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In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

physics
, a force is an influence that can change the
motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its position (mathematics), position over time. Motion is mathematically described in terms of Displacem ...
of an
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at any particular time or pl ...
. A force can cause an object with
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
to change its
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
(e.g. moving from a state of rest), i.e., to
accelerate In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement ( ...

accelerate
. Force can also be described intuitively as a push or a pull. A force has both
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...

magnitude
and
direction Direction may refer to: *Relative direction, for instance left, right, forward, backwards, up, and down ** Anatomical terms of location for those used in anatomy *Cardinal direction Mathematics and science *Direction vector, a unit vector that ...
, making it a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
quantity. It is measured in the
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system The metric system is a that succeeded the decimal ...
of newton (N). Force is represented by the symbol (formerly ). The original form of
Newton's second law In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which a ...
states that the net force acting upon an object is equal to the rate at which its
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

momentum
changes with
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...

time
. If the mass of the object is constant, this law implies that the
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

acceleration
of an object;- is directly
proportional Proportionality, proportion or proportional may refer to: Mathematics * Proportionality (mathematics), the property of two variables being in a multiplicative relation to a constant * Ratio, of one quantity to another, especially of a part compared ...
to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
of the object. Concepts related to force include:
thrust Thrust is a described quantitatively by . When a system expels or in one direction, the accelerated mass will cause a force of equal but opposite direction to be applied to that system. The force applied on a surface in a direction perpen ...

thrust
, which increases the velocity of an object;
drag Drag or The Drag may refer to: Places * Drag, Norway, a village in Tysfjord municipality, Nordland, Norway * ''Drág'', the Hungarian name for Dragu Commune in Sălaj County, Romania * Drag (Austin, Texas), the portion of Guadalupe Street adja ...
, which decreases the velocity of an object; and
torque In physics and mechanics, torque is the rotational equivalent of linear force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

torque
, which produces changes in rotational speed of an object. In an extended body, each part usually applies forces on the adjacent parts; the distribution of such forces through the body is the internal
mechanical stress In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighbouring particles of a continuous material exert on each other, while deformation (mechanics)#Strain, strain is the measure of the deformation of ...
. Such internal mechanical stresses cause no acceleration of that body as the forces balance one another.
Pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

Pressure
, the distribution of many small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (mechanics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
of solid materials, or flow in
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
s.


Development of the concept

Philosophers in
antiquity Antiquity or Antiquities may refer to Historical objects or periods Artifacts * Antiquities, objects or artifacts surviving from ancient cultures Eras Any period before the European Middle Ages In the history of Europe, the Middle Ages ...
used the concept of force in the study of
stationary In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ...

stationary
and moving objects and
simple machine A simple machine is a mechanical device that changes the direction or magnitude of a force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature' ...
s, but thinkers such as
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

Aristotle
and
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

Archimedes
retained fundamental errors in understanding force. In part this was due to an incomplete understanding of the sometimes non-obvious force of
friction Friction is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related en ...

friction
, and a consequently inadequate view of the nature of natural motion. 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 Galileo di Vincenzo Bonaiuti de' Galilei (; 15 February 1564 – 8 January 1642) was an Italian astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the ...

Galileo Galilei
and
Sir Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of s ...

Sir Isaac Newton
. With his mathematical insight,
Sir Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of s ...

Sir Isaac Newton
formulated
laws of motionIn physics, a number of noted theories of the motion of objects have developed. Among the best known are: * Classical mechanics ** Newton's laws of motion ** Euler's laws of motion ** Cauchy momentum equation, Cauchy's equations of motion ** Kepler's ...
that were not improved for nearly three hundred years. By the early 20th century,
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theory ...

Einstein
developed a
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born , widely acknowledged to be one of the greatest physicists of all time ...
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 Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an ob ...

inertia
. With modern insights into
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
and technology that can accelerate particles close to the speed of light,
particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particles, or even household d ...
has devised a
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

Standard Model
to describe forces between particles smaller than atoms. The
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

Standard Model
predicts that exchanged particles called
gauge boson In particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particle ...
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,
electromagnetic Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
, , and
gravitational Gravity (), or gravitation, is a natural phenomenon by which all things with mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or with something, whether as an ...
. High-energy particle physics
observation Observation is the active acquisition of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of its characteristics. Th ...

observation
s made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental
electroweak In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the ...

electroweak
interaction.


Pre-Newtonian concepts

Since antiquity the concept of force has been recognized as integral to the functioning of each of the
simple machine A simple machine is a mechanical device that changes the direction or magnitude of a force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature' ...
s. The
mechanical advantage Mechanical advantage is a measure of the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and ti ...
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 Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking * Work (physics), the product of ...

work
. Analysis of the characteristics of forces ultimately culminated in the work of
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its popula ...

Archimedes
who was especially famous for formulating a treatment of
buoyant force . Buoyancy (), or upthrust, is an upward force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies mat ...
s inherent in
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
s.
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental quest ...

