In physics, a force is an influence that can change the motion of an Physical object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newton (N). Force is represented by the symbol (formerly ).
The original form of Newton's second law states that the net force acting upon an object is equal to the time derivative at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object.
Concepts related to force include: thrust, which increases the velocity of an object; drag, which decreases the velocity of an object; and 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. Such internal mechanical stresses cause no acceleration of that body as the forces balance one another. 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 of solid materials, or flow in .
With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong force, electromagnetic, weak force, and gravitational. Highenergy particle physics made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction.
Aristotle provided a 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 , 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 began facing criticism in medieval science, first by John Philoponus in the 6th century.
The shortcomings of Aristotelian physics would not be fully corrected until the 17th century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus theory. 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 unless acted on by a force, for example friction.Drake, Stillman (1978). Galileo At Work. Chicago: University of Chicago Press.
In the early 17th century, before Newton's Principia, the term "force" (vis) was applied to many physical and nonphysical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named vis viva (live force) by Leibniz. The modern concept of force corresponds to Newton's vis motrix (accelerating force).
For instance, while traveling in a moving vehicle at a constant 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 parabola 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.
's most famous equation is
$\backslash vec\{F\}\; =\; m\backslash vec\{a\}$, he actually wrote down a different form for his second law of motion that did not use differential calculus]]
By the definition of momentum, $$\backslash vec\{F\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash vec\{p\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash left(m\backslash vec\{v\}\backslash right)\}\{\backslash mathrm\{d\}t\},$$ where m is the mass and $\backslash vec\{v\}$ is the 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 (Mv) 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." Emphasis m may be moved outside the derivative operator. The equation then becomes $$\backslash vec\{F\}\; =\; m\backslash frac\{\backslash mathrm\{d\}\backslash vec\{v\}\}\{\backslash mathrm\{d\}t\}.$$ By substituting the definition of acceleration, the algebraic version of Newton's second law is derived: $$\backslash vec\{F\}\; =m\backslash vec\{a\}.$$ 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 measurements. However, while kinematics are welldescribed through reference frame analysis in advanced physics, there are still deep questions that remain as to what is the proper definition of mass. General relativity offers an equivalence between spacetime and mass, but lacking a coherent theory of 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,
Translated by: J.B. Sykes, A.D. Petford, and C.L. Petford. . In section 7, pp. 12–14, this book defines force as dp/dt.Newton's second law can be used to measure the strength of forces. For instance, knowledge of the masses of along with the accelerations of their allows scientists to calculate the gravitational forces on planets.
Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies,"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: $$\backslash vec\{F\}\_\{1,2\}+\backslash vec\{F\}\_\{2,1\}=0.$$ More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.
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 $\backslash vec\{p\}\_1$ is the momentum of object 1 and $\backslash vec\{p\}\_\{2\}$ the momentum of object 2, then $$\backslash frac\{\backslash mathrm\{d\}\backslash vec\{p\}\_1\}\{\backslash mathrm\{d\}t\}\; +\; \backslash frac\{\backslash mathrm\{d\}\backslash vec\{p\}\_2\}\{\backslash mathrm\{d\}t\}=\; \backslash vec\{F\}\_\{1,2\}\; +\; \backslash vec\{F\}\_\{2,1\}\; =\; 0.$$ 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.
The relativistic expression relating force and acceleration for a particle with constant nonzero rest mass $m$ moving in the $x$ direction is: $$\backslash vec\{F\}\; =\; \backslash left(\backslash gamma^3\; m\; a\_x,\; \backslash gamma\; m\; a\_y,\; \backslash gamma\; m\; a\_z\backslash right),$$ where $$\backslash gamma\; =\; \backslash frac\{1\}\{\backslash sqrt\{1\; \; v^2/c^2\}\}.$$ is called the Lorentz factor.
In the early history of relativity, the expressions $\backslash gamma^3\; m$ and $\backslash gamma\; m$ were called longitudinal and transverse mass. Relativistic force does not produce a constant acceleration, but an everdecreasing acceleration as the object approaches the speed of light. Note that $\backslash gamma$ approaches asymptotically an infinite value and is undefined for an object with a nonzero Invariant mass 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 $\backslash gamma$ 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 $$F^\backslash mu\; =\; mA^\backslash mu$$ through the use of fourvectors. This relation is correct in relativity when $F^\backslash mu$ is the fourforce, $m$ is the invariant mass, and $A^\backslash mu$ is the fouracceleration.
Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "Euclidean vector". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the resultant vector. 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 or the two people could be pulling in the same direction. In this simple onedimensional 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 where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction. When two forces act on a point particle, the resulting force, the resultant (also called the net force), can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. 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.
Freebody 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 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 is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one nonzero component. Orthogonal force vectors can be threedimensional with the third component being at rightangles to the other two.
