]] is derived:
<math display="block" qid=Q2397319>\vec{F} =m\vec{a}.</math>

=== Third law ===
{{main|Newton's third law}}
Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if <math>\vec{F}_{1,2}</math> is the force of body 1 on body 2 and <math>\vec{F}_{2,1}</math> that of body 2 on body 1, then
<math display="block">\vec{F}_{1,2}=-\vec{F}_{2,1}.</math>
This law is sometimes referred to as the ''action-reaction law'', with <math> \vec{F}_{1,2}</math> called the ''action'' and <math> -\vec{F}_{2,1}</math> the ''[[Reaction (physics)|reaction]]''.

Newton's Third Law is a result of applying [[symmetry]] to situations where forces can be attributed to the presence of different objects. The third law means that all forces are ''interactions'' between different bodies.<ref>{{cite journal
|title=Newton's third law revisited
|first=C. |last=Hellingman
|journal=Phys. Educ.
|volume=27
|year=1992
|issue=2
|pages=112–115
|quote=Quoting Newton in the ''Principia'': It is not one action by which the Sun attracts Jupiter, and another by which Jupiter attracts the Sun; but it is one action by which the Sun and Jupiter mutually endeavour to come nearer together.
|doi=10.1088/0031-9120/27/2/011 |bibcode=1992PhyEd..27..112H |s2cid=250891975
}}</ref><ref>{{Cite book |last1=Resnick |first1=Robert |title=Physics. 1 |last2=Halliday |first2=David |last3=Krane |first3=Kenneth S. |date=2002 |isbn=978-0-471-32057-9 |edition=5|quote="Any single force is only one aspect of a mutual interaction between ''two'' bodies."}}</ref> and thus that there is no such thing as a unidirectional force or a force that acts on only one body.

In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:
<math display="block">\vec{F}_{1,2}+\vec{F}_{2,1}=0.</math>
More generally, in a [[closed system]] of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the [[center of mass]] of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.<ref name=FeynmanVol1 />{{rp|((19-1))}}<ref name=Kleppner />

Combining Newton's Second and Third Laws, it is possible to show that the [[Conservation of momentum|linear momentum of a system is conserved]] in any [[closed system]]. In a system of two particles, if <math> \vec{p}_1</math> is the momentum of object 1 and <math> \vec{p}_{2}</math> the momentum of object 2, then
<math display="block">\frac{\mathrm{d}\vec{p}_1}{\mathrm{d}t} + \frac{\mathrm{d}\vec{p}_2}{\mathrm{d}t}= \vec{F}_{1,2} + \vec{F}_{2,1} = 0.</math>
Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Defining "force" ===
Some textbooks use Newton's second law as a ''definition'' of force.<ref>{{Cite book |last1=Landau |first1=L. D. |author-link=Lev Landau |last2=Akhiezer |author2-link=Aleksander Akhiezer |first2=A. I. |last3=Lifshitz |first3=A. M. |author3-link=Evgeny Lifshitz |title=General Physics; mechanics and molecular physics |publisher=Pergamon Press |year=1967 |location=Oxford |edition= |isbn=978-0-08-003304-4 |url-access=registration |url=https://archive.org/details/generalphysicsme0000land_d9j0 }}
Translated by: J. B. Sykes, A. D. Petford, and C. L. Petford. {{LCCN|67-30260}}. In section 7, pp. 12–14, this book defines force as ''dp/dt''.</ref><ref>{{Cite book |last1=Kibble |first1=Tom W. B. |last2=Berkshire |first2=Frank H. |title=Classical Mechanics |publisher=Imperial College Press |year=2004 |location=London |edition=5th |isbn=1860944248}}
According to page 12, "[Force] can of course be introduced, by defining it through Newton's second law".</ref><ref>{{Cite book |last1=de Lange |first1=O. L. |last2=Pierrus |first2=J. |title=Solved Problems in Classical Mechanics |publisher=Oxford University Press |year=2010 |location=Oxford |edition= |isbn=978-0-19-958252-5}}
According to page 3, "[Newton's second law of motion] can be regarded as defining force".</ref><ref>{{Cite book|last1=José|first1=Jorge V.|url=https://www.worldcat.org/oclc/857769535|title=Classical dynamics: A Contemporary Approach|last2=Saletan|first2=Eugene J.|date=1998|publisher=Cambridge University Press|isbn=978-1-139-64890-5|location=Cambridge [England]|oclc=857769535|author-link=Jorge V. José |page=9}}</ref> However, for the equation <math>\vec{F} = m\vec{a}</math> for a constant mass <math>m</math> to then have any predictive content, it must be combined with further information.<ref>{{Cite book|last1=Frautschi|first1=Steven C.|title=The Mechanical Universe: Mechanics and Heat|title-link=The Mechanical Universe|last2=Olenick|first2=Richard P.|last3=Apostol|first3=Tom M.|last4=Goodstein|first4=David L.|date=2007|publisher=Cambridge University Press|isbn=978-0-521-71590-4|edition=Advanced|location=Cambridge [Cambridgeshire]|oclc=227002144|author-link=Steven Frautschi|author-link3=Tom M. Apostol|author-link4=David L. Goodstein|page=134}}</ref><ref name=FeynmanVol1 />{{rp|((12-1))}} Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference.<ref name=Kleppner />{{rp|59}} The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways,<ref name="thornton-marion">{{cite book|first1=Stephen T. |last1=Thornton |first2=Jerry B. |last2=Marion |title=Classical Dynamics of Particles and Systems |edition=5th |publisher=Thomson Brooks/Cole |isbn=0-534-40896-6 |year=2004 |pages=49–50}}</ref><ref name=":0">{{cite book |author-last1=Landau |author-first1=Lev D. |title=Mechanics |author-last2=Lifshitz |author-first2=Evgeny M. |date=1969 |publisher=[[Pergamon Press]] |isbn=978-0-080-06466-6 |edition=2nd |series=[[Course of Theoretical Physics]] |volume=1 |translator-last1=Sykes |translator-first1=J. B. |author-link1=Lev Landau |author-link2=Evgeny Lifshitz |translator-last2=Bell |translator-first2=J. S. |translator-link2=John Stewart Bell}}</ref>{{Rp|pages=vii}} which ultimately do not affect how the theory is used in practice.<ref name="thornton-marion"/> Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include [[Ernst Mach]] and [[Walter Noll]].<ref>{{cite book |last=Jammer |first=Max |author-link=Max Jammer |title=Concepts of Force: A study in the foundations of dynamics |year=1999 |publisher=Dover Publications |location=Mineola, NY |isbn=978-0486406893 |pages=220–222 |edition=Facsim.}}</ref><ref>{{cite web |first=Walter |last=Noll |title=On the Concept of Force |url=http://www.math.cmu.edu/~wn0g/Force.pdf |publisher=Carnegie Mellon University |date=April 2007 |access-date=28 October 2013}}</ref>

== Combining forces ==
[[File:Addition av vektorer 003.jpg|thumb|Addition of vectors <math>v_1</math> and <math>v_2</math> results in <math>v</math>]]
Forces act in a particular [[direction (geometry)|direction]] and have [[Magnitude (mathematics)|sizes]] dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "[[Euclidean vector|vector quantities]]". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted [[scalar (physics)|scalar]] quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the [[resultant vector|result]]. If both of these pieces of information are not known for each force, the situation is ambiguous.<ref name="openstax-university-physics"/>{{rp|197}}

Historically, forces were first quantitatively investigated in conditions of [[static equilibrium]] where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive [[Vector (geometric)|vector quantities]]: they have [[magnitude (mathematics)|magnitude]] and direction.<ref name=uniphysics_ch2/> When two forces act on a [[point particle]], the resulting force, the ''resultant'' (also called the ''[[net force]]''), can be determined by following the [[parallelogram rule]] of [[vector addition]]: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