Aristotle
provided a
philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality Reality is the sum or aggregate of all that is real o ...

philosophical
discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four 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, to be in their natural place on the ground and that they will 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. 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 A projectile is a missile propelled by the exertion of a which is allowed to move free under the influence of and . Although any objects in motion through space are projectiles, they are commonly found in and s (for example, a thrown , kicked ...

projectile
s, such as the flight of arrows. The place where the archer moves the projectile was at the start of the flight, and while the projectile sailed through the air, no discernible efficient cause acts on 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 demands a continuum like air for change of place in general.
Aristotelian physics Aristotelian physics is the form of natural science Natural science is a branch of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general) ...
began facing criticism in medieval science, first by
John Philoponus John Philoponus (; ; c. 490 – c. 570), also known as John the Grammarian or John of Alexandria, was a Byzantine Alexandrian philologist, commentaries on Aristotle, Aristotelian commentator and Christian theologian, author of a considerable number ...
in the 6th century. The shortcomings of Aristotelian physics would not be fully corrected until the 17th century work of
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (; 15 February 1564 – 8 January 1642) was an Italian astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the ...

Galileo Galilei
, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the 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 The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
unless acted on by a force, for example
friction Friction is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related en ...

friction
.Drake, Stillman (1978). Galileo At Work. Chicago: University of Chicago Press. In the early 17th century, before Newton's Principia, the term "force" ( 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 (live force) by
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist, and diplomat. He is a promin ...

Leibniz
. The modern concept of force corresponds to Newton's (accelerating force).


Newtonian mechanics

Sir Isaac Newton described the motion of all objects using the concepts of
inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an ob ...

inertia
and force, and in doing so he found they obey certain
conservation laws In physics, a conservation law states that a particular measurable property of an isolated physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...
. In 1687, Newton published his thesis ''
Philosophiæ Naturalis Principia Mathematica (from Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it bec ...
''. This is a recent translation into English by I. Bernard Cohen and Anne Whitman, with help from Julia Budenz. In this work Newton set out three laws of motion that to this day are the way forces are described in physics.


First law

Newton's first law of motion states that objects continue to move in a state of constant velocity unless acted upon by an external
net force In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximate ...

net force
(resultant force). This law is an extension of Galileo's insight that constant velocity was associated with a lack of net force (see a more detailed description of this below). Newton proposed that every object with mass has an innate
inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an ob ...

inertia
that functions as the fundamental equilibrium "natural state" in place of the Aristotelian idea of the "natural state of rest". That is, Newton's empirical first law contradicts the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity. By making ''rest'' physically indistinguishable from ''non-zero constant velocity'', Newton's first law directly connects inertia with the concept of relative velocities. Specifically, in systems where objects are moving with different velocities, it is impossible to determine which object is "in motion" and which object is "at rest". The laws of physics are the same in every
inertial frame of reference In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a const ...
, that is, in all frames related by a
Galilean transformation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
. For instance, while traveling in a moving vehicle at a constant
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
, the laws of physics do not change as a result of its motion. If a person riding within the vehicle throws a ball straight up, that person will observe it rise vertically and fall vertically and not have to apply a force in the direction the vehicle is moving. Another person, observing the moving vehicle pass by, would observe the ball follow a curving in the same direction as the motion of the vehicle. It is the inertia of the ball associated with its constant velocity in the direction of the vehicle's motion that ensures the ball continues to move forward even as it is thrown up and falls back down. From the perspective of the person in the car, the vehicle and everything inside of it is at rest: It is the outside world that is moving with a constant speed in the opposite direction of the vehicle. Since there is no experiment that can distinguish whether it is the vehicle that is at rest or the outside world that is at rest, the two situations are considered to be physically indistinguishable. Inertia therefore applies equally well to constant velocity motion as it does to rest.


Second law

A modern statement of Newton's second law is a vector equation:Newton's ''Principia Mathematica'' actually used a finite difference version of this equation based upon ''impulse''. See ''''. :\vec = \frac, where \vec is the
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

momentum
of the system, and \vec is the net (
vector sum In mathematics, physics and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has Magnitude (mathematics), magnitude (or euclidean norm, length) and Direction (g ...
) force. 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. By the definition of
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...
, :\vec = \frac = \frac, where ''m'' is the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
and \vec is the
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
. If Newton's second law is applied to a system of constant mass,"It is important to note that we ''cannot'' derive a general expression for Newton's second law for variable mass systems by treating the mass in F = ''d''P/''dt'' = ''d''(''M''v) as a ''variable''. ..We ''can'' use F = ''d''P/''dt'' to analyze variable mass systems ''only'' if we apply it to an entire system of constant mass having parts among which there is an interchange of mass." mphasis as in the original ''m'' may be moved outside the derivative operator. The equation then becomes :\vec = m\frac. By substituting the definition of
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

acceleration
, the algebraic version of
Newton's second law In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which a ...
is derived: :\vec =m\vec. Newton never explicitly stated the formula in the reduced form above. Newton's second law asserts the direct proportionality of acceleration to force and the inverse proportionality of acceleration to mass. Accelerations can be defined through
kinematic Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which an object changes its ...
measurements. However, while kinematics are well-described through
reference frame In physics, a frame of reference (or reference frame) consists of an abstract coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, wi ...

reference frame
analysis in advanced physics, there are still deep questions that remain as to what is the proper definition of mass.
General relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
offers an equivalence between
space-time In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
and mass, but lacking a coherent theory of
quantum gravity Quantum gravity (QG) is a field of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, ...

quantum gravity
, it is unclear as to how or whether this connection is relevant on microscales. With some justification, Newton's second law can be taken as a quantitative definition of ''mass'' by writing the law as an equality; the relative units of force and mass then are fixed. Some textbooks use Newton's second law as a ''definition'' of force, but this has been disparaged in other textbooks. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include
Ernst Mach Ernst Waldfried Josef Wenzel Mach (; ; 18 February 1838 – 19 February 1916) was an Austrian physicist and philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from the grc, φιλόσοφος, , t ...

