• Czech: nutit
• Danish: tvinge
• Dutch: dwingen tot, overweldigen
• Finnish: pakottaa
• German: zwingen
• Italian: forzare
• Norwegian: tvinge
• Portuguese: forçar
• Spanish: obligar

• Danish: fremtvinge
• Dutch: afdwingen, forceren
• Finnish: pakottaa
• French: forcer
• German: erzwingen
• Italian: forzare
• Latin: cogo
• Portuguese: forçar
• Spanish: forzar

• ttbc Galician: forzar
• ttbc Interlingua: fortiare
• ttbc Polish: używać siły
• ttbc Romanian: forţa
• ttbc Spanish: forzar
• Volapük: nämön

Derived terms

• enforce
• forceful
• forcible

See also

• Imperial: foot pound
• Metric: newton

French

Pronunciation

• /fɔʀs/, /fORs/
• Rhymes with: -fr|ɔʀs

Etymology

From fortia.

Noun

fr-noun f

1. Force.

In physics, a force is a push or pull that can cause an object with mass to accelerate. Force has both magnitude and direction, making it a vector quantity. According to Newton's Second Law, an object will accelerate in proportion to the net force acting upon it and in inverse proportion to the object's mass. An equivalent formulation is that the net force on an object is equal to the rate of change of momentum it experiences. Forces acting on three-dimensional objects may also cause them to rotate or deform, or result in a change in pressure. The tendency of a force to cause rotation about an axis is termed torque. Deformation and pressure are the result of stress forces within an object.

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. 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. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. The place where forces were applied to projectiles was only at the start of the flight, and while the projectile sailed through the air, no discernible force acts on it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path provided the needed force to continue the projectile moving. This explanation demands that air is needed for projectiles and that, for example, in a vacuum, no projectile would move after the initial push. Additional problems with the explanation include the fact that air resists the motion of the projectiles.

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.

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.

Newton's second law

A modern statement of Newton's second law is a vector differential equation:

\vec = \frac = \frac

where \vec is the momentum of the system. The \vec 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. 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.

The use of Newton's second law as a definition of force has been disparaged in some of the more rigorous textbooks, 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.

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.

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

\vec_=-\vec_.

This law implies that forces always occur in action-reaction pairs. 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_+\vec_=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. 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.

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. 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.

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. When particle A emits (creates) or absorbs (annihilates) particle B, a force accelerates particle A in response to the momentum of particle B, thereby conserving momentum as a whole. This description applies for all forces arising from fundamental interactions. While sophisticated mathematical descriptions are needed to predict, in full detail, the nature of such interactions, there is a conceptually simple way to describe such interactions through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines (similar to waves) and, in the case of virtual particle exchange, are absorbed at an adjacent vertex. When the gauge bosons are represented in a Feynman diagram as existing between two interacting particles, this represents a repulsive force. When the gauge bosons are represented in a Feynman diagram as existing surrounding the two interacting particles, this represents an attractive force.

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". But in order to be conserved, momentum must be redefined as:

\vec = \frac

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:

F_x = \gamma^3 m a_x \,

F_y = \gamma m a_y \,

F_z = \gamma m a_z \,

where the Lorentz factor

\gamma = \frac

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 \gamma 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.

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 F^\mu is the four-force, m is the invariant mass, and A^\mu is the four-acceleration.

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, 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. 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. This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of m will experience a force:

\vec = m\vec

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.

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. 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 m_ due to the gravitational pull of mass m_2 is

\vec=-\frac \hat

where r is the distance between the two objects' centers of mass and \hat is the unit vector pointed in the direction away from the center of the first object toward the center of the second object. were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. These techniques are so powerful that they can be used to predict precisely the motion of celestial bodies to an arbitrary precision at any length of time in the future. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.

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.

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". 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.

Subsequent mathematicians and physicists found the construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force. Thus the electric field anywhere in space is defined as

\vec =

where q is the magnitude of the hypothetical test charge.

Meanwhile, the Lorentz force of magnetism was discovered to exist between two electric currents. It has the same mathematical character as Coulomb's Law with the proviso that like currents attract and unlike currents repel. Similar to the electric field, the magnetic field can be used to determine the magnetic force on an electric current at any point in space. In this case, the magnitude of the magnetic field was determined to be

B =

where I is the magnitude of the hypothetical test current and \ell is the length of hypothetical wire through which the test current flows. The magnetic field exerts a force on all magnets including, for example, those used in compasses. The fact that the 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.

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.

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.

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.

Nuclear forces

There are two "nuclear forces" which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force is the force responsible for the structural integrity of atomic nuclei while the weak nuclear force is responsible for the decay of certain nucleons into leptons and other types of hadrons. 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.

The strong force only acts directly upon elementary particles. However, a residual of the force is observed between hadrons (the best known example being the force that acts between nucleons in atomic nuclei) as the nuclear force. Here the strong force acts indirectly, transmitted as gluons which form part of the virtual pi and rho mesons which classically transmit the nuclear force (see this topic for more). The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called colour confinement.

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.

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

The normal force is the surface force which acts normal to the surface interface between two objects. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force 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. By connecting the same string multiple times to the same object through the use of a set-up that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. However, even though such machines allow for an increase in force, there is a corresponding increase in the length of string that must be displaced in order to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine. This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If \Delta x is the displacement, the force exerted by an ideal spring is equal to:

\vec=-k \Delta \vec

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

\vec = - \frac

where m is the mass of the object, v is the velocity of the object and r is the distance to the center of the circular path and \hat is the unit vector 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. These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.

where \vec is the angular momentum of the particle.

Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques, and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.

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:

\vec=\int

which, by Newton's Second Law, must be equivalent to the change in momentum (yielding the Impulse momentum theorem).

Similarly, integrating with respect to position gives a definition for the work done by a force:

W=\int

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):

P=\frac=\int

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) is defined as that field whose gradient is equal and opposite to the force produced at every point:

\vec=-\vec U.

Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while non-conservative forces are not. 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. Examples of this follow:

For gravity:

\vec = - \frac

where G is the gravitational constant, and m_n is the mass of object n.

For electrostatic forces:

\vec = \frac

where \epsilon_ is electric permittivity of free space, and q_n is the electric charge of object n.

For spring forces:

\vec = - k \vec

where k is the spring constant. 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.

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. 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. 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. The units of slug and poundal are designed to avoid a constant of proportionality in Newton's Second Law.

The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass 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.

References

Bibliography

• Lectures on Physics, Vol 1

External links

• Video lecture on Newton's three laws by Walter Lewin from MIT OpenCourseWare

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