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Encyclopedia > Force
Forces are often described as pushes or pulls. They can be due to phenomena such as gravity, magnetism, or anything else that causes a mass to accelerate.
Look up Force in
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Classical mechanics
$vec{F} = frac{mathrm{d}}{mathrm{d}t}(m vec{v})$
Newton's Second Law
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Since antiquity, scientists have used the concept of force in the study of stationary and moving objects. These studies culminated with the descriptions made by the third century BC philosopher Archimedes of how simple machines functioned. The rules Archimedes determined for how forces interact in simple machines are still a part of modern physics.[5] Earlier descriptions of forces by Aristotle incorporated fundamental misunderstandings, which would not be resolved until the seventeenth century when Isaac Newton correctly described how forces behaved.[4] Newtonian descriptions of forces remained unchanged for nearly three hundred years. Statics is the branch of physics concerned with physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. ... In physics, dynamics is the branch of classical mechanics that is concerned with the effects of forces on the motion of objects. ... For other uses, see Archimedes (disambiguation). ... This article is about the concept in physics. ... For other uses, see Aristotle (disambiguation). ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ...

Current understanding of quantum mechanics and the standard model of particle physics associate forces with the fundamental interactions accompanying the emission or absorption of gauge bosons. Only four fundamental interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational.[3] High-energy particle physics observations, in the 1970s and 1980s, confirmed that the weak and electromagnetic forces are expressions of a unified electroweak interaction.[6] Einstein in his Theory of General Relativity explained that gravity is an attribute of the curvature of space-time, though perceived as a force. For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... In physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... This article covers the physics of gravitation. ... Particle physics is a branch of physics that studies the elementary constituents of matter and radiation, and the interactions between them. ... For other uses, see Observation (disambiguation). ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... Einstein redirects here. ... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ... In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... For other uses of this term, see Spacetime (disambiguation). ...

Aristotle famously described a force as anything which causes an object to undergo "unnatural motion"

These shortcomings would not be fully explained and corrected until the seventeenth century work of 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 early in the seventeenth century. He showed that the bodies were accelerated by gravity to an extent which was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction.[9] Galileo redirects here. ... Inertia is the property of an object to remain at constant velocity unless acted upon by an outside force. ... The Aristotelian theory of gravity was that all bodies move towards their natural place. ... For other uses, see Mass (disambiguation). ... This article is about velocity in physics. ... For other uses, see Friction (disambiguation). ...

## Newtonian mechanics

Isaac Newton is the first person known to explicitly state the first, and the only, mathematical definition of force—as the time-derivative of momentum: F = dp / dt. In 1687, Newton went on to publish his Philosophiae Naturalis Principia Mathematica, which used concepts of inertia, force, and conservation to describe the motion of all objects.[10][4] In this work, Newton set out three laws of motion that to this day are the way forces are described in physics.[10] Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Newtons own copy of his Principia, with handwritten corrections for the second edition. ... This article is about inertia as it applies to local motion. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

Though Sir Isaac Newton's most famous equation is F=ma, he actually wrote down a different form for his second law of motion that used differential calculus.

Godfrey Knellers portrait of Isaac Newton (1689) oil on canvas. ... Godfrey Knellers portrait of Isaac Newton (1689) oil on canvas. ... Sir Isaac Newton in Knellers portrait of 1689. ... Newtons first and second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ...

### Newton's first law

Main article: Newton's first law

For example, while traveling in a moving vehicle at a constant velocity, the laws of physics do not change from being at rest. A person can throw a ball straight up in the air and catch it as it falls down without worrying about applying a force in the direction the vehicle is moving. This is true even though another person who is observing the moving vehicle pass by also observes the ball follow a curving parabolic path 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 every thing inside of it is at rest: It is the outside world that is moving with a constant speed in the opposite direction. 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. In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... This article is about velocity in physics. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane...

The concept of inertia can be further generalized to explain the tendency of objects to continue in many different forms of constant motion, even those that are not strictly constant velocity. The rotational inertia of planet Earth is what fixes the constancy of the length of a day and the length of a year. Albert Einstein extended the principle of inertia further when he explained that reference frames subject to constant acceleration, such as those free-falling toward a gravitating object, were physically equivalent to inertial reference frames. This is why, for example, astronauts experience weightlessness when in free-fall orbit around the Earth, and why Newton's Laws of Motion are more easily discernible in such environments. If an astronaut places an object with mass in mid-air next to herself, it will remain stationary with respect to the astronaut due to its inertia. This is the same thing that would occur if the astronaut and the object were in intergalactic space with no net force of gravity acting on their shared reference frame. This principle of equivalence was one of the foundational underpinnings for the development of the general theory of relativity.[11] Increasing the mass increases the rotational inertia of an object. ... Look up day in Wiktionary, the free dictionary. ... A year (from Old English gÄ“r) is the time between two recurrences of an event related to the orbit of the Earth around the Sun. ... Zero gravity redirects here. ... In relativity the equivalence principle is applied to several related concepts dealing with the uniformity of physical measurements in different frames of reference. ... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ...

### Newton's second law

Main article: Newton's second law

A modern statement of Newton's second law is a vector differential equation:[12] Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... This article is about vectors that have a particular relation to the spatial coordinates. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ...

$vec{F} = frac{dvec{p}}{dt} = frac{d(m vec{v})}{dt}$

where is the momentum of the system. The $vec{F}$ in the equation represents the net (vector sum) force; in equilibrium there is zero net force by definition, but (balanced) forces may be present nevertheless. In contrast, the second law states an unbalanced force acting on an object will result in the object's momentum changing over time. [10] This article is about momentum in physics. ... This article is about vectors that have a particular relation to the spatial coordinates. ...

By assuming mass to be constant, Newton's second law can be expressed approximately as force equaling the product of mass m multiplied by acceleration $vec{a}$. Acceleration is the rate of change of velocity over time: Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... For other uses, see Mass (disambiguation). ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

$vec{F} =mvec{a},$

sometimes called the "second most famous formula in physics".[13] Newton never stated explicitly the F=ma formula for which he is often credited.

Newton's second law asserts the proportionality of acceleration and mass to force. Accelerations can be defined through kinematic measurements. However, while kinematics are well-described 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 space-time 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 are fixed. Kinematics (Greek ÎºÎ¹Î½ÎµÎ¹Î½,kinein, to move) is a branch of mechanics which describes the motion of objects without the consideration of the masses or forces that bring about the motion. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... Quantum gravity is the field of theoretical physics attempting to unify quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ...