Ernst Mach
and
Walter Noll Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University Carnegie Mellon University (CMU) is a private Private or privates may refer to: Music * "In Private "In Private" was t ...
. Newton's second law can be used to measure the strength of forces. For instance, knowledge of the masses of
planet A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and ...

planet
s along with the accelerations of their
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...

orbit
s allows scientists to calculate the gravitational forces on planets.


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 \vec_ is the force of body 1 on body 2 and \vec_ that of body 2 on body 1, then :\vec_=-\vec_. This law is sometimes referred to as the ''action-reaction law'', with \vec_ called the ''action'' and -\vec_ the ''
reaction Reaction may refer to a process or to a response to an action, event, or exposure: Physics and chemistry *Chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformat ...
''. Newton's Third Law is a result of applying
symmetry Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...

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,"Any single force is only one aspect of a mutual interaction between ''two'' bodies." 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: :\vec_+\vec_=0. More generally, in a
closed system A closed system is a physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment (systems), environm ...
of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the
center of mass In physics, the center of mass of a distribution of mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of " ...
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. Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved. In a system of two particles, if \vec_1 is the momentum of object 1 and \vec_ the momentum of object 2, then :\frac + \frac= \vec_ + \vec_ = 0. 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.


Special theory of relativity

In the special theory of relativity, mass and
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

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 :\vec = \frac remains valid because it is a mathematical definition. But for relativistic momentum to be conserved, it must be redefined as: : \vec = \frac, where m_0 is the
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
and c the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
. The relativistic expression relating force and acceleration for a particle with constant non-zero
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
m moving in the x direction is: :\vec = \left(\gamma^3 m a_x, \gamma m a_y, \gamma m a_z\right), where : \gamma = \frac. is called the
Lorentz factor The Lorentz factor or Lorentz term is a quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assi ...

Lorentz factor
. In the early history of relativity, the expressions \gamma^3 m and \gamma m were called longitudinal and transverse mass. Relativistic force does not produce a constant acceleration, but an ever-decreasing acceleration as the object approaches the speed of light. Note that \gamma approaches asymptotically an infinite value and is undefined for an object with a non-zero
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
as it approaches the speed of light, and the theory yields no prediction at that speed. If v is very small compared to c, then \gamma is very close to 1 and :F = m a is a close approximation. Even for use in relativity, however, one can restore the form of :F^\mu = mA^\mu \, through the use of
four-vectors In special relativity, a four-vector (also known as a 4-vector) is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space consid ...
. This relation is correct in relativity when F^\mu is the
four-forceIn the special theory of relativity, four-force is a four-vector that replaces the classical force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'natu ...
, m is the
invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
, and A^\mu is the
four-accelerationIn the theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity. Special relativity applies to all physical phenomena in the absence of gravity. Gen ...
.


Descriptions

Since forces are perceived as pushes or pulls, this can provide an intuitive understanding for describing forces. As with other physical concepts (e.g.
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

temperature
), the intuitive understanding of forces is quantified using precise
operational definition An operational definition specifies concrete, replicable procedures designed to represent a construct. In the words of American psychologist S.S. Stevens (1935), "An operation is the performance which we execute in order to make known a concept." ...
s that are consistent with direct
observations Observation is the active acquisition of information Information can be thought of as the resolution of uncertainty; it answers the question of "What an entity is" and thus defines both its essence and the nature of its characteristics. Th ...
and . Through experimentation, it is determined that laboratory measurements of forces are fully consistent with the conceptual definition of force offered by
Newtonian mechanics Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
. Forces act in a particular
direction Direction may refer to: *Relative direction, for instance left, right, forward, backwards, up, and down ** Anatomical terms of location for those used in anatomy *Cardinal direction Mathematics and science *Direction vector, a unit vector that ...
and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as " vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
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
result A result (also called upshot) is the final consequence of a sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes a ...
. If both of these pieces of information are not known for each force, the situation is ambiguous. For example, if you know that two people are pulling on the same rope with known magnitudes of force but you do not know which direction either person is pulling, it is impossible to determine what the acceleration of the rope will be. The two people could be pulling against each other as in
tug of war Tug of war (also known as tug o' war, tug war, rope war, rope pulling, or tugging war) is a sport that pits two teams against each other in a test of strength: teams pull on opposite ends of a rope A rope is a group of yarn Yarn is a lo ...