There are two kinds of equilibrium: static equilibrium and dynamic equilibrium.
The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration.
Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as and . For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his Three Laws of Motion.
Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity.
A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a nonzero velocity, it continues to move with a nonzero 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.
This becomes different only in the framework of quantum field theory, 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 angular momentum variable called the "spin", and there is the Pauli exclusion principle relating the space and the spin variables. Depending on the value of the spin, identical particles split into two different classes, and . 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.
The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron beta decay into an electron, proton, and neutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.
The fundamental theories for forces developed from the unification of different ideas. For example, Sir Isaac Newton unified, with his universal theory of gravitation, the force responsible for objects falling near the surface of the Earth with the force responsible for the falling of celestial bodies about the Earth (the Moon) and around the Sun (the planets). Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter () interacting by exchanging called . This Standard Model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory, which was subsequently confirmed by observation. The complete formulation of the Standard Model predicts an as yet unobserved Higgs mechanism, but observations such as neutrino oscillations suggest that the Standard Model is incomplete. A Grand Unified Theory 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 proposed to accommodate some of the outstanding unsolved problems in physics. Physicists are still attempting to develop selfconsistent unification models that would combine all four fundamental interactions into a theory of everything. Einstein tried and failed at this endeavor, but currently the most popular approach to answering this question is string theory.
For an object in freefall, this force is unopposed and the net force on the object is its weight. For objects not in freefall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.
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 that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion.
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body. Combining these ideas gives a formula that relates the mass ($m\_\backslash oplus$) and the radius ($R\_\backslash oplus$) of the Earth to the gravitational acceleration: $$\backslash vec\{g\}=\backslash frac\{Gm\_\backslash oplus\}\{r^2\}\; \backslash hat\{r\}$$ where $r$ is the distance between the two objects' centers of mass and $\backslash hat\{r\}$ 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 were invented to calculate the deviations of due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed. Mercury's orbit, however, did not match that predicted by Newton's Law of Gravitation. Some astrophysicists predicted the existence of another planet (Vulcan) that would explain the discrepancies; however no such planet could be found. When Albert Einstein formulated his theory of general relativity (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 geodesic through curved spacetime – defined as the shortest spacetime path between two spacetime events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of spacetime can be observed and the force is inferred from the object's curved path. Thus, the straight line path in spacetime is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with the radius of curvature of the order of few ). The time derivative of the changing momentum of the object is what we label as "gravitational force".
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 $$\backslash vec\{E\}\; =\; \{\backslash vec\{F\}\; \backslash over\{q\}\}$$ where $q$ is the magnitude of the hypothetical test charge.
Meanwhile, the Lorentz force of magnetism was discovered to exist between two . 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 $$B\; =\; \{F\; \backslash over\{I\; \backslash ell\}\}$$ where $I$ is the magnitude of the hypothetical test current and $\backslash ell$ is the length of hypothetical wire through which the test current flows. The magnetic field exerts a force on all including, for example, those used in . The fact that the geomagnetism is aligned closely with the orientation of the Earth's rotation 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 called Lorentz force 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: $$\backslash vec\{F\}\; =\; q\backslash left(\backslash vec\{E\}\; +\; \backslash vec\{v\}\; \backslash times\; \backslash vec\{B\}\backslash right)$$ where $\backslash vec\{F\}$ is the electromagnetic force, $q$ is the magnitude of the charge of the particle, $\backslash vec\{E\}$ is the electric field, $\backslash vec\{v\}$ is the velocity of the particle that is cross product with the magnetic field ($\backslash vec\{B\}$).
The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs.
, Chapter 2, p. 19 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 "selfgenerating" through a wave that traveled at a speed that he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.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 (or QED), which fully describes all electromagnetic phenomena as being mediated by wave–particles known as . 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
The strong force is today understood to represent the interactions between and as detailed by the theory of quantum chromodynamics (QCD). The strong force is the fundamental force mediated by gluons, acting upon quarks, antiparticle, 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 (notably, the in atomic nuclei), known as the nuclear force. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rho , the classical transmitters of the nuclear force. The failure of many searches for has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement.
The static friction force ($F\_\{\backslash mathrm\{sf\}\}$) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction ($\backslash mu\_\{\backslash mathrm\{sf\}\}$) multiplied by the normal force ($F\_N$). In other words, the magnitude of the static friction force satisfies the inequality: $$0\; \backslash le\; F\_\{\backslash mathrm\{sf\}\}\; \backslash le\; \backslash mu\_\{\backslash mathrm\{sf\}\}\; F\_\backslash mathrm\{N\}.$$
The kinetic friction force ($F\_\{\backslash mathrm\{kf\}\}$) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals: $$F\_\{\backslash mathrm\{kf\}\}\; =\; \backslash mu\_\{\backslash mathrm\{kf\}\}\; F\_\backslash mathrm\{N\},$$
where $\backslash mu\_\{\backslash mathrm\{kf\}\}$ is the coefficient of kinetic friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction.