The use of Newton's second law as a definition of force has been disparaged in some of the more rigorous textbooks,[3][14] because it is essentially a mathematical truism. The equality between the abstract idea of a "force" and the abstract idea of a "changing momentum vector" ultimately has no observational significance because one cannot be defined without simultaneously defining the other. What a "force" or "changing momentum" is must either be referred to an intuitive understanding of our direct perception, or be defined implicitly through a set of self-consistent mathematical formulas. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of "force" include Ernst Mach, Clifford Truesdell and Walter Noll.[15] A truism is a claim that is so obvious or self-evident as to be hardly worth mentioning, except as a reminder or as a rhetorical or literary device. ... Ernst Mach Ernst Mach (February 18, 1838 â€“ February 19, 1916) was an Austrian-Czech physicist and philosopher and is the namesake for the Mach number and the optical illusion known as Mach bands. ... Clifford Ambrose Truesdell III, (February 18, 1919 â€“ January 14, 2000) was an American mathematician, natural philosopher, historian of mathematics, and polemicist. ...

Newton's second law can be used to measure the strength of forces. For instance, knowledge of the masses of planets along with the accelerations of their orbits allows scientists to calculate the gravitational forces on planets. This article is about the astronomical term. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ...

### Newton's third law

Main article: Newton's third law

Newton's third law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. For any two objects (call them 1 and 2), Newton's third law states that Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... Sphere symmetry group o. ...

$vec{F}_{mathrm{1 on 2}}=-vec{F}_{mathrm{2 on 1}}.$

This law implies that forces always occur in action-reaction pairs.[10] Any force that is applied to object 1 due to the action of object 2 is automatically accompanied by a force applied to object 2 due to the action of object 1.[16] If object 1 and object 2 are considered to be in the same system, then the net force on the system due to the interactions between objects 1 and 2 is zero since

$vec{F}_{mathrm{1 on 2}}+vec{F}_{mathrm{2 on 1}}=0$.

This means that in a closed system of particles, there are no internal forces that are unbalanced. That is, action-reaction pairs of forces shared between any two objects in a closed system will not cause the center of mass of the system to accelerate. The constituent objects only accelerate with respect to each other, the system itself remains unaccelerated. Alternatively, if an external force acts on the system, then the center of mass will experience an acceleration proportional to the magnitude of the external force divided by the mass of the system.[3] In thermodynamics, a closed system, as contrasted with an isolated system, can exchange heat and work, but not matter, with its surroundings. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

Combining Newton's second and third laws, it is possible to show that the linear momentum of a system is conserved. Using In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

$vec{F}_{mathrm{1 on 2}} = frac{dvec{p}_{mathrm{1 on 2}}}{dt} = -vec{F}_{mathrm{2 on 1}} = -frac{dvec{p}_{mathrm{2 on 1}}}{dt}$

$Delta{vec{p}_{mathrm{1 on 2}}} = - Delta{vec{p}_{mathrm{2 on 1}}}$

is obtained. For a system which includes objects 1 and 2,

$sum{Delta{vec{p}}}=Delta{vec{p}_{mathrm{1 on 2}}} + Delta{vec{p}_{mathrm{2 on 1}}} = 0$

which is the conservation of linear momentum.[17] Using the similar arguments, it is possible to generalizing this to a system of an arbitrary number of particles. This shows that exchanging momentum between constituent objects will not affect the net momentum of a system. 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.[3]

## Descriptions

Free-body diagrams of an object on a flat surface and an inclined plane. Forces are resolved and added together to determine their magnitudes and the resultant.

Since forces are perceived as pushes or pulls, this can provide an intuitive understanding for describing forces.[4] As with other physical concepts (e.g. temperature), the intuitive understanding of forces is quantified using precise operational definitions that are consistent with direct observations and compared to a standard measurement scale. Through experimentation, it is determined that laboratory measurements of forces are fully consistent with the conceptual definition of force offered by Newtonian mechanics. A free body diagram is a pictorial representation often used by physicists to show all contact and non-contact forces acting on the given free body. ... The inclined plane is one of the classical simple machines; as the name suggests, it is a flat surface whose endpoints are at different heights. ... For other uses, see Temperature (disambiguation). ... An operational definition is a showing of somethingâ€”such as a variable, term, or objectâ€”in terms of the specific process or set of validation tests used to determine its presence and quantity. ... In psychology and the cognitive sciences, perception is the process of acquiring, interpreting, selecting, and organizing sensory information. ... Measurement is the estimation of the magnitude of some attribute of an object, such as its length or weight, relative to a unit of measurement. ... A conceptual definition is an element of the scientific research process, in which a specific concept is defined as a measurable occurrence. ...

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 "vector quantities". This means that forces follow a different set of mathematical rules that 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 result. 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 a certain amount of strength 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 one-dimensional 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. The term direction can be applied to various topics. ... In science, a magnitude is the numerical size of something: see orders of magnitude. ... Look up vector in Wiktionary, the free dictionary. ... A scalar may be: Look up scalar in Wiktionary, the free dictionary. ... For the technique in organ building, see Resultant (organ). ... Tug of war Tug of war, also known as rope pulling, is a sport that directly pits two teams against each other in a test of strength. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...

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 resultant.[18] A free body diagram is a pictorial representation often used by physicists to show all contact and non-contact forces acting on the given free body. ... This article is about vectors that have a particular relation to the spatial coordinates. ...

As well as being added, forces can also be resolved into independent components at right angles 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 vectors is often a more mathematically clean way to describe forces than using magnitudes and directions.[19] 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; forces acting at ninety degrees to each other have no effect on each 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. Force vectors can also be three-dimensional, with the third component at right-angles to the two other components.[3] This article is about angles in geometry. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

### Equilibria

Equilibrium occurs when the resultant force acting on an object is zero (that is, the vector sum of all forces is zero). There are two kinds of equilibrium: static equilibrium and dynamic equilibrium. A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ... In physics, static equilibrium, or neutral balance, exists when the forces (actions), and torques, on all components of a defined system are balanced such that no component is undergoing an acceleration relative to the designated frame of reference. ... A dynamic equilibrium occurs when two reversible processes occur at the same rate. ...

#### Static equilibrium

Main article: statics

Static equilibrium was understood well before the invention of classical mechanics. Objects which are at rest have zero net force acting on them.[20] Statics is the branch of physics concerned with physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at rest under the action of external forces of equilibrium. ...

The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, any object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, surface forces resist the downward force with equal upward force (called the normal force) and result in the object having a non-zero weight. The situation is one of zero net force and no acceleration.[4] Fn represents the normal force. ... For other uses, see Weight (disambiguation). ...

Pushing against an object 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.[4] Determining the Coefficient of Friction. ...