[[File:Freebodydiagram3 pn.svg|thumb|[[Free body diagram]]s of a block on a flat surface and an [[inclined plane]]. Forces are resolved and added together to determine their magnitudes and the net force.]]
[[Free-body diagram]]s can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that [[Vector (geometric)|graphical vector addition]] can be done to determine the net force.<ref>{{cite web
|title=Introduction to Free Body Diagrams
|work=Physics Tutorial Menu
|publisher=[[University of Guelph]]
|url=http://eta.physics.uoguelph.ca/tutorials/fbd/intro.html
|access-date=2008-01-02
|url-status=dead
|archive-url=https://web.archive.org/web/20080116042455/http://eta.physics.uoguelph.ca/tutorials/fbd/intro.html
|archive-date=2008-01-16
}}</ref>

As well as being added, forces can also be resolved into independent components at [[right angle]]s to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of [[basis vector]]s is often a more mathematically clean way to describe forces than using magnitudes and directions.<ref>{{cite web
|first=Tom
|last=Henderson
|title=The Physics Classroom
|work=The Physics Classroom and Mathsoft Engineering & Education, Inc.
|year=2004
|url=http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/vectors/u3l1b.html
|access-date=2008-01-02
|url-status=dead
|archive-url=https://web.archive.org/web/20080101141103/http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l1b.html
|archive-date=2008-01-01
}}</ref> This is because, for [[orthogonal]] components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right angles to the other two.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

===Equilibrium===
When all the forces that act upon an object are balanced, then the object is said to be in a state of [[Mechanical equilibrium|equilibrium]].<ref name="openstax-university-physics">{{cite book|title=University Physics, Volume 1 |last1=Ling |first1=Samuel J. |last2=Sanny |first2=Jeff |last3=Moebs |first3=William |display-authors=etal |publisher=[[OpenStax]] |url=https://openstax.org/details/books/university-physics-volume-1 |year=2021 |isbn=978-1-947-17220-3}}</ref>{{rp|566}} Hence, equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extended body, it is also necessary that the net torque be zero. A body is in ''static equilibrium'' with respect to a frame of reference if it at rest and not accelerating, whereas a body in ''dynamic equilibrium'' is moving at a constant speed in a straight line, i.e., moving but not accelerating. What one observer sees as static equilibrium, another can see as dynamic equilibrium and vice versa.<ref name="openstax-university-physics"/>{{rp|566}}

==== Static <span class="anchor" id="Static equilibrium"></span> ====
{{main|Statics|Static equilibrium}}
Static equilibrium was understood well before the invention of classical mechanics. Objects that are at rest have zero net force acting on them.<ref>{{cite web
|title=Static Equilibrium
|work=Physics Static Equilibrium (forces and torques)
|publisher=[[University of the Virgin Islands]]
|url=http://www.uvi.edu/Physics/SCI3xxWeb/Structure/StaticEq.html
|access-date=2008-01-02 |archive-url=https://web.archive.org/web/20071019054156/http://www.uvi.edu/Physics/SCI3xxWeb/Structure/StaticEq.html |archive-date=October 19, 2007}}</ref>

The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a [[normal force]]). The situation produces zero net force and hence no acceleration.<ref name=uniphysics_ch2/>

Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by [[static friction]], generated between the object and the table surface. For a situation with no movement, the static friction force ''exactly'' balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.<ref name=uniphysics_ch2/>

A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as [[weighing scale]]s and [[spring balance]]s. For example, an object suspended on a vertical [[spring scale]] experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant [[density]] (widely exploited for millennia to define standard weights); [[Archimedes' principle]] for buoyancy; Archimedes' analysis of the [[lever]]; [[Boyle's law]] for gas pressure; and [[Hooke's law]] for springs. These were all formulated and experimentally verified before Isaac Newton expounded his [[Newton's Laws of Motion|Three Laws of Motion]].<ref name=uniphysics_ch2/><ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

==== Dynamic <span class="anchor" id="Dynamical equilibrium"></span><span class="anchor" id="Dynamic equilibrium"></span> ====
{{main|Dynamics (physics)}}
[[File:Galileo.arp.300pix.jpg|thumb|upright|[[Galileo Galilei]] was the first to point out the inherent contradictions contained in Aristotle's description of forces.]]
Dynamic equilibrium was first described by [[Galileo]] who noticed that certain assumptions of Aristotelian physics were contradicted by observations and [[logic]]. Galileo realized that [[Galilean relativity|simple velocity addition]] demands that the concept of an "absolute [[rest frame]]" did not exist. Galileo concluded that motion in a constant [[velocity]] was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. When this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.<ref name="Galileo"/>

Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity. A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with [[kinetic friction]]. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. When kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

== Examples of forces in classical mechanics <span class="anchor" id="Non-fundamental forces"></span> ==
Some forces are consequences of the fundamental ones. In such situations, idealized models can be used to gain physical insight. For example, each solid object is considered a [[rigid body]].{{Citation needed|date=January 2024}}

=== Gravitational ===
{{main|Gravity}}
[[File:Falling ball.jpg|upright|thumb|Images of a freely falling basketball taken with a [[stroboscope]] at 20 flashes per second. The distance units on the right are multiples of about 12&nbsp;millimeters. The basketball starts at rest. At the time of the first flash (distance zero) it is released, after which the number of units fallen is equal to the square of the number of flashes.]]
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the [[acceleration]] of every object in [[free-fall]] was constant and independent of the mass of the object. Today, this [[Gravitational acceleration|acceleration due to gravity]] towards the surface of the Earth is usually designated as <math> \vec{g}</math> and has a magnitude of about 9.81 [[meter]]s per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth.<ref>{{cite journal |last=Cook |first=A. H. |journal=Nature |title=A New Absolute Determination of the Acceleration due to Gravity at the National Physical Laboratory |date=1965 |doi=10.1038/208279a0 |page=279 |volume=208 |bibcode=1965Natur.208..279C |issue=5007 |s2cid=4242827 |doi-access=free }}</ref> This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of <math>m</math> will experience a force:
<math display="block">\vec{F} = m\vec{g}.</math>

For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a [[normal force]] (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a [[Newton's law of gravity|law of gravity]] that could account for the celestial motions that had been described earlier using [[Kepler's laws of planetary motion]].<ref name=uniphysics_ch4 />

Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an [[inverse square law]]. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body.<ref name=uniphysics_ch4 /> Combining these ideas gives a formula that relates the mass (<math> m_\oplus</math>) and the radius (<math> R_\oplus</math>) of the Earth to the gravitational acceleration:
<math display="block" qid=Q30006>\vec{g}=-\frac{Gm_\oplus}{{R_\oplus}^2} \hat{r},</math>
where the vector direction is given by <math>\hat{r}</math>, is the [[unit vector]] directed outward from the center of the Earth.<ref name="Principia"/>

In this equation, a dimensional constant <math>G</math> is used to describe the relative strength of gravity. This constant has come to be known as the [[Newtonian constant of gravitation]], though its value was unknown in Newton's lifetime. Not until 1798 was [[Henry Cavendish]] able to make the first measurement of <math>G</math> using a [[torsion balance]]; this was widely reported in the press as a measurement of the mass of the Earth since knowing <math>G</math> could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed the same [[Kepler's laws|laws of motion]], his law of gravity had to be universal. Succinctly stated, [[Newton's law of gravitation]] states that the force on a spherical object of mass <math>m_1</math> due to the gravitational pull of mass <math>m_2</math> is
<math display="block" qid=Q11412>\vec{F}=-\frac{Gm_{1}m_{2}}{r^2} \hat{r},</math>
where <math>r</math> is the distance between the two objects' centers of mass and <math>\hat{r}</math> is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.<ref name="Principia"/>