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 for fluids suspended in gravitational fields, winds in atmospheric science, and the 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 socalled "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction: $$\backslash vec\{F\}\_\backslash mathrm\{d\}\; =\; \; b\; \backslash vec\{v\}$$ where:
More formally, forces in continuum mechanics are fully described by a stress–tensor with terms that are roughly defined as $$\backslash sigma\; =\; \backslash frac\{F\}\{A\}$$
where $A$ is the relevant crosssectional area for the volume for which the stresstensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the crosssectional area (the of the tensor) as well as Shear stress terms associated with forces that act parallel to the crosssectional area (the offdiagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also and compressions. University Physics, Sears, Young & Zemansky, pp. 18–38
In general relativity, 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 ascribe electromagnetism and the other fundamental forces respectively to the curvature of differently scaled dimensions, which would ultimately imply that all forces are fictitious.
Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. As a consequence of Newton's First Law of Motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's Second Law of Motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body: $$\backslash vec\{\backslash tau\}\; =\; I\backslash vec\{\backslash alpha\}$$
where
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 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: $$\backslash vec\{\backslash tau\}\; =\; \backslash frac\{\backslash mathrm\{d\}\backslash vec\{L\}\}\{\backslash mathrm\{dt\}\},$$ where $\backslash vec\{L\}$ 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 revolutions through the action of internal torques.
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 $\backslash hat\{r\}$ is the unit vector 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.
Similarly, integrating with respect to position gives a definition for the work done by a force: $$W=\; \backslash int\_\{\backslash vec\{x\}\_1\}^\{\backslash vec\{x\}\_2\}\; \{\backslash vec\{F\}\; \backslash cdot\; \{\backslash mathrm\{d\}\backslash vec\{x\}\}\},$$
which is equivalent to changes in kinetic energy (yielding the work energy theorem).
Power P is the rate of change d W/d t of the work W, as the trajectory is extended by a position change $d\backslash vec\{x\}$ in a time interval d t: $$\backslash mathrm\{d\}W\; =\; \backslash frac\{\backslash mathrm\{d\}W\}\{\backslash mathrm\{d\}\backslash vec\{x\}\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash vec\{x\}\; =\; \backslash vec\{F\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash vec\{x\},$$ so $$P\; =\; \backslash frac\{\backslash mathrm\{d\}W\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash mathrm\{d\}W\}\{\backslash mathrm\{d\}\backslash vec\{x\}\}\; \backslash cdot\; \backslash frac\{\backslash mathrm\{d\}\backslash vec\{x\}\}\{\backslash mathrm\{d\}t\}\; =\; \backslash vec\{F\}\; \backslash cdot\; \backslash vec\{v\},$$ with $\backslash vec\{v\}\; =\; \backslash mathrm\{d\}\backslash vec\{x\}/\backslash mathrm\{d\}t$ the velocity.
Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.
Conservative forces include gravity, the Electromagnetism force, and the spring force. Each of these forces has models that are dependent on a position often given as a radius $\backslash vec\{r\}$ emanating from spherically symmetric potentials. Examples of this follow:
For gravity: $$\backslash vec\{F\}\_g\; =\; \; \backslash frac\{G\; m\_1\; m\_2\}\{r^2\}\; \backslash hat\{r\}$$ where $G$ is the gravitational constant, and $m\_n$ is the mass of object n.
For electrostatic forces: $$\backslash vec\{F\}\_e\; =\; \backslash frac\{q\_1\; q\_2\}\{4\; \backslash pi\; \backslash varepsilon\_\{0\}\; r^2\}\; \backslash hat\{r\}$$ where $\backslash varepsilon\_\{0\}$ is Permittivity, and $q\_n$ is the electric charge of object n.
For spring forces: $$\backslash vec\{F\}\_s\; =\; \; k\; r\; \backslash hat\{r\}$$ where $k$ is the spring constant.
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 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.
The gravitational footpoundsecond English unit of force is the poundforce (lbf), defined as the force exerted by gravity on a poundmass in the Standard gravity field of . The poundforce provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one poundforce.
An alternative unit of force in a different footpoundsecond system, the absolute fps system, is the poundal, defined as the force required to accelerate a onepound mass at a rate of one foot per second squared. The units of slug and poundal are designed to avoid a constant of proportionality in Newton's Second Law.
The poundforce has a metric counterpart, less commonly used than the newton: the kilogramforce (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogramforce 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 kilogramforce 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, which is equivalent to 1000 lbf.
See also Tonforce.