A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. 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 is equal to 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.[3][4] Digital kitchen scales. ... Spring BAlance- it is a machine that measured the gravitational forces of earth which is Nine point eight Per second squared. ... Spring scale. ... For other uses, see Density (disambiguation). ... In physics, buoyancy is an upward force on an object immersed in a fluid (i. ... For other uses, see Archimedes (disambiguation). ... Leverage redirects here. ... Boyles law (sometimes referred to as the Boyle-Mariotte law) is one of the gas laws and basis of derivation for the ideal gas law, which describes the relationship between the product pressure and volume within a closed system as constant when temperature and moles remain at a fixed... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... Sir Isaac Newton FRS (4 January 1643 â€“ 31 March 1727) [ OS: 25 December 1642 â€“ 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

#### Dynamical equilibrium

Main article: Dynamics (physics)
Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's description of forces.

Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamical equilibrium: when all the forces on an object balance but it still moves at a constant velocity.

A simple case of dynamical 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 a net zero 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.[3] Kinetic friction is the type of friction that an object is subject to after it is in motion. ...

### Feynman diagrams

Main article: Feynman diagrams
A Feynman diagram for the decay of a neutron into a proton. The W boson is between two vertices indicating a repulsion.

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 decays into an electron, proton, and neutrino: an interaction mediated by the same gauge boson that is responsible for the weak nuclear force. While the Feynman diagram for this interaction has similar features to a repulsive interaction, the decay is more complicated than a simple "repulsive force".[21] A fundamental interaction or fundamental force is a mechanism by which particles interact with each other, and which cannot be explained in terms of another interaction. ... This article or section does not adequately cite its references or sources. ... In nuclear physics, beta decay (sometimes called neutron decay) is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted. ... For other uses, see Electron (disambiguation). ... For other uses, see Proton (disambiguation). ... For other uses, see Neutrino (disambiguation). ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ...

### Special relativity

In the special theory of relativity mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's second law $vec{F} = mathrm{d}vec{p}/mathrm{d}t$ remains formally valid.[22] But in order to be conserved, momentum must be redefined as: Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ...

$vec{p} = frac{mvec{v}}{sqrt{1 - v^2/c^2}}$

where

v is the velocity and
c is the speed of light.

The relativistic expression relating force and acceleration for a particle with non-zero rest mass $m,$ moving in the $x,$ direction is: A line showing the speed of light on a scale model of Earth and the Moon, taking about 1â…“ seconds to traverse that distance. ... The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ...

$F_x = gamma^3 m a_x ,$
$F_y = gamma m a_y ,$
$F_z = gamma m a_z ,$

where the Lorentz factor It has been suggested that Lorentz term be merged into this article or section. ...

$gamma = frac{1}{sqrt{1 - v^2/c^2}}$[23]

Here a constant force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. Note that γ is undefined for an object with a non zero rest mass at the speed of light, and the theory yields no prediction at that speed. For the album by Hux Flux, see Division by Zero (album). ... The invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. ...

One can however restore the form of

$F^mu = mA^mu ,$

for use in relativity through the use of four-vectors. This relation is correct in relativity when Fμ is the four-force, m is the invariant mass, and Aμ is the four-acceleration.[24] In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ... In the special theory of relativity four-force is a four-vector that replaces the classical force; the four-force of the four-vector a is defined as the change in four-momentum over the particles own time: . Since where m0 is the rest mass and Ua is the... The invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. ... In special relativity, four-acceleration is a four-vector and is defined as the change in four-velocity over the particles proper time: where and and is the Lorentz factor for the speed . ...

## Fundamental models

All the forces in the universe are based on four fundamental forces. The strong and weak forces act only at very short distances, and are responsible for holding certain nucleons and compound nuclei together. The electromagnetic force acts between electric charges and the gravitational force acts between masses. All other forces are based on the existence of the four fundamental interactions. For example, friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces, and the Pauli Exclusion Principle,[25] which does not allow atoms to pass through each other. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces and the Exclusion Principle acting together to return the object to its equilibrium position. Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating frames of reference.[3] Nucleon is the common name used in nuclear chemistry to refer to a neutron or a proton, the components of an atoms nucleus. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... For other uses, see Mass (disambiguation). ... For other uses, see Friction (disambiguation). ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, composed of the electric field and the magnetic field. ... For other uses, see Atom (disambiguation). ... An open surface with X-, Y-, and Z-contours shown. ... For other uses, see Spring. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...

The development of fundamental theories for forces proceeded along the lines of unification of disparate ideas. For example, Isaac Newton unified the force responsible for objects falling at the surface of the Earth with the force responsible for the orbits of celestial mechanics in his universal theory of gravitation. Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through one consistent theory of electromagnetism. In the twentieth century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons.[26] This standard model of particle physics posits a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory subsequently confirmed by observation. The complete formulation of the standard model predicts an as yet unobserved Higgs mechanism, but observations such as neutrino oscillations indicate that the standard model is incomplete. A grand unified theory allowing 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 self-consistent 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.[6] This article does not cite any references or sources. ... Michael Faraday, FRS (September 22, 1791 â€“ August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... In the description of the interaction between elementary particles in quantum field theory, a virtual particle is a temporary elementary particle, used to describe an intermediate stage in the interaction. ... Gauge bosons are bosonic particles which act as carriers of the fundamental forces of Nature. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... This box:      The Higgs mechanism, also called the Brout-Englert-Higgs mechanism, Higgs-Kibble mechanism or Anderson-Higgs mechanism, was proposed in 1964 by Robert Brout and Francois Englert [1], independently by Peter Higgs [2] and by Gerald Guralnik, C. R. Hagen, and Tom Kibble [3] following earlier work by... The solar neutrino problem was a major discrepancy between measurements of the neutrinos flowing through the Earth and theoretical models of the solar interior, lasting from the mid-1960s to about 2002. ... Grand unification, grand unified theory, or GUT is a theory in physics that unifies the strong interaction and electroweak interaction. ... This article or section is in need of attention from an expert on the subject. ... This is a list of some of the unsolved problems in physics. ... This page discusses Theories of Everything in physics. ... This box:      String theory is a still developing mathematical approach to theoretical physics, whose original building blocks are one-dimensional extended objects called strings. ...

### Gravity

Main article: Gravity
An initially stationary object which is allowed to fall freely under gravity drops a distance which is proportional to the square of the elapsed time. An image was taken 20 flashes per second. During the first 1/20th of a second the ball drops one unit of distance (here, a unit is about 12 mm); by 2/20ths it has dropped a total of 4 units; by 3/20ths, 9 units and so on.

What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as $vec{g}$ and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth.[27] 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: Gravity is a force of attraction that acts between bodies that have mass. ... Image File history File links Metadata Size of this preview: 178 Ã— 598 pixelsFull resolution (819 Ã— 2751 pixel, file size: 363 KB, MIME type: image/jpeg) Original photograph File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links Metadata Size of this preview: 178 Ã— 598 pixelsFull resolution (819 Ã— 2751 pixel, file size: 363 KB, MIME type: image/jpeg) Original photograph File historyClick on a date/time to view the file as it appeared at that time. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... Free Fall opens with one of the most stunning first paragraphs I have ever, or am ever likely to, read. ... For other uses, see Mass (disambiguation). ... The nominal acceleration due to gravity at sea level on the Earths surface, also known as standard gravity, is defined as exactly 9. ... The metre, or meter (symbol: m) is the SI base unit of length. ... This article is about the unit of time. ...