This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the 20th century. During that time, sophisticated methods of [[perturbation analysis]]<ref>{{cite web |last=Watkins |first=Thayer |title=Perturbation Analysis, Regular and Singular |work=Department of Economics |publisher=San José State University |url=http://www.sjsu.edu/faculty/watkins/perturb.htm |access-date=2008-01-05 |archive-date=2011-02-10 |archive-url=https://web.archive.org/web/20110210010802/http://www.sjsu.edu/faculty/watkins/perturb.htm |url-status=dead }}</ref> were invented to calculate the deviations of [[orbit]]s due to the influence of multiple bodies on a [[planet]], [[moon]], [[comet]], or [[asteroid]]. The formalism was exact enough to allow mathematicians to predict the existence of the planet [[Neptune]] before it was observed.<ref name='Neptdisc'>{{cite web |url=http://www.ucl.ac.uk/sts/nk/neptune/index.htm |title=Neptune's Discovery. The British Case for Co-Prediction. |access-date=2007-03-19 |last=Kollerstrom |first=Nick |year=2001 |publisher=University College London |archive-url= https://web.archive.org/web/20051111190351/http://www.ucl.ac.uk/sts/nk/neptune/index.htm |archive-date=2005-11-11}}</ref>

=== Electromagnetic ===
{{main|Electromagnetic force}}
The [[electrostatic force]] was first described in 1784 by Coulomb as a force that existed intrinsically between two [[electric charge|charges]].<ref name=Cutnell/>{{rp|519}} The properties of the electrostatic force were that it varied as an [[inverse square law]] directed in the [[polar coordinates|radial direction]], was both attractive and repulsive (there was intrinsic [[Electrical polarity|polarity]]), was independent of the mass of the charged objects, and followed the [[superposition principle]]. [[Coulomb's law]] unifies all these observations into one succinct statement.<ref name="Coulomb">{{cite journal |first=Charles |last=Coulomb |journal=Histoire de l'Académie Royale des Sciences |year=1784 |title=Recherches théoriques et expérimentales sur la force de torsion et sur l'élasticité des fils de metal |pages=229–269}}</ref>

Subsequent mathematicians and physicists found the construct of the ''[[electric field]]'' to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "[[test charge]]" anywhere in space and then using Coulomb's Law to determine the electrostatic force.<ref name=FeynmanVol2/>{{rp|((4-6–4-8))}} Thus the electric field anywhere in space is defined as
<math display="block">\vec{E} = {\vec{F} \over{q}},</math>
where <math>q</math> is the magnitude of the hypothetical test charge. Similarly, the idea of the ''[[magnetic field]]'' was introduced to express how magnets can influence one another at a distance. The [[Lorentz force|Lorentz force law]] gives the force upon a body with charge <math>q</math> due to electric and magnetic fields:
<math display="block" qid=Q849919>\vec{F} = q\left(\vec{E} + \vec{v} \times \vec{B}\right),</math>
where <math> \vec{F}</math> is the electromagnetic force, <math> \vec{E}</math> is the electric field at the body's location, <math>\vec{B}</math> is the magnetic field, and <math> \vec{v}</math> is the [[velocity]] of the particle. The magnetic contribution to the Lorentz force is the [[cross product]] of the velocity vector with the magnetic field.<ref>{{Cite book|last=Tonnelat|first=Marie-Antoinette|url=https://www.worldcat.org/oclc/844001|title=The principles of electromagnetic theory and of relativity.|date=1966|publisher=D. Reidel|isbn=90-277-0107-5|location=Dordrecht|oclc=844001|author-link=Marie-Antoinette Tonnelat |page=85}}</ref><ref name="openstax-university-physics2">{{cite book|title=University Physics, Volume 2 |url=https://openstax.org/details/books/university-physics-volume-2 |publisher=[[OpenStax]] |year=2021 |first1=Samuel J. |last1=Ling |first2=Jeff |last2=Sanny |first3=William |last3=Moebs |isbn=978-1-947-17221-0}}</ref>{{rp|482}}

The origin of electric and magnetic fields would not be fully explained until 1864 when [[James Clerk Maxwell]] unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by [[Oliver Heaviside]] and [[Josiah Willard Gibbs]].<ref>{{cite book
|title=Polarized light in liquid crystals and polymers
|first1=Toralf
|last1=Scharf
|publisher=John Wiley and Sons
|year=2007
|isbn=978-0-471-74064-3
|page=19
|chapter=Chapter 2
|chapter-url=https://books.google.com/books?id=CQNE13opFucC&pg=PA19}}</ref> These "[[Maxwell's equations]]" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a [[wave]] that traveled at a speed that he calculated to be the [[speed of light]]. This insight united the nascent fields of electromagnetic theory with [[optics]] and led directly to a complete description of the [[electromagnetic spectrum]].<ref>
{{cite book
 |first=William
 |last=Duffin
 |title=Electricity and Magnetism
 |publisher=McGraw-Hill
 |pages=[https://archive.org/details/electricitymagn00duff/page/364 364–383]
 |year=1980
 |edition=3rd
 |isbn=978-0-07-084111-6
 |url=https://archive.org/details/electricitymagn00duff/page/364
}}</ref>

=== Normal ===
[[File:Incline.svg|right|thumb|''F''<sub>N</sub> represents the [[normal force]] exerted on the object.]]
{{main|Normal force}}
When objects are in contact, the force directly between them is called the normal force, the component of the total force in the system exerted normal to the interface between the objects.<ref name=Cutnell>{{Cite book |last1=Cutnell |first1=John D. |title=Physics |last2=Johnson |first2=Kenneth W. |date=2004 |publisher=Wiley |isbn=978-0-471-44895-2 |edition=6th|location=Hoboken, NJ}}</ref>{{rp|264}} The normal force is closely related to Newton's third law. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Friction ===
{{main|Friction}}
Friction is a force that opposes relative motion of two bodies. At the macroscopic scale, the frictional force is directly related to the normal force at the point of contact. There are two broad classifications of frictional forces: [[static friction]] and [[kinetic friction]].<ref name="openstax-university-physics"/>{{rp|267}}

The static friction force (<math>F_{\mathrm{sf}}</math>) will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the [[coefficient of static friction]] (<math>\mu_{\mathrm{sf}}</math>) multiplied by the normal force (<math>F_\text{N}</math>). In other words, the magnitude of the static friction force satisfies the inequality:
<math display="block">0 \le F_{\mathrm{sf}} \le \mu_{\mathrm{sf}} F_\mathrm{N}.</math>

The kinetic friction force (<math>F_{\mathrm{kf}}</math>) is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:
<math display="block">F_{\mathrm{kf}} = \mu_{\mathrm{kf}} F_\mathrm{N},</math>

where <math>\mu_{\mathrm{kf}}</math> is the [[coefficient of kinetic friction]]. The coefficient of kinetic friction is normally less than the coefficient of static friction.<ref name="openstax-university-physics"/>{{rp|267–271}}

=== Tension ===
{{main|Tension (physics)}}
Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with ideal [[pulley]]s, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action–reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object.<ref>{{cite web
|title=Tension Force
|work=Non-Calculus Based Physics I
|url=http://www.mtsu.edu/~phys2010/Lectures/Part_2__L6_-_L11/Lecture_9/Tension_Force/tension_force.html
|access-date=2008-01-04
|archive-date=2007-12-27
|archive-url=https://web.archive.org/web/20071227065923/http://www.mtsu.edu/~phys2010/Lectures/Part_2__L6_-_L11/Lecture_9/Tension_Force/tension_force.html
|url-status=dead
}}</ref> By connecting the same string multiple times to the same object through the use of a configuration that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. Such machines allow a [[mechanical advantage]] for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in the [[conservation of energy|conservation of mechanical energy]] since the [[#Kinematic integrals|work done on the load]] is the same no matter how complicated the machine.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner /><ref>{{cite web |last=Fitzpatrick |first=Richard |title=Strings, pulleys, and inclines |date=2006-02-02 |url=http://farside.ph.utexas.edu/teaching/301/lectures/node48.html |access-date=2008-01-04}}</ref>