tug of war
or the two people could be pulling in the same direction. In this simple
one-dimensional In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
example, without knowing the direction of the forces it is impossible to decide whether the net force is the result of adding the two force magnitudes or subtracting one from the other. Associating forces with vectors avoids such problems. Historically, forces were first quantitatively investigated in conditions of
static equilibrium In classical mechanics, a particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, phys ...
where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
and direction. When two forces act on a
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, l ...
, the resulting force, the ''resultant'' (also called the ''
net force In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximate ...

net force
''), can be determined by following the
parallelogram rule A parallelogram. The sides are shown in blue and the diagonals in red. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
of
vector addition In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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. However, if the forces are acting on an extended body, their respective lines of application must also be specified in order to account for their effects on the motion of the body. Free-body diagrams 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 graphical vector addition can be done to determine the net force. As well as being added, forces can also be resolved into independent components at
right angle In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for
orthogonal In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bili ...
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.


Equilibrium

When all the forces that act upon an object are balanced, then the object is said to be in a state of
equilibrium List of types of equilibrium, the condition of a system in which all competing influences are balanced, in a wide variety of contexts. Equilibrium may also refer to: Film and television * Equilibrium (film), ''Equilibrium'' (film), a 2002 scien ...
. 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. There are two kinds of equilibrium:
static equilibrium In classical mechanics, a particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, phys ...
and
dynamic equilibrium In chemistry, and in physics, a dynamic equilibrium exists once a reversible reaction occurs. Substances transition between the reactants and products at equal rates, meaning there is no net change. Reactants and products are formed at such a rate ...
.


Static

Static equilibrium was understood well before the invention of classical mechanics. Objects that are at rest have zero net force acting on them. 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 In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (ve ...

normal force
). The situation produces zero net force and hence no acceleration. 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 Friction is the resisting the relative motion of solid surfaces, fluid layers, and material elements against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of two solid surfa ...

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. A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as
weighing scale A scale or balance is a device to measure weight or mass. These are also known as mass scales, weight scales, mass balances, and weight balances. The traditional scale consists of two plates or bowls suspended at equal distances from a Lever ...

weighing scale
s and
spring balance A spring scale or spring balance or newton meter is a type of mechanical force gauge or weighing scale A scale or balance is a device to measure weight or mass. These are also known as mass scales, weight scales, mass balances, weight balanc ...
s. For example, an object suspended on a vertical
spring scale A spring scale or spring balance or newton meter is a type of mechanical force gauge A force gauge (also called a force meter) is a measuring instrument used to measure forces. Applications exist in research and development, laboratory, quality, ...
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 The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

density
(widely exploited for millennia to define standard weights);
Archimedes' principle Archimedes' principle states that the upward buoyant force that is exerted on a body immersed in a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external f ...
for buoyancy; Archimedes' analysis of the
lever A lever ( or ) is a simple machine A simple machine is a mechanical device that changes the direction or magnitude of a force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge o ...

lever
; Boyle's law for gas pressure; and
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
for springs. These were all formulated and experimentally verified before Isaac Newton expounded his Three Laws of Motion.


Dynamic

Dynamic equilibrium was first described by
Galileo Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a in the field of who focuses their studies on a specific question or field o ...

Galileo
who noticed that certain assumptions of Aristotelian physics were contradicted by observations and
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

logic
. Galileo realized that simple velocity addition demands that the concept of an "absolute
rest frameIn special relativity the rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest. The rest frame of compound objects (such as a fluid, or a solid made of many vibrating atoms) is taken to be the frame ...
" did not exist. Galileo concluded that motion in a constant
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

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. However, 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. 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 Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative late ...
. 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. However, when kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.


Forces in quantum mechanics

The notion "force" keeps its meaning in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, though one is now dealing with operators instead of classical variables and though the physics is now described by the
Schrödinger equation The Schrödinger equation is a linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a ma ...
instead of Newtonian equations. This has the consequence that the results of a measurement are now sometimes "quantized", i.e. they appear in discrete portions. This is, of course, difficult to imagine in the context of "forces". However, the potentials ''V''(''x'',''y'',''z'') or
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
, from which the forces generally can be derived, are treated similarly to classical position variables, i.e., V(x,y,z)\to (\hat x,\hat y,\hat z). This becomes different only in the framework of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, where these fields are also quantized. However, already in quantum mechanics there is one "caveat", namely the particles acting onto each other do not only possess the spatial variable, but also a discrete intrinsic variable called the "
spin Spin or spinning may refer to: Businesses * or South Pacific Island Network * , an American scooter-sharing system * , a chain of table tennis lounges Computing * , 's tool for formal verification of distributed software systems * , a Mach-like ...
", and there is the
Pauli exclusion principle The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermion In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc ...
relating the space and the spin variables. Depending on the value of the spin, identical particles split into two different classes,
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
s and
boson In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinkin ...

boson
s. If two identical fermions (e.g. electrons) have a ''symmetric'' spin function (e.g. parallel spins) the spatial variables must be ''antisymmetric'' (i.e. they exclude each other from their places much as if there was a repulsive force), and vice versa, i.e. for antiparallel ''spins'' the ''position variables'' must be symmetric (i.e. the apparent force must be attractive). Thus in the case of two fermions there is a strictly negative correlation between spatial and spin variables, whereas for two bosons (e.g. quanta of electromagnetic waves, photons) the correlation is strictly positive. Thus the notion "force" loses already part of its meaning.