$vec{F} = mvec{g}$

In free-fall, this force is unopposed and therefore the net force on the object is the force of gravity. For objects not in free-fall, the force of gravity is opposed by the weight of the object. For example, a person standing on the ground experiences zero net force, since the force of gravity is balanced by the weight of the person that is manifested by a normal force exerted on the person by the ground.[3] For other uses, see Weight (disambiguation). ... Fn represents the normal force. ...

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.[28] Gravitation is the tendency of masses to move toward each other. ... Illustration of Keplers three laws with two planetary orbits. ...

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 mass of the gravitating object directly affected the acceleration due to gravity.[28] Combining these ideas gives a formula that relates the mass of the Earth ($M_oplus$), the radius of the Earth ($R_oplus$) to the acceleration due to gravity: In physics, an inverse-square law is any physical law stating that some quantity is inversely proportional to the square of the distance from a point. ...

$vec{g}=-frac{GM_oplus}{{R_oplus}^2} hat{r}$

where the vector direction is given by $hat{r}$ which is the unit vector directed outward from the center of the Earth.[10] In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ...

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,[29] though it was of an unknown value in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of G using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing the 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 motion, his law of gravity had to be universal. Succinctly stated, Newton's Law of Gravitation states that the force on an object of mass m1 due to the gravitational pull of mass m2 is The gravitational constant G is a key element in Newtons law of universal gravitation. ... For other persons named Henry Cavendish, see Henry Cavendish (disambiguation). ... A torsion spring is a ribbon, bar, or coil that reacts against twisting motion. ... Johannes Keplers primary contributions to astronomy/ astrophysics were the three laws of planetary motion. ... The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ...

$vec{F}=-frac{Gm_{1}m_{2}}{r^2} hat{r}$

where r is the distance between the two objects' centers of mass and $hat{r}$ is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.[10] In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

It was only the orbit of the planet Mercury that Newton's Law of Gravitation seemed not to fully explain. Some astrophysicists predicted the existence of another planet (Vulcan) that would explain the discrepancies; however, despite some early indications, no such planet could be found. When Albert Einstein finally 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 less correct than an alternative.[32] This article is about the planet. ... Vulcan was the name given to a small planet proposed to exist in an orbit between Mercury and the Sun in a 19th-century hypothesis. ... â€œEinsteinâ€ redirects here. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... Tests of Einsteins general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. ...

Since then, and so far, general relativity has been acknowledged as the theory which 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 straight lines through curved space-time – 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 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 space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly curved (with the 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".[3] In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ... In gravitational theory, gravity can deflect and modify the behaviour of light, causing spatial distances (measured by light) to be progressively modified or warped. ... External ballistics is the part of the science of ballistics that deals with the behaviour of a non-powered projectile in flight. ... Mathematically the term trajectory refers to the ordered set of states which are assumed by a dynamical system over time (see e. ... This article is about the sport. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: Ï€Î±ÏÎ±Î²Î¿Î»Î®) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... The distance from the center of a sphere or ellipsoid to its surface is its radius. ... A light-year, symbol ly, is the distance light travels in one year: exactly 9. ...

### Electromagnetic forces

Main article: Electromagnetic force

The electrostatic force was first described in 1784 by Coulomb as a force which existed intrinsically between two charges.[33] The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the law of superposition. Coulomb's Law unifies all these observations into one succinct statement.[34] In physics, the electromagnetic force is the force that the electromagnetic field exerts on electrically charged particles. ... In physics, the electrostatic force is the force arising between static (that is, non-moving) electric charges. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... In physics, an inverse-square law is any physical law stating that some quantity is inversely proportional to the square of the distance from a point. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... The polarity of an object is, in general, its physical alignment of atoms. ... See here for the superposition principle of physics. ... This box:      Coulombs torsion balance Coulombs law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows: The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each...

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.[35] Thus the electric field anywhere in space is defined as In physics, the space surrounding an electric charge or in the presence of a time-varying magnetic field has a property called an electric field. ... A test charge is an object (usually a point particle) that has negligible charge; one can ignore the electrical field generated by the object itself. ...

$vec{E} = {vec{F} over{q}}$

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 Lorentz force. ... For other senses of this word, see magnetism (disambiguation). ... This box:      Electric current is the flow (movement) of electric charge. ... For the indie-pop band, see The Magnetic Fields. ...

$B = {F over{I ell}}$

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 Earth's magnetic field is aligned closely with the orientation of the Earth's axis causes compass magnets to become oriented because of the magnetic force pulling on the needle. For other uses, see Magnet (disambiguation). ... This article is about the navigational instrument. ... The cause of Earths magnetic field (the surface magnetic field) is not known for certain, but is possibly explained by dynamo theory. ... This article is about rotation as a movement of a physical body. ... Orientation can refer to different things. ...

Through combining the definition of electric current as the time rate of change of electric charge, a rule of vector multiplication called Lorentz's Law describes the force on a charge moving in an magnetic field.[35] 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:

$vec{F} = q(vec{E} + vec{v} times vec{B})$

where $vec{F}$ is the electromagnetic force, q is the magnitude of the charge of the particle, $vec{E}$ is the electric field, $vec{v}$ is the velocity of the particle which is crossed with the magnetic field ($vec{B}$). This article is about velocity in physics. ... For the cross product in algebraic topology, see KÃ¼nneth theorem. ...

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 succinct set of four equations. 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 which 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.[36] James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... The wave equation is an important partial differential equation which generally describes all kinds of waves, such as sound waves, light waves and water waves. ... A line showing the speed of light on a scale model of Earth and the Moon, taking about 1â…“ seconds to traverse that distance. ... For the book by Sir Isaac Newton, see Opticks. ... Although some radiations are marked as N for no in the diagram, some waves do in fact penetrate the atmosphere, although extremely minimally compared to the other radiations The electromagnetic (EM) spectrum is the range of all possible electromagnetic radiation. ...