=== Spring ===
{{main|Elasticity (physics)|Hooke's law}}
[[File:Mass-spring-system.png|upright|thumb|''F<sub>k</sub>'' is the force that responds to the load on the spring]]
A simple elastic force acts to return a [[Spring (device)|spring]] to its natural length. An [[ideal spring]] is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the [[displacement field (mechanics)|displacement]] of the spring from its equilibrium position.<ref>{{cite web |last=Nave |first=Carl Rod |title=Elasticity |work=HyperPhysics |publisher=University of Guelph |url=http://hyperphysics.phy-astr.gsu.edu/hbase/permot2.html |access-date=2013-10-28}}</ref> This linear relationship was described by [[Robert Hooke]] in 1676, for whom [[Hooke's law]] is named. If <math>\Delta x</math> is the displacement, the force exerted by an ideal spring equals:
<math display="block" qid=Q170282>\vec{F}=-k \Delta \vec{x},</math>
where <math>k</math> is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to the applied load.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Centripetal ===
{{main|Centripetal force}}
For an object in [[uniform circular motion]], the net force acting on the object equals:<ref>{{cite web |last=Nave |first=Carl Rod |title=Centripetal Force |work=HyperPhysics |publisher=University of Guelph |url=http://hyperphysics.phy-astr.gsu.edu/hbase/cf.html |access-date=2013-10-28}}</ref>
<math display="block" qid=Q172881>\vec{F} = - \frac{mv^2 \hat{r}}{r},</math>
where <math>m</math> is the mass of the object, <math>v</math> is the velocity of the object and <math>r</math> is the distance to the center of the circular path and <math> \hat{r}</math> is the [[unit vector]] pointing in the radial direction outwards from the center. This means that the net force felt by the object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the [[speed]] of the object (magnitude of the velocity), but only the direction of the velocity vector. More generally, the net force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Continuum mechanics ===
[[File:Stokes sphere.svg|thumb|upright|When the drag force (<math>F_\text{d}</math>) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (<math>F_\text{g}</math>), the object reaches a state of [[#Dynamic equilibrium|dynamic equilibrium]] at [[terminal velocity]].]]
{{main|Pressure|Drag (physics)|Stress (mechanics)}}
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized [[point particle]]s rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of [[continuum mechanics]] describe the way forces affect the material. For example, in extended [[fluid mechanics|fluids]], differences in [[pressure]] result in forces being directed along the pressure [[gradient]]s as follows:
<math display="block">\frac{\vec{F}}{V} = - \vec{\nabla} P,</math>

where <math>V</math> is the volume of the object in the fluid and <math>P</math> is the [[scalar function]] that describes the pressure at all locations in space. Pressure gradients and differentials result in the [[buoyancy|buoyant force]] for fluids suspended in gravitational fields, winds in [[atmospheric science]], and the [[lift (physics)|lift]] associated with [[aerodynamics]] and [[flight]].<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

A specific instance of such a force that is associated with [[dynamic pressure]] is fluid resistance: a body force that resists the motion of an object through a fluid due to [[viscosity]]. For so-called "[[Drag (physics)#Very low Reynolds numbers – Stokes' drag|Stokes' drag]]" the force is approximately proportional to the velocity, but opposite in direction:
<math display="block" qid=Q824561>\vec{F}_\mathrm{d} = - b \vec{v}, </math>
where:
* <math>b</math> is a constant that depends on the properties of the fluid and the dimensions of the object (usually the [[Cross section (geometry)|cross-sectional area]]), and
* <math> \vec{v}</math> is the velocity of the object.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

More formally, forces in [[continuum mechanics]] are fully described by a [[Stress (mechanics)|stress]] [[tensor]] with terms that are roughly defined as
<math display="block" qid=Q206175>\sigma = \frac{F}{A},</math>
where <math>A</math> is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the [[matrix diagonal]]s of the tensor) as well as [[Shear stress|shear]] terms associated with forces that act [[Parallel (geometry)|parallel]] to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all [[strain (physics)|strains]] (deformations) including also [[tensile stress]]es and [[compression (physical)|compressions]].<ref name=uniphysics_ch2>{{cite book|title=University Physics |last1=Sears |first1=Francis W. |last2=Zemansky |first2=Mark W. |last3=Young |first3=Hugh D. |author-link1=Francis Sears |author-link2=Mark Zemansky |author-link3=Hugh D. Young |title-link=University Physics |pages=18–38 |publisher=Addison-Wesley |edition=6th |year=1982 |isbn=0-201-07199-1}}</ref><ref name=Kleppner>{{cite book |last1=Kleppner |first1=Daniel |last2=Kolenkow |first2=Robert J. |title=An Introduction to Mechanics|year=2014|publisher=Cambridge University Press|location=Cambridge|isbn=978-0521198110|edition=2nd|chapter=Chapter 3: Forces and equations of motion|chapter-url=https://archive.org/details/KleppnerD.KolenkowR.J.IntroductionToMechanics2014/page/n102}}</ref>{{rp|133–134}}<ref name=FeynmanVol2>{{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |title=The Feynman lectures on physics. Vol. II: Mainly electromagnetism and matter |year=2010 |publisher=Basic Books |location=New York |isbn=978-0465024940 |edition=New millennium |title-link=The Feynman Lectures on Physics |author-link1=Richard Feynman |author-link2=Robert B. Leighton |author-link3=Matthew Sands}}</ref>{{rp|((38-1–38-11))}}

=== Fictitious ===
{{main|Fictitious forces}}
There are forces that are [[frame dependent]], meaning that they appear due to the adoption of non-Newtonian (that is, [[non-inertial frame|non-inertial]]) [[Frame of reference|reference frames]]. Such forces include the [[Centrifugal force (rotating reference frame)|centrifugal force]] and the [[Coriolis force]].<ref>{{cite web |last=Mallette |first=Vincent |title= The Coriolis Force |work=Publications in Science and Mathematics, Computing and the Humanities |publisher=Inwit Publishing, Inc. |date=1982–2008 |url=http://www.algorithm.com/inwit/writings/coriolisforce.html |access-date=2008-01-04}}</ref> These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner /> Because these forces are not genuine they are also referred to as "pseudo forces".<ref name=FeynmanVol1 />{{rp|((12-11))}}

In [[general relativity]], [[gravity]] becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.<ref>{{Cite book |last=Choquet-Bruhat |first=Yvonne |url=https://www.worldcat.org/oclc/317496332 |title=General Relativity and the Einstein Equations |date=2009 |publisher=Oxford University Press |isbn=978-0-19-155226-7 |location=Oxford |oclc=317496332 |author-link=Yvonne Choquet-Bruhat |page=39}}</ref>

== Concepts derived from force ==
=== Rotation and torque ===
[[File:Torque animation.gif|frame|right|Relationship between force (''F''), torque (''τ''), and [[angular momentum|momentum]] vectors (''p'' and ''L'') in a rotating system.]]
{{main|Torque}}
Forces that cause extended objects to rotate are associated with [[torque]]s. Mathematically, the torque of a force <math> \vec{F}</math> is defined relative to an arbitrary reference point as the [[cross product]]:
<math display="block" qid=Q48103>\vec{\tau} = \vec{r} \times \vec{F},</math>
where <math> \vec{r}</math> is the [[position vector]] of the force application point relative to the reference point.<ref name="openstax-university-physics"/>{{rp|497}}

Torque is the rotation equivalent of force in the same way that [[angle]] is the rotational equivalent for [[position (vector)|position]], [[angular velocity]] for [[velocity]], and [[angular momentum]] for [[momentum]]. As a consequence of Newton's first law of motion, there exists [[rotational inertia]] that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous [[angular acceleration]] of the rigid body:
<math display="block">\vec{\tau} = I\vec{\alpha},</math>
where
* <math>I</math> is the [[moment of inertia]] of the body
* <math> \vec{\alpha}</math> is the angular acceleration of the body.<ref name="openstax-university-physics"/>{{rp|502}}

This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the [[Moment of inertia tensor|tensor]] that, when properly analyzed, fully determines the characteristics of rotations including [[precession]] and [[nutation]].<ref name=":0" />{{Rp|pages=96–113}}

Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:<ref>{{cite web |last=Nave |first=Carl Rod |title=Newton's 2nd Law: Rotation |work=HyperPhysics |publisher=University of Guelph |url=http://hyperphysics.phy-astr.gsu.edu/HBASE/n2r.html |access-date=2013-10-28}}</ref>
<math display="block">\vec{\tau} = \frac{\mathrm{d}\vec{L}}{\mathrm{dt}},</math>
where <math> \vec{L}</math> is the angular momentum of the particle.

Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques,<ref>{{cite web |last=Fitzpatrick |first=Richard |title=Newton's third law of motion |date=2007-01-07 |url=http://farside.ph.utexas.edu/teaching/336k/lectures/node26.html |access-date=2008-01-04}}</ref> and therefore also directly implies the [[conservation of angular momentum]] for closed systems that experience rotations and [[revolution (geometry)|revolution]]s through the action of internal torques.

=== Kinematic integrals ===
{{main|Impulse (physics)|l1=Impulse|Mechanical work|Power (physics)}}
Forces can be used to define a number of physical concepts by [[integration (calculus)|integrating]] with respect to [[kinematics|kinematic variables]]. For example, integrating with respect to time gives the definition of [[Impulse (physics)|impulse]]:<ref>{{Cite book
|title=Engineering Mechanics
|first1=Russell C.
|last1=Hibbeler
|publisher=Pearson Prentice Hall
|year=2010
|edition=12th
|isbn=978-0-13-607791-6
|page=222
}}</ref>
<math display="block">\vec{J}=\int_{t_1}^{t_2}{\vec{F} \, \mathrm{d}t},</math>
which by Newton's Second Law must be equivalent to the change in momentum (yielding the [[Impulse momentum theorem]]).

Similarly, integrating with respect to position gives a definition for the [[work (physics)|work done]] by a force:<ref name=FeynmanVol1/>{{rp|((13-3))}}
<math display="block" qid=Q42213>W= \int_{\vec{x}_1}^{\vec{x}_2} {\vec{F} \cdot {\mathrm{d}\vec{x}}},</math>
which is equivalent to changes in [[kinetic energy]] (yielding the [[work energy theorem]]).<ref name=FeynmanVol1/>{{rp|((13-3))}}

[[Power (physics)|Power]] ''P'' is the rate of change d''W''/d''t'' of the work ''W'', as the [[trajectory]] is extended by a position change <math> d\vec{x}</math> in a time interval d''t'':<ref name=FeynmanVol1/>{{rp|((13-2))}}
<math display="block"> \mathrm{d}W = \frac{\mathrm{d}W}{\mathrm{d}\vec{x}} \cdot \mathrm{d}\vec{x} = \vec{F} \cdot \mathrm{d}\vec{x},</math>
so
<math display="block">P = \frac{\mathrm{d}W}{\mathrm{d}t} = \frac{\mathrm{d}W}{\mathrm{d}\vec{x}} \cdot \frac{\mathrm{d}\vec{x}}{\mathrm{d}t} = \vec{F} \cdot \vec{v},
</math>
with <math qid=Q11465>\vec{v} = \mathrm{d}\vec{x}/\mathrm{d}t</math> the [[velocity]].

=== Potential energy ===
{{main|Potential energy}}
Instead of a force, often the mathematically related concept of a [[potential energy]] field is used. For instance, the gravitational force acting upon an object can be seen as the action of the [[gravitational field]] that is present at the object's location. Restating mathematically the definition of energy (via the definition of [[Mechanical work|work]]), a potential [[scalar field]] <math>U(\vec{r})</math> is defined as that field whose [[gradient]] is equal and opposite to the force produced at every point:
<math display="block">\vec{F}=-\vec{\nabla} U.</math>

Forces can be classified as [[Conservative force|conservative]] or nonconservative. Conservative forces are equivalent to the gradient of a [[potential]] while nonconservative forces are not.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Conservation ===
{{main|Conservative force}}
A conservative force that acts on a [[closed system]] has an associated mechanical work that allows energy to convert only between [[kinetic energy|kinetic]] or [[potential energy|potential]] forms. This means that for a closed system, the net [[mechanical energy]] is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space,<ref>{{cite web
|last=Singh
|first=Sunil Kumar
|title=Conservative force
|work=Connexions
|date=2007-08-25
|url=http://cnx.org/content/m14104/latest/
|access-date=2008-01-04}}</ref> and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the [[contour map]] of the elevation of an area.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

Conservative forces include [[gravity]], the [[Electromagnetism|electromagnetic]] force, and the [[Hooke's law|spring]] force. Each of these forces has models that are dependent on a position often given as a [[radius|radial vector]] <math> \vec{r}</math> emanating from [[spherical symmetry|spherically symmetric]] potentials.<ref>{{cite web
|last=Davis
|first=Doug
|title=Conservation of Energy
|work=General physics
|url=http://www.ux1.eiu.edu/~cfadd/1350/08PotEng/ConsF.html
|access-date=2008-01-04}}</ref> Examples of this follow:

For gravity:
<math display="block">\vec{F}_\text{g} = - \frac{G m_1 m_2}{r^2} \hat{r},</math>
where <math>G</math> is the [[gravitational constant]], and <math>m_n</math> is the mass of object ''n''.

For electrostatic forces:
<math display="block">\vec{F}_\text{e} = \frac{q_1 q_2}{4 \pi \varepsilon_{0} r^2} \hat{r},</math>
where <math>\varepsilon_{0}</math> is [[Permittivity|electric permittivity of free space]], and <math>q_n</math> is the [[electric charge]] of object ''n''.

For spring forces:
<math display="block">\vec{F}_\text{s} = - k r \hat{r},</math>
where <math>k</math> is the [[spring constant]].<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of [[Microstate (statistical mechanics)|microstates]]. For example, static friction is caused by the gradients of numerous electrostatic potentials between the [[atom]]s, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other [[contact force]]s, [[Tension (physics)|tension]], [[Physical compression|compression]], and [[drag (physics)|drag]]. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with [[statistical mechanics]]. In macroscopic closed systems, nonconservative forces act to change the [[internal energy|internal energies]] of the system, and are often associated with the transfer of heat. According to the [[Second law of thermodynamics]], nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as [[entropy]] increases.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

== Units ==
The [[SI]] unit of force is the [[Newton (unit)|newton]] (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg·m·s<sup>−2</sup>.The corresponding [[CGS]] unit is the [[dyne]], the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s<sup>−2</sup>. A newton is thus equal to 100,000&nbsp;dynes.<ref name= metric_units>{{cite book |first1=Cornelius |last1=Wandmacher |first2=Arnold |last2=Johnson |title=Metric Units in Engineering |page=[https://archive.org/details/metricunitsineng0000wand/page/15 15] |year=1995 |publisher=ASCE Publications |isbn=978-0-7844-0070-8 |url=https://archive.org/details/metricunitsineng0000wand/page/15 }}</ref>

The gravitational [[foot-pound-second]] [[English unit]] of force is the [[pound-force]] (lbf), defined as the force exerted by gravity on a [[pound-mass]] in the [[Standard gravity|standard gravitational]] field of 9.80665&nbsp;m·s<sup>−2</sup>.<ref name=metric_units/> The pound-force provides an alternative unit of mass: one [[slug (unit)|slug]] is the mass that will accelerate by one foot per second squared when acted on by one pound-force.<ref name=metric_units/> An alternative unit of force in a different foot–pound–second system, the absolute fps system, is the [[poundal]], defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared.<ref name=metric_units/>

The pound-force has a metric counterpart, less commonly used than the newton: the [[kilogram-force]] (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the [[metric slug]] (sometimes mug or hyl) is that mass that accelerates at 1&nbsp;m·s<sup>−2</sup> when subjected to a force of 1&nbsp;kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated, sometimes used for expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque.<ref name= metric_units/>

{{units of force|center=yes|cat=no}}

: See also ''[[Ton-force]]''.