Feynman diagrams

In modern
particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particles, or even household d ...
, forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying
gauge boson In particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particle ...
s. With the development of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
and
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, it was realized that force is a redundant concept arising from
conservation of momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a direction. If is an object's ma ...
( 4-momentum in relativity and momentum of
virtual particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...
s in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativity theory, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum m ...
). The conservation of momentum can be directly derived from the homogeneity or
symmetry Symmetry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
of
space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameter A parameter (from the Ancient Greek language, Ancient Gre ...

space
and so is usually considered more fundamental than the concept of a force. Thus the currently known
fundamental forces In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
are considered more accurately to be "
fundamental interactions #REDIRECT Fundamental interaction In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physic ...
". When particle A emits (creates) or absorbs (annihilates) virtual particle B, a momentum conservation results in recoil of particle A making impression of repulsion or attraction between particles A A' exchanging by B. This description applies to all forces arising from fundamental interactions. While sophisticated mathematical descriptions are needed to predict, in full detail, the accurate result of such interactions, there is a conceptually simple way to describe such interactions through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see
world line The world line (or worldline) of an object is the path (topology), path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is disti ...

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 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. The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of
fundamental interaction #REDIRECT Fundamental interaction In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physic ...
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 The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

neutron
into an
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

electron
,
proton A proton is a subatomic particle, symbol or , with a positive electric charge of +1''e'' elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approximately one atomic mass unit, are collecti ...

proton
, and
neutrino A neutrino ( or ) (denoted by the Greek letter ) is a fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. T ...

neutrino
, an interaction mediated by the same gauge boson that is responsible for the
weak nuclear force In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which ...
.


Fundamental forces

All of the known forces of the universe are classified into four
fundamental interaction #REDIRECT Fundamental interaction In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physic ...
s. The strong and the forces act only at very short distances, and are responsible for the interactions between
subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All ...
s, including
nucleon In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during ...
s and compound
nuclei ''Nucleus'' (plural nuclei) is a Latin word for the seed inside a fruit. It most often refers to: *Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA ...
. The
electromagnetic force Electromagnetism is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ...
acts between
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
s, and the
gravitational force Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is t ...
acts between
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
es. All other forces in nature derive from these four fundamental interactions. For example,
friction Friction is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related en ...

friction
is a manifestation of the electromagnetic force acting between
atoms An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

atoms
of two surfaces, and the
Pauli exclusion principle The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermion In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc ...
, which does not permit atoms to pass through each other. Similarly, the forces in
springs Spring(s) may refer to: Common uses * Spring (season), a season of the year * Spring (device), a mechanical device that stores energy * Spring (hydrology), a natural source of water * Spring (mathematics), a geometric surface in the shape of a heli ...
, modeled by
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
, are the result of electromagnetic forces and the Pauli exclusion principle acting together to return an object to its
equilibrium List of types of equilibrium, the condition of a system in which all competing influences are balanced, in a wide variety of contexts. Equilibrium may also refer to: Film and television * Equilibrium (film), ''Equilibrium'' (film), a 2002 scien ...
position. Centrifugal forces are
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

acceleration
forces that arise simply from the acceleration of
rotating A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

rotating
frames of reference In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
. The fundamental theories for forces developed from the unification of different ideas. For example, Sir
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

Isaac Newton
unified, with his universal theory of
gravitation Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is t ...

gravitation
, the force responsible for objects falling near the surface of the
Earth Earth is the third planet from the Sun and the only astronomical object known to harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wi ...

Earth
with the force responsible for the falling of celestial bodies about the Earth (the
Moon The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

Moon
) and around the Sun (the planets).
Michael Faraday Michael Faraday (; 22 September 1791 – 25 August 1867) was an English scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge ...

Michael Faraday
and
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as num ...

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 Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
s) interacting by exchanging
virtual particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...
s called
gauge boson In particle physics Particle physics (also known as high energy physics) is a branch of that studies the nature of the particles that constitute and . Although the word ' can refer to various types of very small objects (e.g. , gas particle ...
s. This
Standard Model The Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsi ...

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 In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the ...

electroweak
theory, which was subsequently confirmed by observation. The complete formulation of the Standard Model predicts an as yet unobserved
Higgs mechanism In the Standard Model of particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ...
, but observations such as
neutrino oscillation Neutrino oscillation is a quantum mechanics, quantum mechanical phenomenon in which a neutrino created with a specific lepton lepton number, family number ("lepton flavor": electron, muon, or tau lepton, tau) can later be Quantum measurement, meas ...
s suggest that the Standard Model is incomplete. A
Grand Unified Theory A Grand Unified Theory (GUT) is a model in particle physics in which, at high energies, the three gauge interactions of the Standard Model comprising the electromagnetism, electromagnetic, weak interaction, weak, and strong interaction, strong ...
that allows for the combination of the electroweak interaction with the strong force is held out as a possibility with candidate theories such as
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical physics, theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of super ...
proposed to accommodate some of the outstanding
unsolved problems in physics The following is a list of notable unsolved problems grouped into broad areas of physics. Some of the major unsolved problems in physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature ...
. Physicists are still attempting to develop self-consistent unification models that would combine all four fundamental interactions into a
theory of everything A theory of everything (TOE or ToE), final theory, ultimate theory, or master theory is a hypothetical single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the univers ...
. Einstein tried and failed at this endeavor, but currently the most popular approach to answering this question is
string theory In physics, string theory is a Mathematical theory, theoretical framework in which the Point particle, point-like particles of particle physics are replaced by Dimension (mathematics and physics), one-dimensional objects called String (physic ...

string theory
.