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 photons. In QED, photons are the fundamental exchange particle which described all interactions relating to electromagnetism including the electromagnetic force.[37] A diagram illustrating the emission of electrons from a metal plate, requiring energy gained from an incoming photon to be more than the work function of the material. ... The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

It is a common misconception to ascribe the stiffness and rigidity of solid matter to the repulsion of like charges under the influence of the electromagnetic force. However, these characteristics actually result from the Pauli Exclusion Principle. Since electrons are fermions, they cannot occupy the same quantum mechanical state as other electrons. When the electrons in a material are densely packed together, there are not enough lower energy quantum mechanical states for them all, so some of them must be in higher energy states. This means that it takes energy to pack them together. While this effect is manifested macroscopically as a structural "force", it is technically only the result of the existence of a finite set of electron states. Solid-state physics, the largest branch of condensed matter physics, is the study of rigid matter, or solids. ... The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925. ... For other uses, see Electron (disambiguation). ... In particle physics, fermions are particles with half-integer spin, such as protons and electrons. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

### Nuclear forces

Main article: Nuclear force

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[38] is the force responsible for the structural integrity of atomic nuclei while the weak nuclear force[39] is responsible for the decay of certain nucleons into leptons and other types of hadrons.[3] This article is about the force sometimes called the residual strong force. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ... In physics a nucleon is a collective name for two baryons: the neutron and the proton. ... For the former Greek currency unit, see Greek drachma. ... A hadron, in particle physics, is a subatomic particle which experiences the nuclear force. ...

The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD).[40] The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The strong interaction is the most powerful of the four fundamental forces. For other uses, see Interaction (disambiguation). ... For other uses, see Quark (disambiguation). ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ... Quantum chromodynamics (abbreviated as QCD) is the theory of the strong interaction (color force), a fundamental force describing the interactions of the quarks and gluons found in hadrons (such as the proton, neutron or pion). ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... In physics, gluons are the bosonic particles which are responsible for the strong nuclear force. ... For other uses, see Quark (disambiguation). ... Corresponding to most kinds of particle, there is an associated antiparticle with the same mass and opposite charges. ... In particle physics, gluons are subatomic particles that cause quarks to interact, and are indirectly responsible for the binding of protons and neutrons together in atomic nuclei. ...

The weak force is due to the exchange of the heavy W and Z bosons. Its most familiar effect is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. 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 Kelvin. Such temperatures have been probed in modern particle accelerators and show the conditions of the universe in the early moments of the Big Bang. In physics, the W and Z bosons are the elementary particles that mediate the weak nuclear force. ... In nuclear physics, beta decay (sometimes called neutron decay) is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted. ... This article or section does not adequately cite its references or sources. ... The nucleus of an atom is the very small dense region, of positive charge, in its centre consisting of nucleons (protons and neutrons). ... Radioactivity may mean: Look up radioactivity in Wiktionary, the free dictionary. ... The strong nuclear force or strong interaction (also called color force or colour force) is a fundamental force of nature which affects only quarks and antiquarks, and is mediated by gluons in a similar fashion to how the electromagnetic force is mediated by photons. ... In physics, the electroweak theory presents a unified description of two of the four fundamental forces of nature: electromagnetism and the weak nuclear force. ... For other uses, see Kelvin (disambiguation). ... Atom Smasher redirects here. ... For other uses, see Universe (disambiguation). ... For other uses, see Big Bang (disambiguation). ...

## Non-fundamental models

Some forces can be modeled by making simplifying assumptions about the physical conditions. In such situations, idealized models can be utilized to gain physical insight.

### Normal force

Fn represents the normal force exerted on the object.
Main article: Normal force

The normal force is the surface force which acts normal to the surface interface between two objects.[41] 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 of an object crashing into an immobile surface. This force is proportional to the square of that object's velocity due to the conservation of energy and the work energy theorem when applied to completely inelastic collisions.[3] Fn represents the normal force. ... Fn represents the normal force. ... A normal vector is a vector which is perpendicular to a surface or manifold. ... This article is about the law of conservation of energy in physics. ... An inelastic collision is a collision in which some of the kinetic energy of the colliding bodies is converted into internal energy in at least one body such that kinetic energy is not conserved. ...

### Friction

Main article: Friction

Friction is a surface force that opposes motion. The frictional force is directly related to the normal force which acts to keep two solid objects separated at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction. For other uses, see Friction (disambiguation). ... Fn represents the normal force. ... Determining the Coefficient of Friction. ... Kinetic friction is the type of friction that an object is subject to after it is in motion. ...

The static friction force (Fsf) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction (μsf) multiplied by the normal force (FN). In other words the magnitude of the static friction force satisfies the inequality: The coefficient of static friction is a physical concept that determines how much force is required before an inert object, of a given material, at rest on another known substance, can be put into motion. ...

$0 le F_{sf} le mu_{sf} F_N$.

The kinetic friction force (Fkf) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force is equal to

Fkf = μkfFN,

where μkf is the coefficient of kinetic friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction.[3]

### Continuum mechanics

When the drag force (Fd) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (Fg), the object reaches a state of dynamical equilibrium at terminal velocity.

$frac{vec{F}}{V} = - vec{nabla} P$

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 buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.[3] In mathematics and physics, a scalar field associates a scalar value, which can be either mathematical in definition, or physical, to every point in space. ... In physics, buoyancy is the upward force on an object produced by the surrounding fluid (i. ... For other uses, see Wind (disambiguation). ... Meteorology is the scientific study of the atmosphere that focuses on weather processes and forecasting. ... The lift force, lifting force or simply lift is a mechanical force generated by a solid object moving through a fluid. ... For the Daft Punk song, see Aerodynamic (song). ... For other uses, see Flight (disambiguation). ...

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 "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction: Velocity pressure is also called fluid dynamic pressure or Q given by the equation. ... For other uses, see Viscosity (disambiguation). ... An object moving through a gas or liquid experiences a force in direction opposite to its motion. ...

$vec{F}_d = - b vec{v} ,$

where:

b is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and
$vec{v}$ is the velocity of the object.[3]

More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as Cross section may refer to the following In geometry, Cross section is the intersection of a 3-dimensional body with a plane. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Stress is a measure of force per unit area within a body. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

$sigma = frac{F}{A}$

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 terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all deformations including also tensile stresses and compressions. In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ... Parallel may refer to: Parallel (geometry) Parallel (latitude), an imaginary east-west line circling a globe Parallelism (grammar), a balance of two or more similar words, phrases, or clauses Parallel (manga), a shÅnen manga by Toshihiko Kobayashi Parallel (video), a video album by R.E.M. The Parallel, an... In any branch of science dealing with materials and their behaviour, strain is the geometrical expression of deformation caused by the action of stress on a physical body. ... Tensile stress (or tension) is the stress state leading to expansion; that is, the length of a material tends to increase in the tensile direction. ... Bold text Wiktionary has related dictionary definitions, such as: compressor, compression inthe wkjhrlfidhb;g/df == Compressor may refer to: Gas compressor, a mechanical device that compresses a gas e. ...

### Tension

Main article: Tension (physics)

### Elastic force

Fk is the force that responds to the load on the spring.