== Revisions of the force concept ==
At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly.

=== Special theory of relativity ===
{{Main | Relativistic mechanics#Force}}
In the [[special theory of relativity]], mass and [[energy]] are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law,
<math display="block">\vec{F} = \frac{\mathrm{d}\vec{p}}{\mathrm{d}t},</math>
remains valid because it is a mathematical definition.<ref name=Cutnell/>{{rp|855–876}} But for momentum to be conserved at  relativistic relative velocity, <math>v</math>, momentum must be redefined as:
<math display="block"> \vec{p} = \frac{m_0\vec{v}}{\sqrt{1 - v^2/c^2}},</math>
where <math>m_0</math> is the [[rest mass]] and <math>c</math> the [[speed of light]].

The expression relating force and acceleration for a particle with constant non-zero [[rest mass]] <math>m</math> moving in the <math>x</math> direction at velocity <math>v</math> is:<ref name=French1972>{{Cite book |last=French |first=A. P. |title=Special Relativity |date=1972 |publisher=Chapman & Hall |isbn=978-0-17-771075-9 |edition=reprint |series=The MIT introductory physics series |location=London}}</ref>{{rp|216}}
<math display="block">\vec{F} = \left(\gamma^3 m a_x, \gamma m a_y, \gamma m a_z\right),</math>
where
<math display="block" qid=Q599404> \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.</math>
is called the [[Lorentz factor]]. The Lorentz factor increases steeply as the relative velocity approaches the speed of light.  Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach <math>c</math>.<ref name=French1972/>{{rp|26}}<ref name=FeynmanVol1/>{{rp|at=§15–8}}
If <math>v</math> is very small compared to <math>c</math>, then <math>\gamma</math> is very close to 1 and
<math display="block">F = m a</math>
is a close approximation. Even for use in relativity, one can restore the form of
<math display="block">F^\mu = mA^\mu </math>
through the use of [[four-vectors]]. This relation is correct in relativity when <math>F^\mu</math> is the [[four-force]], <math>m</math> is the [[invariant mass]], and <math>A^\mu</math> is the [[four-acceleration]].<ref>{{cite web
|first=John B.
|last=Wilson
|title=Four-Vectors (4-Vectors) of Special Relativity: A Study of Elegant Physics
|work=The Science Realm: John's Virtual Sci-Tech Universe
|url=http://SciRealm.com/4Vectors.html
|archive-url=https://web.archive.org/web/20090626152836/http://www.austininc.com/SciRealm/4Vectors.html
|archive-date=26 June 2009
|url-status=dead
|access-date=2008-01-04
}}</ref>

The [[general relativity|''general'' theory of relativity]] incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully [[#Gravitational|below]].

=== Quantum mechanics ===
{{main|Quantum mechanics|Pauli exclusion principle}}
[[Quantum mechanics]] is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object ''has'', one considers what result might ''appear'' when a [[Measurement in quantum mechanics|measurement]] of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.<ref>{{cite journal|last=Mermin|first=N. David|author-link=N. David Mermin|year=1993|title=Hidden variables and the two theorems of John Bell|journal=[[Reviews of Modern Physics]]|volume=65|issue=3|pages=803&ndash;815|arxiv=1802.10119|bibcode=1993RvMP...65..803M|doi=10.1103/RevModPhys.65.803|s2cid=119546199 |quote=It is a fundamental quantum doctrine that a measurement does not, in general, reveal a pre-existing value of the measured property.}}</ref><ref>{{Cite journal|last1=Schaffer|first1=Kathryn|last2=Barreto Lemos|first2=Gabriela|date=24 May 2019|title=Obliterating Thingness: An Introduction to the "What" and the "So What" of Quantum Physics|journal=[[Foundations of Science]] |volume=26 |pages=7–26 |language=en|arxiv=1908.07936|doi=10.1007/s10699-019-09608-5|issn=1233-1821|s2cid=182656563}}</ref> The [[Expectation value (quantum mechanics)|expectation value]] for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.<ref>{{Cite journal|last1=Marshman|first1=Emily|last2=Singh|first2=Chandralekha|author-link2=Chandralekha Singh|date=2017-03-01|title=Investigating and improving student understanding of the probability distributions for measuring physical observables in quantum mechanics|journal=[[European Journal of Physics]]|volume=38|issue=2|pages=025705|doi=10.1088/1361-6404/aa57d1|bibcode=2017EJPh...38b5705M |s2cid=126311599 |issn=0143-0807|doi-access=free}}</ref>

In quantum mechanics, interactions are typically described in terms of energy rather than force. The [[Ehrenfest theorem]] provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the [[Born rule]] is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.<ref name="Cohen-Tannoudji">{{cite book|last1=Cohen-Tannoudji |first1=Claude |last2=Diu |first2=Bernard |last3=Laloë |first3=Franck |title=Quantum Mechanics |author-link1=Claude Cohen-Tannoudji |publisher=John Wiley & Sons |year=2005 |isbn=0-471-16433-X |translator-first1=Susan Reid |translator-last1=Hemley |translator-first2=Nicole |translator-last2=Ostrowsky |translator-first3=Dan |translator-last3=Ostrowsky |page=242}}</ref><ref>{{cite book|last=Peres|first=Asher|author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods|publisher=[[Kluwer]]|year=1993|isbn=0-7923-2549-4|oclc=28854083 |page=302}}</ref>

Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the [[uncertainty principle]] limits the minimum extent of an electron probability distribution<ref name=Lieb-RMP>{{Cite journal |last=Lieb |first=Elliott H. |date=1976-10-01 |title=The stability of matter |url=https://link.aps.org/doi/10.1103/RevModPhys.48.553 |journal=Reviews of Modern Physics |language=en |volume=48 |issue=4 |pages=553–569 |doi=10.1103/RevModPhys.48.553 |issn=0034-6861 |quote=the fact that if one tries to compress a wave function ''anywhere'' then the kinetic energy will increase. This principle was provided by Sobolev (1938)...}}</ref> and the [[Pauli exclusion principle]] prevents electrons from sharing the same probability distribution.<ref name=Lieb-Bulletin>{{Cite journal |last=Lieb |first=Elliott H. |date=1990 |title=The stability of matter: from atoms to stars |url=https://www.ams.org/bull/1990-22-01/S0273-0979-1990-15831-8/ |journal=Bulletin of the American Mathematical Society |language=en |volume=22 |issue=1 |pages=1–49 |doi=10.1090/S0273-0979-1990-15831-8 |issn=0273-0979 |quote= bulk matter is stable, and has a volume proportional to the number of particles, because of the Pauli exclusion principle for fermions (Le., the electrons). Effectively the electrons behave like a fluid with energy density <math>\rho^{5/3}</math>, and this limits the compression caused by the attractive electrostatic forces. |doi-access=free }}</ref> This gives rise to an emergent pressure known as [[electron degeneracy pressure|degeneracy pressure]]. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids [[Stability of matter|stability]].<ref>{{cite book|last=Griffiths|title=Introduction to Quantum Mechanics, Second Edition|year=2005|publisher=[[Prentice Hall]]|location=London, UK|isbn=0131244051|pages=221–223}}</ref>