Gravitational

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 In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

acceleration
of every object in
free-fall In Newtonian physics, free fall is any motion of a body where gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, stars, galaxy, galaxies, ...

free-fall
was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as \vec and has a magnitude of about 9.81
meter The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...

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. 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 m will experience a force: :\vec = m\vec 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 In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (ve ...

normal force
(a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward. 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
law of gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—including planets, stars, galaxy, galaxies, and even light—are brought toward (or ''gravitate'' toward) one another. ...
that could account for the celestial motions that had been described earlier using
Kepler's laws of planetary motion In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the Copernican heliocentrism, heliocentric theory of Nicolaus Copernicus, repl ...
. 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 420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light source and is constant with in ...

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. Combining these ideas gives a formula that relates the mass ( m_\oplus) and the radius ( R_\oplus) of the Earth to the gravitational acceleration: :\vec=-\frac \hat where the vector direction is given by \hat, is the
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
directed outward from the center of the Earth. In this equation, a dimensional constant G is used to describe the relative strength of gravity. This constant has come to be known as Newton's Universal Gravitation Constant, though its value was unknown in Newton's lifetime. Not until 1798 was
Henry Cavendish Henry Cavendish FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resourc ...
able to make the first measurement of G using a
torsion balance :'' Torsion coefficient links here.'' A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy In physical sciences, mechanical energy is the sum of potent ...

torsion balance
; this was widely reported in the press as a measurement of the mass of the Earth since knowing G could allow one to solve for the Earth's mass given the above equation. Newton, however, realized that since all celestial bodies followed the same
laws of motionIn physics, a number of noted theories of the motion of objects have developed. Among the best known are: * Classical mechanics ** Newton's laws of motion ** Euler's laws of motion ** Cauchy momentum equation, Cauchy's equations of motion ** Kepler's ...
, his law of gravity had to be universal. Succinctly stated,
Newton's Law of Gravitation Newton's law of universal gravitation is usually stated as that every particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can ...
states that the force on a spherical object of mass m_1 due to the gravitational pull of mass m_2 is :\vec=-\frac \hat where r is the distance between the two objects' centers of mass and \hat is the unit vector pointed in the direction away from the center of the first object toward the center of the second object. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
were invented to calculate the deviations of
orbit In celestial mechanics, an orbit is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...

orbit
s due to the influence of multiple bodies on a
planet A planet is an astronomical body orbiting a star or Stellar evolution#Stellar remnants, stellar remnant that is massive enough to be Hydrostatic equilibrium, rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and ...

planet
,
moon The Moon is Earth's only natural satellite. At about one-quarter the diameter of Earth (comparable to the width of Australia (continent), Australia), it is the largest natural satellite in the Solar System relative to the size of its plane ...

moon
,
comet A comet is an icy, small Solar System body A small Solar System body (SSSB) is an object in the Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astr ...

comet
, or
asteroid An asteroid is a minor planet of the Solar System#Inner solar system, inner Solar System. Historically, these terms have been applied to any astronomical object orbiting the Sun that did not resolve into a disc in a telescope and was not observ ...

asteroid
. The formalism was exact enough to allow mathematicians to predict the existence of the planet
Neptune Neptune is the eighth and farthest-known Solar planet from the Sun. In the Solar System, it is the fourth-largest planet by diameter, the third-most-massive planet, and the densest giant planet. It is 17 times the mass of Earth, slightly mo ...

Neptune
before it was observed.
Mercury Mercury usually refers to: * Mercury (planet) Mercury is the smallest planet in the Solar System and the closest to the Sun. Its orbit around the Sun takes 87.97 Earth days, the shortest of all the Sun's planets. It is named after the Roman g ...

Mercury
's orbit, however, did not match that predicted by Newton's Law of Gravitation. Some astrophysicists predicted the existence of another planet (
Vulcan Vulcan may refer to: Mythology * Vulcan (mythology), the god of fire, volcanoes, metalworking, and the forge in Roman mythology Arts, entertainment and media Film and television * Vulcan (Star Trek), Vulcan (''Star Trek''), name of a fictional rac ...
) that would explain the discrepancies; however no such planet could be found. When
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ...

Albert Einstein
formulated his theory of
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
(GR) he turned his attention to the problem of Mercury's orbit and found that his theory added 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. 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 through
curved space-time General relativity, also known as the general theory of relativity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern ph ...
– defined as the shortest space-time path between two space-time 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 space-time can be observed and the force is inferred from the object's curved path. Thus, the straight line path in space-time 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 space-time trajectory is almost a straight line, slightly curved (with the radius of curvature (applications), radius of curvature of the order of few light-years). The time derivative of the changing momentum of the object is what we label as "gravitational force".


