A force is an influence that can cause an Physical object to change its velocity, i.e., to accelerate, meaning a change in speed or direction, unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a Euclidean vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol .
Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include elastic, , Normal force, and gravity. The rotational version of force is torque, which produces changes in the rotational speed of an object. In an extended body, each part often applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. In equilibrium these stresses cause no acceleration of the body as the forces balance one another. If these are not in equilibrium they can cause deformation of solid materials, or flow in .
In modern physics, which includes relativity and quantum mechanics, the laws governing motion are revised to rely on fundamental interactions as the ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes.
By the early 20th century, Albert Einstein developed a theory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light and also provided insight into the forces produced by gravitation and inertia. With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong force, electromagnetic, weak force, and gravitational.
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, were in their natural place when on the ground, and that they stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of , such as the flight of arrows. An archer causes the arrow to move at the start of the flight, and it then sails through the air even though no discernible efficient cause acts upon it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation requires a continuous medium such as air to sustain the motion.
Though Aristotelian physics was criticized as early as the 6th century,
In the early 17th century, before Newton's Principia, the term "force" (vis) was applied to many physical and non-physical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named vis viva (live force) by Leibniz. The modern concept of force corresponds to Newton's vis motrix (accelerating force).
A modern statement of Newton's second law is a vector equation: where is the momentum of the system, and is the net (vector sum) force. If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time.
In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum,
where m is the mass and is the velocity. If Newton's second law is applied to a system of constant mass, m may be moved outside the derivative operator. The equation then becomes
By substituting the definition of acceleration, the algebraic version of Newton's second law is derived:
Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies.
In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:
More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system.
Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved in any closed system. In a system of two particles, if is the momentum of object 1 and the momentum of object 2, then
Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.
Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction. When two forces act on a point particle, the resulting force, the resultant (also called the net force), can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action.
Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the net force.
As well as being added, forces can also be resolved into independent components at to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right angles to the other two.
The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration.
Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object.
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as and . For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his Three Laws of Motion.
Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity. A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. When kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion.
For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.
Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion.
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body. Combining these ideas gives a formula that relates the mass () and the radius () of the Earth to the gravitational acceleration:
where is the distance between the two objects' centers of mass and is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.
This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the 20th century. During that time, sophisticated methods of perturbation analysis were invented to calculate the deviations of due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed.
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
where is the magnitude of the hypothetical test charge. Similarly, the idea of the magnetic field was introduced to express how magnets can influence one another at a distance. The Lorentz force gives the force upon a body with charge due to electric and magnetic fields:
where is the electromagnetic force, is the electric field at the body's location, is the magnetic field, and is the velocity of the particle. The magnetic contribution to the Lorentz force is the cross product of the velocity vector with the magnetic field.
The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs.
The static friction force () will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the coefficient of static friction () multiplied by the normal force (). In other words, the magnitude of the static friction force satisfies the inequality:
The kinetic friction force () is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:
where is the coefficient of kinetic friction. The coefficient of kinetic friction is normally less than the coefficient of static friction.
where is the volume of the object in the fluid and is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyancy for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight.
A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:
where:
More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as
where 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 of the tensor) as well as Shear stress 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 strains (deformations) including also and compressions.
In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.
Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. As a consequence of Newton's first law of motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body:
where
This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation.
Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:
where 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.
Similarly, integrating with respect to position gives a definition for the work done by a force:
which is equivalent to changes in kinetic energy (yielding the work energy theorem).
Power P is the rate of change d W/d t of the work W, as the trajectory is extended by a position change in a time interval d t:
so
with the velocity.
Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.
Conservative forces include gravity, the Electromagnetism force, and the spring force. Each of these forces has models that are dependent on a position often given as a radius emanating from spherically symmetric potentials. Examples of this follow:
For gravity:
where is the gravitational constant, and is the mass of object n.
For electrostatic forces:
where is Permittivity, and is the electric charge of object n.
For spring forces:
where is the spring constant.
For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of microstates. For example, static friction is caused by the gradients of numerous electrostatic potentials between the , but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other , tension, compression, and drag. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials.
The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energy of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.
The gravitational foot-pound-second English unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the Standard gravity field of 9.80665 m·s−2. The pound-force provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force. An alternative unit of force in a different foot–pound–second system, the absolute fps system, is the poundal, defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared.
The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass that accelerates at 1 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, sometimes used for expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque.
The expression relating force and acceleration for a particle with constant non-zero rest mass moving in the direction at velocity is:
where
is called the Lorentz factor. The Lorentz factor increases steeply as the relative velocity approaches the speed of light. Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach .
If is very small compared to , then is very close to 1 and
is a close approximation. Even for use in relativity, one can restore the form of
through the use of four-vectors. This relation is correct in relativity when is the four-force, is the invariant mass, and is the four-acceleration.
The general theory of relativity incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully below.
In quantum mechanics, interactions are typically described in terms of energy rather than force. The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.
Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the uncertainty principle limits the minimum extent of an electron probability distribution and the Pauli exclusion principle prevents electrons from sharing the same probability distribution. This gives rise to an emergent pressure known as degeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids stability.
While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of . 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 and, in the case of virtual particle exchange, are absorbed at an adjacent vertex. The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron beta decay into an electron, proton, and neutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force.
The fundamental theories for forces developed from the unification of different ideas. For example, Newton's universal theory of gravitation showed that the force responsible for objects falling near the surface of the Earth is also the force responsible for the falling of celestial bodies about the Earth (the Moon) and around the Sun (the planets). Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter () interacting by exchanging called . This Standard Model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory, which was subsequently confirmed by observation.
Since then, general relativity has been acknowledged as the theory that best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in geodesic through curved spacetime – defined as the shortest spacetime path between two spacetime events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of spacetime can be observed and the force is inferred from the object's curved path. Thus, the straight line path in spacetime is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with the radius of curvature of the order of few ). The time derivative of the changing momentum of the object is what we label as "gravitational force".
The strong force is today understood to represent the interactions between and as detailed by the theory of quantum chromodynamics (QCD). The strong force is the fundamental force mediated by gluons, acting upon quarks, antiparticle, and the gluons themselves. The strong force only acts directly upon elementary particles. A residual is observed between (notably, the in atomic nuclei), known as the nuclear force. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rho , the classical transmitters of the nuclear force. The failure of many searches for has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement.
Third law
Defining "force"
Combining forces
Equilibrium
Static
Dynamic
Examples of forces in classical mechanics
Gravitational
Electromagnetic
Normal
Friction
Tension
Spring
Centripetal
Continuum mechanics
Fictitious
Concepts derived from force
Rotation and torque
Yank
The term is used in biomechanical analysis, athletic assessment and robotic control. The second (called "tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.
Kinematic integrals
Potential energy
Conservation
Units
Revisions of the force concept
Special theory of relativity
Quantum mechanics
Quantum field theory
Fundamental interactions
Gravitational
Electromagnetic
Strong nuclear
Weak nuclear
See also
External links
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