=== Quantum field theory ===
{{main|Quantum field theory}}
[[File:Beta Negative Decay.svg|thumb|Feynman diagram for the decay of a neutron into a proton. The [[W boson]] is between two vertices indicating a repulsion.]]
In modern [[particle physics]], forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying [[gauge boson]]s. With the development of [[quantum field theory]] and [[general relativity]], it was realized that force is a redundant concept arising from [[conservation of momentum]] ([[4-momentum]] in relativity and momentum of [[virtual particle]]s in [[quantum electrodynamics]]). The conservation of momentum can be directly derived from the homogeneity or [[Symmetry in physics|symmetry]] of [[space]] and so is usually considered more fundamental than the concept of a force. Thus the currently known [[fundamental forces]] are considered more accurately to be "[[fundamental interactions]]".<ref name="final theory">{{cite book |last=Weinberg |first=S. |year=1994 |title=Dreams of a Final Theory |publisher=Vintage Books |isbn=978-0-679-74408-5}}</ref>{{rp|199–128}}

While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of [[Feynman diagram]]s. In a Feynman diagram, each matter particle is represented as a straight line (see [[world line]]) traveling through time, which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction [[Vertex (graph theory)|vertices]], and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines and, in the case of virtual particle exchange, are absorbed at an adjacent vertex.<ref name=Shifman>{{cite book |first=Mikhail |last=Shifman |title=ITEP lectures on particle physics and field theory |publisher=World Scientific |year=1999 |isbn=978-981-02-2639-8}}</ref> The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of [[fundamental interaction]]s but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a [[neutron]] [[beta decay|decays]] into an [[electron]], [[proton]], and [[neutrino|antineutrino]], an interaction mediated by the same gauge boson that is responsible for the [[weak nuclear force]].<ref name="Shifman"/>

== Fundamental interactions ==
{{main|Fundamental interaction}}

All of the known forces of the universe are classified into four [[fundamental interaction]]s. The [[strong force|strong]] and the [[weak force|weak]] forces act only at very short distances, and are responsible for the interactions between [[subatomic particle]]s, including [[nucleon]]s and compound [[Atomic nucleus|nuclei]]. The [[electromagnetic force]] acts between [[electric charge]]s, and the [[gravitational force]] acts between [[mass]]es. All other forces in nature derive from these four fundamental interactions operating within [[quantum mechanics]], including the constraints introduced by the [[Schrödinger equation]] and the [[Pauli exclusion principle]].<ref name=Lieb-Bulletin/> For example, [[friction]] is a manifestation of the electromagnetic force acting between [[atoms]] of two surfaces. The forces in [[spring (device)|springs]], modeled by [[Hooke's law]], are also the result of electromagnetic forces. [[Centrifugal force (fictitious)|Centrifugal forces]] are [[acceleration]] forces that arise simply from the acceleration of [[rotation|rotating]] [[frames of reference]].<ref name=FeynmanVol1/>{{rp|((12-11))}}<ref name=Kleppner/>{{rp|359}}

The fundamental theories for forces developed from the [[Unified field theory|unification]] of different ideas. For example, Newton's universal theory of [[gravitation]] showed that the force responsible for objects falling near the surface of the [[Earth]] is also the force responsible for the falling of celestial bodies about the Earth (the [[Moon]]) and around the Sun (the planets). [[Michael Faraday]] and [[James Clerk Maxwell]] demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of [[quantum mechanics]] led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter ([[fermion]]s) interacting by exchanging [[virtual particle]]s called [[gauge boson]]s.<ref>{{cite web |title=Fermions & Bosons |work=The Particle Adventure |url=http://particleadventure.org/frameless/fermibos.html |access-date=2008-01-04 |url-status=dead |archive-url=https://web.archive.org/web/20071218074732/http://particleadventure.org/frameless/fermibos.html |archive-date=2007-12-18 }}</ref> This [[Standard Model]] of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in [[electroweak]] theory, which was subsequently confirmed by observation.<ref>{{cite web|url=https://www.nobelprize.org/prizes/physics/1999/advanced-information/ |title=Additional background material on the Nobel Prize in Physics 1999 |date=1999-10-12 |access-date=2023-07-26 |website=Nobel Prize |first=Cecilia |last=Jarlskog |author-link=Cecilia Jarlskog}}</ref>

{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ '''The four fundamental forces of nature'''<ref>{{cite web |url=http://www.cpepphysics.org/cpep_sm_large.html |title=Standard model of particles and interactions |publisher=Contemporary Physics Education Project |date=2000 |access-date=2 January 2017 |archive-date=2 January 2017 |archive-url=https://web.archive.org/web/20170102180203/http://www.cpepphysics.org/cpep_sm_large.html |url-status=dead }}</ref>
!rowspan="2" style="text-align: center;"| Property/Interaction
!rowspan="2" style="text-align: center;background-color:#8585C2"|Gravitation
!style="background-color:#F012F0"|Weak
!style="background-color:#FF4D4D"|Electromagnetic
!colspan="2" style="text-align: center;background-color:#99B280"|Strong
|-
!colspan="2" style="text-align: center;background-color:#FF9999"| <small>(Electroweak)</small>
!style="background-color:#CCD8C0"|<small>Fundamental</small>
!style="background-color:#F0F3EC"|<small>Residual</small>
|-
|style="background-color:#FFFFF6"|Acts on:
|align="center"|Mass - Energy
|align="center"|Flavor
|align="center"|Electric charge
|align="center"|Color charge
|align="center"|Atomic nuclei
|-
|style="background-color:#FFFFF6"|Particles experiencing:
|align="center"|All
|align="center"|Quarks, leptons
|align="center"|Electrically charged
|align="center"|Quarks, Gluons
|align="center"|Hadrons
|-
|style="background-color:#FFFFF6"|Particles mediating:
|align="center"|Graviton <br /><small>(not yet observed)</small>
|align="center"|W<sup>+</sup> W<sup>−</sup> Z<sup>0</sup>
|align="center"|γ
|align="center"|Gluons
|align="center"|Mesons
|-
|style="background-color:#FFFFF6"|Strength in the scale of quarks:
|align="center"|{{val||e=-41}}
|align="center"|{{val||e=-4}}
|align="center"|1
|align="center"|60
|<small>Not applicable <br />to quarks</small>
|-
|style="background-color:#FFFFF6"|Strength in the scale of <br />protons/neutrons:
|align="center"|{{val||e=-36}}
|align="center"|{{val||e=-7}}
|align="center"|1
|align="center"|<small>Not applicable <br />to hadrons</small>
|align="center"|20
|}