An elastic force acts to return a 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 of the spring from its equilibrium position.[44] This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If Δx is the displacement, the force exerted by an ideal spring is equal to: Elasticity is a branch of physics which studies the properties of elastic materials. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... For other uses, see Spring. ... Look up displacement in Wiktionary, the free dictionary. ... Robert Hooke, FRS (July 18, 1635 â€“ March 3, 1703) was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...

$vec{F}=-k Delta vec{x}$

where k is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the elastic force to act in opposition to the applied load.[3]

### Centripetal force

Main article: Centripetal force

For an object accelerating in circular motion, the unbalanced force acting on the object is equal to[45] The centripetal force is the external force required to make a body follow a circular path at constant speed. ...

$vec{F} = - frac{mv^2 hat{r}}{r}$

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 $hat{r}$ 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.[3] In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length, (or magnitude) is 1. ... This article does not cite any references or sources. ...

### Fictitious forces

Main article: Fictitious forces

There are forces which are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force.[46] These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.[3] 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. In the first part of this article, up until the section on general relativity, the words and expressions are used with their daily life meaning in mind, and they should be read as such. ... Refers to reference frame dependance. ... This article or section is in need of attention from an expert on the subject. ... In physics, the Coriolis effect is an inertial force first described by Gaspard-Gustave Coriolis, a French scientist, in 1835. ... For a generally accessible and less technical introduction to the topic, see Introduction to general relativity. ... Kaluza-Klein theory (or KK theory, for short) is a model which sought to unify classical gravity and electromagnetism. ... This box:      String theory is a still developing mathematical approach to theoretical physics, whose original building blocks are one-dimensional extended objects called strings. ...

## Rotations and torque

Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system.
Main article: Torque

Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque on a particle is defined as the cross-product: Image File history File links Torque_animation. ... This box:      This gyroscope remains upright while spinning due to its angular momentum. ... For other senses of this word, see torque (disambiguation). ... This article is about rotation as a movement of a physical body. ... For other senses of this word, see torque (disambiguation). ... For the cross product in algebraic topology, see KÃ¼nneth theorem. ...

$vec{tau} = vec{r} times vec{F}$

where

$vec{r}$ is the particle's position vector relative to a pivot
$vec{F}$ is the force acting on the particle.

$vec{tau} = Ivec{alpha}$

where

I is the moment of inertia of the particle
$vec{alpha}$ is the angular acceleration of the particle.

This provides a definition for the moment of inertia which is the rotational equivalent for mass. In more advanced treatments of mechanics, the moment of inertia acts as a tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation. Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... For other uses, see Mass (disambiguation). ... Moment of inertia, also called mass moment of inertia or the angular mass, (SI units kg m2, Former British units slug ft2), is the rotational analog of mass. ... Precession redirects here. ... Rotation (green), Precession (blue) and Nutation (red) of the Earth Nutation is a slight irregular motion (etymologically a nodding) in the axis of rotation of a largely axially symmetric object, such as a gyroscope or a planet. ...

Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:

$vec{tau} = frac{dvec{L}}{dt}$[47]

where $vec{L}$ 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,[48] and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques. In physics, angular momentum intuitively measures how much the linear momentum is directed around a certain point called the origin; the moment of momentum. ... -1...

## Kinematic integrals

Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse: In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... Kinematics (Greek ÎºÎ¹Î½ÎµÎ¹Î½,kinein, to move) is a branch of mechanics which describes the motion of objects without the consideration of the masses or forces that bring about the motion. ... For other uses, see Impulse (disambiguation). ...

$vec{I}=int{vec{F} dt}$

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 done by a force: Work (abbreviated W) is the energy transferred in applying force over a distance. ...

$W=int{vec{F} cdot{dvec{x}}}$

which, in a system where all the forces are conservative (see below) is equivalent to changes in kinetic and potential energy (yielding the Work energy theorem). The time derivative of the definition of work gives a definition for power in term of force and the velocity ($vec{v}$): The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... Potential energy can be thought of as energy stored within a physical system. ... In physics, power (symbol: P) is the rate at which work is performed or energy is transmitted, or the amount of energy required or expended for a given unit of time. ...

$P=frac{dW}{dt}=int{vec{F} cdot{dvec{v}}}$

## Potential energy

Main article: 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 work), a potential scalar field $U(vec{r})$ is defined as that field whose gradient is equal and opposite to the force produced at every point: Potential energy can be thought of as energy stored within a physical system. ... Potential energy can be thought of as energy stored within a physical system. ... A gravitational field is a model used within physics to explain how gravity exists in the universe. ... In physics, mechanical work is the amount of energy transferred by a force. ... In mathematics and physics, a scalar field associates a scalar to every point in space. ... For other uses, see Gradient (disambiguation). ...

$vec{F}=-vec{nabla} U.$

Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while non-conservative forces are not.[3] A conservative force is a force which is path-independent. ... For other uses, see Gradient (disambiguation). ... In physics, a potential may refer to the scalar potential or to the vector potential. ...

### Conservative forces

Main article: Conservative force

A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or 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,[49] 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.[3] A conservative force is a force which is path-independent. ... In thermodynamics, a closed system, as contrasted with an isolated system, can exchange heat and work, but not matter, with its surroundings. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... Potential energy can be thought of as energy stored within a physical system. ... In physics, mechanical energy describes the potential energy and kinetic energy present in the components of a mechanical system. ... Example of a topographic map with contour lines Topographic maps, also called contour maps, topo maps or topo quads (for quadrangles), are maps that show topography, or land contours, by means of contour lines. ...

Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models which are dependent on a position often given as a radial vector $vec{r}$ emanating from spherically symmetric potentials.[50] Examples of this follow: Gravity is a force of attraction that acts between bodies that have mass. ... This box:      Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... This article is about an authentication, authorization, and accounting protocol. ... In mechanics and geometry, the rotation group is the set of all rotations of 3-dimensional Euclidean space, R3. ...

For gravity:

$vec{F} = - frac{G m_1 m_2 vec{r}}{r^3}$

where G is the gravitational constant, and mn is the mass of object n. The gravitational constant G is a key element in Newtons law of universal gravitation. ...

For electrostatic forces:

$vec{F} = frac{q_{1} q_{2} vec{r}}{4 pi epsilon_{0} r^3}$

where ε0 is electric permittivity of free space, and qn is the electric charge of object n. Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... This box:      Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ...

For spring forces:

$vec{F} = - k vec{r}$

where k is the spring constant.[3] In physics, Hookes law of elasticity states that if a force (F) is applied to an elastic spring or prismatic rod (with length L and cross section A), its extension is linearly proportional to its tensile stress Ïƒ and modulus of elasticity (E): or It is named after the 17th...