Electromagnetic

The electrostatic force was first described in 1784 by Coulomb as a force that existed intrinsically between two electric charge, charges. The properties of the electrostatic force were that it varied as an
inverse square law 420px, S represents the light source, while r represents the measured points. The lines represent the flux emanating from the sources and fluxes. The total number of flux lines depends on the strength of the light source and is constant with in ...

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. 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. Thus the electric field anywhere in space is defined as :\vec = where q is the magnitude of the hypothetical test charge. Meanwhile, the Lorentz force of magnetism was discovered to exist between two electric currents. It has the same mathematical character as Coulomb's Law with the proviso that like currents attract and unlike currents repel. Similar to the electric field, the magnetic field can be used to determine the magnetic force on an electric current at any point in space. In this case, the magnitude of the magnetic field was determined to be :B = where I is the magnitude of the hypothetical test current and \ell is the length of hypothetical wire through which the test current flows. The magnetic field exerts a force on all magnets including, for example, those used in compasses. The fact that the geomagnetism, Earth's magnetic field is aligned closely with the orientation of the Earth's rotation, axis causes compass magnets to become oriented because of the magnetic force pulling on the needle. Through combining the definition of electric current as the time rate of change of electric charge, a rule of Cross product, vector multiplication called Lorentz force, Lorentz's Law describes the force on a charge moving in a magnetic field. The connection between electricity and magnetism allows for the description of a unified ''electromagnetic force'' that acts on a charge. This force can be written as a sum of the electrostatic force (due to the electric field) and the magnetic force (due to the magnetic field). Fully stated, this is the law: :\vec = q(\vec + \vec \times \vec) where \vec is the electromagnetic force, q is the magnitude of the charge of the particle, \vec is the electric field, \vec is the
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
of the particle that is cross product, crossed with the magnetic field ( \vec). The origin of electric and magnetic fields would not be fully explained until 1864 when
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as num ...

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. These "Maxwell 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 The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum. However, attempting to reconcile electromagnetic theory with two observations, the photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome. Through the work of leading theoretical physicists, a new theory of electromagnetism was developed using quantum mechanics. This final modification to electromagnetic theory ultimately led to
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativity theory, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum m ...
(or QED), which fully describes all electromagnetic phenomena as being mediated by wave–particles known as photons. In QED, photons are the fundamental exchange particle, which described all interactions relating to electromagnetism including the electromagnetic force.For a complete library on quantum mechanics see Quantum mechanics#References, Quantum mechanics – References


Strong nuclear

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 while the
weak nuclear force In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which ...
is responsible for the decay of certain
nucleon In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during ...
s into leptons and other types of hadrons. The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). The strong force is the fundamental force mediated by gluons, acting upon quarks, antiparticle, antiquarks, and the gluons themselves. The (aptly named) strong interaction is the "strongest" of the four fundamental forces. The strong force only acts ''directly'' upon elementary particles. However, a residual of the force is observed between hadrons (the best known example being the force that acts between
nucleon In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during ...
s in atomic nuclei) as the nuclear force. Here the strong force acts indirectly, transmitted as gluons, which form part of the virtual pi and rho mesons, which classically transmit the nuclear force (see this topic for more). The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement.


Weak nuclear

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+ and W bosons, and neutral current, involving electrically neutral Z0 bosons. The most familiar effect of weak interaction is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. This is a type of charged-current interaction. The word "weak" derives from the fact that the field strength is some 1013 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 1015 kelvins. Such temperatures have been probed in modern particle accelerators and show the conditions of the universe in the early moments of the Big Bang.


Non-fundamental forces

Some forces are consequences of the fundamental ones. In such situations, idealized models can be utilized to gain physical insight.


Normal force

The normal force is due to repulsive forces of interaction between atoms at close contact. When their electron clouds overlap, Pauli repulsion (due to
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
ic nature of
electron The electron is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

electron
s) follows resulting in the force that acts in a direction Normal (geometry), normal to the surface interface between two objects. 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.


Friction

Friction is a surface force that opposes relative motion. The frictional force is directly related to the normal force that acts to keep two solid objects separated at the point of contact. There are two broad classifications of frictional forces:
static friction Friction is the resisting the relative motion of solid surfaces, fluid layers, and material elements against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of two solid surfa ...

static friction
and
kinetic friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding (motion), sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative late ...
. The static friction force (F_) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction (\mu_) multiplied by the normal force (F_N). In other words, the magnitude of the static friction force satisfies the inequality: :0 \le F_ \le \mu_ F_\mathrm. The kinetic friction force (F_) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals: :F_ = \mu_ F_\mathrm, where \mu_ is the coefficient of kinetic friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction.


Tension

Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and unstretchable. They can be combined with ideal pulleys, 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. By connecting the same string multiple times to the same object through the use of a set-up 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. However, even though such machines allow for an mechanical advantage, increase in force, there is a corresponding increase in the length of string that must be displaced in order 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.