=== Gravitational ===
[[File:GRAVITY A powerful new probe of black holes.jpg|thumb|Instruments like GRAVITY provide a powerful probe for gravity force detection.<ref>{{cite web |title=Powerful New Black Hole Probe Arrives at Paranal |url=http://www.eso.org/public/announcements/ann15061/ |access-date=13 August 2015}}</ref>]]
Newton's law of gravitation is an example of ''action at a distance'': one body, like the Sun, exerts an influence upon any other body, like the Earth, no matter how far apart they are. Moreover, this action at a distance is ''instantaneous.'' According to Newton's theory, the one body shifting position changes the gravitational pulls felt by all other bodies, all at the same instant of time. [[Albert Einstein]] recognized that this was inconsistent with special relativity and its prediction that influences cannot travel faster than the [[speed of light]]. So, he sought a new theory of gravitation that would be relativistically consistent.<ref>
{{cite book
 | last1=Misner |first1=Charles W. |author-link1=Charles W. Misner
 | last2=Thorne |first2=Kip S. |author-link2=Kip Thorne
 | last3=Wheeler |first3=John Archibald |author-link3=John Archibald Wheeler
 | year=1973
 | title=Gravitation
 | title-link=Gravitation (book)
 | publisher=[[W. H. Freeman]]
 | location=San Francisco
 | isbn=978-0-7167-0344-0
 | pages=3–5
}}</ref><ref>
{{Cite book
 |last=Choquet-Bruhat |first=Yvonne |author-link=Yvonne Choquet-Bruhat
 |url=https://www.worldcat.org/oclc/317496332
 |title=General Relativity and the Einstein Equations
 |date=2009
 |publisher=Oxford University Press
 |isbn=978-0-19-155226-7 |location=Oxford |oclc=317496332
 |pages=37–39
}}</ref> [[Mercury (planet)|Mercury]]'s orbit did not match that predicted by Newton's law of gravitation. Some astrophysicists predicted the existence of an undiscovered planet ([[Vulcan (hypothetical planet)|Vulcan]]) that could explain the discrepancies. When Einstein formulated his theory of [[general relativity]] (GR) he focused on Mercury's problematic orbit and found that his theory added [[Perihelion precession of Mercury|a correction, which could account for the discrepancy]]. This was the first time that Newton's theory of gravity had been shown to be inexact.<ref>{{cite news |last1=Siegel |first1=Ethan |title=When Did Isaac Newton Finally Fail? |url=https://www.forbes.com/sites/startswithabang/2016/05/20/when-did-isaac-newton-finally-fail/#6fdc279675f5 |access-date=3 January 2017 |work=Forbes |date=20 May 2016}}</ref>

Since then, general relativity has been acknowledged as the theory that best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in [[geodesic|straight lines]] through [[curved spacetime]] – defined as the shortest spacetime path between two spacetime events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of spacetime can be observed and the force is inferred from the object's curved path. Thus, the straight line path in spacetime is seen as a curved line in space, and it is called the ''[[external ballistics|ballistic]] [[trajectory]]'' of the object. For example, a [[Basketball (ball)|basketball]] thrown from the ground moves in a [[parabola]], as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with the [[radius of curvature (applications)|radius of curvature]] of the order of few [[light-year]]s). The time derivative of the changing momentum of the object is what we label as "gravitational force".<ref name=Kleppner />

=== Electromagnetic ===

[[Maxwell's equations]] and the set of techniques built around them adequately describe a wide range of physics involving force in electricity and magnetism. This classical theory already includes relativity effects.<ref>{{Cite book |last1=Panofsky |first1=Wolfgang K. |title=Classical electricity and magnetism |last2=Phillips |first2=Melba |date=2005 |publisher=Dover Publ |isbn=978-0-486-43924-2 |edition=2 |location=Mineola, NY}}</ref>  Understanding quantized electromagnetic interactions between elementary particles requires [[quantum electrodynamics]] (or QED). In QED, photons are fundamental exchange particles, describing all interactions relating to electromagnetism including the electromagnetic force.<ref>{{cite book|first=Anthony |last=Zee |author-link=Anthony Zee |title=Quantum Field Theory in a Nutshell |title-link=Quantum Field Theory in a Nutshell |edition=2nd |isbn=978-0-691-14034-6 |year=2010 |page=29 |publisher=Princeton University Press}}</ref>

=== Strong nuclear ===
{{main|Strong interaction}}
There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. The [[strong nuclear force]] is the force responsible for the structural integrity of [[atomic nuclei]], and gains its name from its ability to overpower the electromagnetic repulsion between protons.<ref name=Cutnell/>{{rp|940}}<ref>{{cite OED|strong, 7.g ''physics'' |1058721983}}</ref>

The strong force is today understood to represent the [[Fundamental interaction|interaction]]s between [[quark]]s and [[gluon]]s as detailed by the theory of [[quantum chromodynamics]] (QCD).<ref>{{cite web |last=Stevens |first=Tab |title=Quantum-Chromodynamics: A Definition – Science Articles |date=10 July 2003 |url=http://www.physicspost.com/science-article-168.html |archive-url=https://web.archive.org/web/20111016103116/http://www.physicspost.com/science-article-168.html |archive-date=2011-10-16 |access-date=2008-01-04}}</ref> The strong force is the [[fundamental force]] mediated by gluons, acting upon quarks, [[antiparticle|antiquarks]], and the gluons themselves. The strong force only acts ''directly'' upon elementary particles. A residual is observed between [[hadron]]s (notably, the [[nucleon]]s in atomic nuclei), known as the [[nuclear force]]. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rho [[meson]]s, the classical transmitters of the nuclear force. The failure of many searches for [[free quark]]s has shown that the elementary particles affected are not directly observable. This phenomenon is called [[color confinement]].<ref>{{Cite book |last=Goldberg |first=Dave |title=The Standard Model in a Nutshell |date=2017 |publisher=Princeton University Press |isbn=978-0-691-16759-6}}</ref>{{Rp|page=232}}

=== Weak nuclear ===
{{Main|Weak interaction}}
Unique among the fundamental interactions, the weak nuclear force creates no bound states.<ref name=GreinerMuller>{{Cite book |last1=Greiner |first1=Walter |title=Gauge theory of weak interactions: with 75 worked examples and exercises |last2=Müller |first2=Berndt |last3=Greiner |first3=Walter |date=2009 |publisher=Springer |isbn=978-3-540-87842-1 |edition=4|location=Heidelberg}}</ref> The weak force is due to the exchange of the heavy [[W and Z bosons]]. Since the weak force is mediated by two types of bosons, it can be divided into two types of interaction or "[[Feynman diagram|vertices]]" — [[charged current]], involving the electrically charged W<sup>+</sup> and W<sup>−</sup> bosons, and [[neutral current]], involving electrically neutral Z<sup>0</sup> bosons. The most familiar effect of weak interaction is [[beta decay]] (of neutrons in atomic nuclei) and the associated [[radioactivity]].<ref name=Cutnell/>{{rp|951}} This is a type of charged-current interaction. The word "weak" derives from the fact that the field strength is some 10<sup>13</sup> times less than that of the [[strong force]]. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed, which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately {{val|e=15|ul=K}}.<ref>{{Cite book |last=Durrer |first=Ruth |title=The Cosmic Microwave Background |date=2008 |publisher=Cambridge Pniversity Press |isbn=978-0-521-84704-9 |pages=41–42 |author-link=Ruth Durrer}}</ref> Such temperatures occurred in the plasma collisions in the early moments of the [[Big Bang]].<ref name=GreinerMuller/>{{rp|201}}

== See also ==
{{portal|Physics}}
* {{annotated link|Contact force}}
* {{annotated link|Force control}}
* {{annotated link|Force gauge}}
* {{annotated link|Orders of magnitude (force)}}
* {{annotated link|Parallel force system}}
* {{annotated link|Rigid body}}
* {{annotated link|Specific force}}

== References ==
{{Reflist|30em|refs=
<ref name=uniphysics_ch4>Young, Hugh; Freedman, Roger; Sears, Francis; and Zemansky, Mark (1949) ''[[University Physics]]''. Pearson Education. pp. 59–82.
</ref>
}}

== External links ==
{{Wiktionary}}
{{Commons category|Forces (physics)}}
* {{cite web|title=Classical Mechanics, Week 2: Newton's Laws |website=[[MIT OpenCourseWare]] |access-date=2023-08-09 |url=https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/pages/week-2-newtons-laws/}}
* {{cite web|title=Fundamentals of Physics I, Lecture 3: Newton's Laws of Motion |website=[[Open Yale Courses]] |access-date=2023-08-09 |url=https://oyc.yale.edu/physics/phys-200/lecture-3}}

{{Fundamental interactions}}
{{Classical mechanics SI units}}
{{Authority control}}
{{good article}}

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