### 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 which yield forces as arising from a macroscopic statistical average of microstates. For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model which is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and 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.[3] In statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system, that the system visits in the course of its thermal fluctuations. ... For other uses, see Friction (disambiguation). ... Properties For alternative meanings see atom (disambiguation). ... For other uses, see Friction (disambiguation). ... In physics, a contact force is a force between two objects (or an object and a surface) that are in contact with each other. ... Tension is a reaction force applied by a stretched string (rope or a similar object) on the objects which stretch it. ... Physical compression is the result of the subjection of a material to compressive stress, resulting in reduction of volume. ... An object moving through a gas or liquid experiences a force in direction opposite to its motion. ...

The connection between macroscopic non-conservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the 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.[3] Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... The second law of thermodynamics is an expression of the universal law of increasing entropy. ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ...

## Units of measurement

The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg•m•s−2.[51] The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g•cm•s−2. 1 newton is thus equal to 100,000 dyne. Look up si, Si, SI in Wiktionary, the free dictionary. ... For other uses, see Newton (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... In physics, the dyne is a unit of force specified in the centimetre-gram-second (cgs) system of units, symbol dyn. One dyne is equal to exactly 10-5 newtons. ...

The foot-pound-second Imperial unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m•s−2.[51] The pound-force provides an alternate unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force.[51] An alternate unit of force in the same system is the poundal, defined as the force required to accelerate a one pound mass at a rate of one foot per second squared.[51] The units of slug and poundal are designed to avoid a constant of proportionality in Newton's Second Law. The Imperial units or the Imperial system is a collection of English units, first defined in the Weights and Measures Act of 1824, later refined (until 1959) and reduced. ... This article is about post-1824 imperial units, see also English unit, U.S. customary units or Avoirdupois. ... The pound-force is a non-SI unit of force or weight (properly abbreviated lbf or lbf). The pound-force is equal to a mass of one pound multiplied by the standard acceleration due to gravity on Earth (which is defined as exactly 9. ... Officially the pound is the name for at least three different units of mass: The pound (avoirdupois). ... g (also gee, g-force or g-load) is a non-SI unit of acceleration defined as exactly 9. ... The slug is an English unit of mass. ... The poundal is a non-SI unit of force. ... The slug is an English unit of mass. ... The poundal is a non-SI unit of force. ... Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ...

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.[51] The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass which accelerates at 1 m•s−2 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 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. The unit kilogram-force (kgf, often just kg) or kilopond (kp) is defined as the force exerted by one kilogram of mass in standard Earth gravity. ... The deprecated unit kilogram-force (kgf) or kilopond (kp) is the force exerted by one kilogram of mass in standard Earth gravity (defined as exactly 9. ... For other meanings, see Slug (disambiguation) The slug is an English and U.S. customary unit of mass. ... This article does not cite any references or sources. ... For other meanings, see Slug (disambiguation) The slug is an English and U.S. customary unit of mass. ... Look up si, Si, SI in Wiktionary, the free dictionary. ...

Units of force
newton
(SI unit)
dyne kilogram-force,
kilopond
pound-force poundal
1 N ≡ 1 kg·m/s² = 105 dyn ≈ 0.10197 kp ≈ 0.22481 lbf ≈ 7.2330 pdl
1 dyn = 10−5 N ≡ 1 g·cm/s² ≈ 1.0197×10−6 kp ≈ 2.2481×10−6 lbf ≈ 7.2330×10−5 pdl
1 kp = 9.80665 N = 980665 dyn gn·(1 kg) ≈ 2.2046 lbf ≈ 70.932 pdl
1 lbf ≈ 4.448222 N ≈ 444822 dyn ≈ 0.45359 kp gn·(1 lb) ≈ 32.174 pdl
1 pdl ≈ 0.138255 N ≈ 13825 dyn ≈ 0.014098 kp ≈ 0.031081 lbf ≡ 1 lb·ft/s²
The value of gn as used in the official definition of the kilogram-force is used here for all gravitational units.

For other uses, see Newton (disambiguation). ... Look up si, Si, SI in Wiktionary, the free dictionary. ... In physics, the dyne is a unit of force specified in the centimetre-gram-second (cgs) system of units, symbol dyn. One dyne is equal to exactly 10-5 newtons. ... The unit kilogram-force (kgf, often just kg) or kilopond (kp) is defined as the force exerted by one kilogram of mass in standard Earth gravity. ... The pound-force is a non-SI unit of force or weight (properly abbreviated lbf or lbf). The pound-force is equal to a mass of one pound multiplied by the standard acceleration due to gravity on Earth (which is defined as exactly 9. ... The poundal is a non-SI unit of force. ... Look up pound in Wiktionary, the free dictionary. ... A foot (plural: feet or foot;[1] symbol or abbreviation: ft or, sometimes, â€² â€“ a prime) is a unit of length, in a number of different systems, including English units, Imperial units, and United States customary units. ... g (also gee, g-force or g-load) is a non-SI unit of acceleration defined as exactly 9. ...