Elastic force

An 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. This linear relationship was described by Robert Hooke in 1676, for whom
Hooke's law The balance wheel at the core of many mechanical clocks and watches depends on Hooke's law. Since the torque generated by the coiled spring is proportional to the angle turned by the wheel, its oscillations have a nearly constant period. Hooke ...
is named. If \Delta x is the displacement, the force exerted by an ideal spring equals: :\vec=-k \Delta \vec where k 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.


Continuum mechanics

Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized
point particle A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, l ...
s rather than three-dimensional objects. However, 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 gradients as follows: :\frac = - \vec P where V is the volume of the object in the fluid and P 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. 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: :\vec_\mathrm = - b \vec where: *b 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 * \vec is the velocity of the object. More formally, forces in continuum mechanics are fully described by a Stress (mechanics), stress–tensor with terms that are roughly defined as :\sigma = \frac where A 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 diagonals 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 stresses and compression (physical), compressions.''University Physics'', Sears, Young & Zemansky, pp. 18–38


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. These forces are considered fictitious because they do not exist in frames of reference that are not accelerating. Because these forces are not genuine they are also referred to as "pseudo forces". In
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry. As an extension, Kaluza–Klein theory and
string theory In physics, string theory is a Mathematical theory, theoretical framework in which the Point particle, point-like particles of particle physics are replaced by Dimension (mathematics and physics), one-dimensional objects called String (physic ...

string theory
ascribe electromagnetism and the other Fundamental interaction, fundamental forces respectively to the curvature of differently scaled dimensions, which would ultimately imply that all forces are fictitious.


Rotations and torque

Forces that cause extended objects to rotate are associated with
torque In physics and mechanics, torque is the rotational equivalent of linear force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

torque
s. Mathematically, the torque of a force \vec is defined relative to an arbitrary reference point as the cross-product: :\vec = \vec \times \vec where \vec is the position vector of the force application point relative to the reference point. 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 The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
, and angular momentum for
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

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: :\vec = I\vec where *I is the moment of inertia of the body * \vec is the angular acceleration of the body. 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. Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque: :\vec = \frac, where \vec 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, and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolution (geometry), revolutions through the action of internal torques.


Centripetal force

For an object accelerating in circular motion, the unbalanced force acting on the object equals: :\vec = - \frac where m is the mass of the object, v is the velocity of the object and r is the distance to the center of the circular path and \hat is the
unit vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
pointing in the radial direction outwards from the center. This means that the unbalanced centripetal force felt by any 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. The unbalanced 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.


Kinematic integrals

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: :\vec=\int_^, 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: :W= \int_^ , which is equivalent to changes in kinetic energy (yielding the work energy theorem). 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 d\vec in a time interval d''t'': : \textW\, =\, \frac\, \cdot\, \text\vec\, =\, \vec\, \cdot\, \text\vec, so :P\, =\, \frac\, =\, \frac\, \cdot\, \frac\, =\, \vec\, \cdot\, \vec, with \vec = \text\vec/\textt the
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

velocity
.


Potential energy

Instead of a force, often the mathematically related concept of a potential energy field can be used for convenience. 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 U(\vec) is defined as that field whose gradient is equal and opposite to the force produced at every point: :\vec=-\vec U. 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.


Conservative forces

A conservative force that acts on a
closed system A closed system is a physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment (systems), environm ...
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, 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. 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 \vec emanating from spherical symmetry, spherically symmetric potentials. Examples of this follow: For gravity: :\vec_g = - \frac \hat where G is the gravitational constant, and m_n is the mass of object ''n''. For electrostatic forces: :\vec_e = \frac \hat where \varepsilon_ is Permittivity, electric permittivity of free space, and q_n is the
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
of object ''n''. For spring forces: :\vec_s = - k r \hat where k is the spring constant.


Nonconservative forces

For certain physical scenarios, it is impossible to model forces as being due to gradient of potentials. This is often due to macrophysical considerations that yield forces as arising from a macroscopic statistical average of Microstate (statistical mechanics), microstates. For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, Tension (physics), tension, Physical compression, compression, and drag (physics), drag. However, 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. 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.


Units of measurement

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 . The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or . A newton is thus equal to 100,000 dynes. 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 . 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. 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. The units of slug (unit), slug and poundal are designed to avoid a constant of proportionality in Newton's Second Law. 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 when subjected to a force of 1 kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated; however it still sees use for some purposes as expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque. Other arcane units of force include the sthène, which is equivalent to 1000 N, and the kip (unit), kip, which is equivalent to 1000 lbf. See also Ton-force.


Force measurement

See force gauge,
spring scale A spring scale or spring balance or newton meter is a type of mechanical force gauge A force gauge (also called a force meter) is a measuring instrument used to measure forces. Applications exist in research and development, laboratory, quality, ...
, load cell


See also

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Notes


References


Further reading

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External links


Video lecture on Newton's three laws
by Walter Lewin from MIT OpenCourseWare
A Java simulation on vector addition of forces

Force demonstrated as any influence on an object that changes the object's shape or motion (video)
{{good article Force, Natural philosophy Classical mechanics Vector physical quantities Temporal rates