## References

1. ^ glossary. Earth Observatory. NASA. Retrieved on 2008-04-09. “Force: Any external agent that causes a change in the motion of a free body, or that causes stress in a fixed body.”
2. ^ See for example pages 9-1 and 9-2 of Feynman, Leighton and Sands (1963).
3. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z e.g. Feynman, R. P., Leighton, R. B., Sands, M. (1963). Lectures on Physics, Vol 1. Addison-Wesley. ; Kleppner, D., Kolenkow, R. J. (1973). An introduction to mechanics. McGraw-Hill. .
4. ^ a b c d e f g h University Physics, Sears, Young & Zemansky, pp18–38
5. ^ a b Heath,T.L.. The Works of Archimedes (1897). The unabridged work in PDF form (19 MB). Archive.org. Retrieved on 2007-10-14.
6. ^ a b c Weinberg, S. (1994). Dreams of a Final Theory. Vintage Books USA. ISBN 0-679-74408-8
7. ^ Land, Helen The Order of Nature in Aristotle's Physics: Place and the Elements (1998)
8. ^ Hetherington, Norriss S. (1993). Cosmology: Historical, Literary, Philosophical, Religious, and Scientific Perspectives. Garland Reference Library of the Humanities, 100.
9. ^ a b Drake, Stillman (1978). Galileo At Work. Chicago: University of Chicago Press. ISBN 0-226-16226-5
10. ^ a b c d e f g Newton, Isaac (1999). The Principia Mathematical Principles of Natural Philosophy. Berkeley: University of California Press. ISBN 0-520-08817-4.  This is a recent translation into English by I. Bernard Cohen and Anne Whitman, with help from Julia Budenz.
11. ^ DiSalle, Robert (2002-03-30). Space and Time: Inertial Frames. Stanford Encyclopedia of Philosophy. Retrieved on 2008-03-24.
12. ^ Newton's Principia Mathematica actually used a finite difference version of this equation based upon impulse. See Impulse.
13. ^ For example, by Rob Knop PhD in his Galactic Interactions blog on February 26, 2007 at 9:29 a.m. [1]
14. ^ One exception to this rule is: Landau, L. D. (1967). General Physics; mechanics and molecular physics, First English, Oxford: Pergamon Press.  Translated by: J. B. Sykes, A. D. Petford, and C. L. Petford. Library of Congress Catalog Number 67-30260. In section 7, pages 12–14, this book defines force as dp/dt.
15. ^ e.g. W. Noll, “On the Concept of Force”, in part B of Walter Noll's website..
16. ^ Henderson, Tom (1996-2007). Lesson 4: Newton's Third Law of Motion. The Physics Classroom. Retrieved on 2008-01-04.
17. ^ Dr. Nikitin (2007). Dynamics of translational motion. Retrieved on 2008-01-04.
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19. ^ Henderson, Tom (2004). The Physics Classroom. The Physics Classroom and Mathsoft Engineering & Education, Inc.. Retrieved on 2008-01-02.
20. ^ Static Equilibrium. Physics Static Equilibrium (forces and torques). University of the Virgin Islands. Retrieved on 2008-01-02.
21. ^ a b Shifman, Mikhail. ITEP LECTURES ON PARTICLE PHYSICS AND FIELD THEORY. World Scientific.
22. ^ Cutnell. Physics, Sixth Edition, 855–876.
23. ^ Seminar: Visualizing Special Relativity. THE RELATIVISTIC RAYTRACER. Retrieved on 2008-01-04.
24. ^ Wilson, John B.. Four-Vectors (4-Vectors) of Special Relativity: A Study of Elegant Physica. The Science Realm: John's Virtual Sci-Tech Universe. Retrieved on 2008-01-04.
25. ^ Nave, R. Pauli Exclusion Principle. HyperPhysics***** Quantum Physics. Retrieved on 2008-01-02.
26. ^ Fermions & Bosons. The Particle Adventure. Retrieved on 2008-01-04.
27. ^ Cook, A. H. (16-160-1965). "A New Absolute Determination of the Acceleration due to Gravity at the National Physical Laboratory". Nature. doi:10.1038/208279a0.
28. ^ a b University Physics, Sears, Young & Zemansky, pp59–82
29. ^ Sir Isaac Newton: The Universal Law of Gravitation. Astronomy 161 The Solar System. Retrieved on 2008-01-04.
30. ^ Watkins, Thayer. Perturbation Analysis, Regular and Singular. Department of Economics. San José State University.
31. ^ Kollerstrom, Nick (2001). Neptune's Discovery. The British Case for Co-Prediction.. University College London. Archived from the original on 2005-11-11. Retrieved on 2007-03-19.
32. ^ Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik 49: 769-822.
33. ^ Cutnell. Physics, Sixth Edition, 519.
34. ^ Coulomb, Charles (1784). "Recherches théoriques et expérimentales sur la force de torsion et sur l'élasticité des fils de metal". Histoire de l’Académie Royale des Sciences.
35. ^ a b Feynman, Leighton and Sands (2006). The Feynman Lectures on Physics The Definitive Edition Volume II. Pearson Addison Wesley. ISBN 0-8053-9047-2.
36. ^ Duffin, William (1980). Electricity and Magnetism, 3rd Ed.. McGraw-Hill, 364–383. ISBN 0-07-084111-X.
37. ^ For a complete library on quantum mechanics see Quantum_mechanics#References
38. ^ Cutnell. Physics, Sixth Edition, 940.
39. ^ Cutnell. Physics, Sixth Edition, 951.
40. ^ Stevens, Tab (10/07/2003). Quantum-Chromodynamics: A Definition - Science Articles. Retrieved on 2008-01-04.
41. ^ Cutnell. Physics, Sixth Edition, 93.
42. ^ Tension Force. Non-Calculus Based Physics I. Retrieved on 2008-01-04.
43. ^ Fitzpatrick, Richard (2006-02-02). Strings, pulleys, and inclines. Retrieved on 2008-01-04.
44. ^ Elasticity, Periodic Motion. HyperPhysics. Georgia State University. Retrieved on 2008-01-04.
45. ^ Nave, R. Centripetal Force. HyperPhysics***** Mechanics ***** Rotation.
46. ^ Mallette, Vincent (1982-2008). Inwit Publishing, Inc. and Inwit, LLC -- Writings, Links and Software Distributions - The Coriolis Force. Publications in Science and Mathematics, Computing and the Humanities. Inwit Publishing, Inc.. Retrieved on 2008-01-04.
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51. ^ a b c d e Wandmacher, Cornelius (1995). Metric Units in Engineering. ASCE Publications, 15. ISBN 0784400709.

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PDF is an abbreviation with several meanings: Portable Document Format Post-doctoral fellowship Probability density function There also is an electronic design automation company named PDF Solutions. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 33rd day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... Year 2007 (MMVII) was a common year starting on Monday of the Gregorian calendar in the 21st century. ... is the 7th day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ... 2008 (MMVIII) is the current year, a leap year that started on Tuesday of the Anno Domini (or common era), in accordance with the Gregorian calendar. ... is the 4th day of the year in the Gregorian calendar. ...

## Bibliography

• Corbell, H.C.; Philip Stehle (1994). Classical Mechanics p 28,. New York: Dover publications. ISBN 0-486-68063-0.
• Cutnell, John d.; Johnson, Kenneth W. (2004). Physics, Sixth Edition. Hoboken, NJ: John Wiley & Sons Inc.. ISBN 041-44895-8.
• Feynman, R. P., Leighton, R. B., Sands, M. (1963). Lectures on Physics, Vol 1. Addison-Wesley. ISBN 0-201-02116-1.
• Halliday, David; Robert Resnick; Kenneth S. Krane (2001). Physics v. 1. New York: John Wiley & Sons. ISBN 0-471-32057-9.
• Parker, Sybil (1993). Encyclopedia of Physics, p 443,. Ohio: McGraw-Hill. ISBN 0-07-051400-3.
• Sears F., Zemansky M. & Young H. (1982). University Physics. Reading, MA: Addison-Wesley. ISBN 0-201-07199-1.
• Serway, Raymond A. (2003). Physics for Scientists and Engineers. Philadelphia: Saunders College Publishing. ISBN 0-534-40842-7.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics, 5th ed., W. H. Freeman. ISBN 0-7167-0809-4.
• Verma, H.C. (2004). Concepts of Physics Vol 1., 2004 Reprint, Bharti Bhavan. ISBN 81-7709-187-5.

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 Force - Wikipedia, the free encyclopedia (2260 words) Force is a vector quantity defined as the rate of change of the momentum of the body that would be induced by that force acting alone. The SI unit of force is the newton, while the English unit of force is the pound-force. Conservative forces are equivalent to the gradient of a potential, and include gravity, electromagnetic force, and spring force.
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