Forces can be described as a push or pull on an object. They can be due to phenomena such asgravity,magnetism, or anything that might cause a mass to accelerate.
Inphysics, aforce is an action (usually a push or a pull) that can cause anobject to change itsvelocity or its shape, or to resist other forces, or to cause changes of pressure in a fluid. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because themagnitude anddirection of a force are both important, force is avector quantity (force vector). TheSI unit of force is thenewton (N), and force is often represented by the symbolF.
Force plays an important role in classical mechanics. The concept of force is central to all three ofNewton's laws of motion. Types of forces often encountered inclassical mechanics includeelastic,frictional,contact or "normal" forces, andgravitational. The rotational version of force istorque, which produceschanges in the rotational speed of an object. In an extended body, each part applies forces on the adjacent parts; the distribution of such forces through the body is the internalmechanical stress. In the case of multiple forces, if the net force on an extended body is zero the body is in equilibrium.
Inmodern physics, which includesrelativity andquantum mechanics, the laws governing motion are revised to rely onfundamental interactions as the ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes.[1]
Philosophers inantiquity used the concept of force in the study ofstationary andmoving objects andsimple machines, but thinkers such asAristotle andArchimedes retained fundamental errors in understanding force. In part, this was due to an incomplete understanding of the sometimes non-obvious force offriction and a consequently inadequate view of the nature of natural motion.[2] A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected byGalileo Galilei andSir Isaac Newton. With his mathematical insight, Newton formulatedlaws of motion that were not improved for over two hundred years.[3]
By the early 20th century,Einstein developed atheory 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 andinertia. With modern insights intoquantum mechanics and technology that can accelerate particles close to the speed of light,particle physics has devised aStandard Model to describe forces between particles smaller than atoms. TheStandard Model predicts that exchanged particles calledgauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are:strong,electromagnetic,weak, andgravitational.[4]: 2–10 [5]: 79 High-energy particle physicsobservations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamentalelectroweak interaction.[6]
Aristotle famously described a force as anything that causes an object to undergo "unnatural motion"
Since antiquity the concept of force has been recognized as integral to the functioning of each of thesimple machines. Themechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount ofwork. Analysis of the characteristics of forces ultimately culminated in the work ofArchimedes who was especially famous for formulating a treatment ofbuoyant forces inherent influids.[2]
Aristotle provided aphilosophical discussion of the concept of a force as an integral part ofAristotelian cosmology. In Aristotle's view, the terrestrial sphere contained fourelements 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.[7] 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 ofprojectiles, 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.[8]
ThoughAristotelian physics was criticized as early as the 6th century,[9][10] its shortcomings would not be corrected until the 17th century work ofGalileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force ofimpetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove theAristotelian theory of motion. He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain theirvelocity unless acted on by a force, for examplefriction.[11] Galileo's idea that force is needed to change motion rather than to sustain it, further improved upon byIsaac Beeckman,René Descartes, andPierre Gassendi, became a key principle of Newtonian physics.[12]
In the early 17th century, before Newton'sPrincipia, the term "force" (Latin: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 namedvis viva (live force) byLeibniz. The modern concept of force corresponds to Newton'svis motrix (accelerating force).[13]
Sir Isaac Newton described the motion of all objects using the concepts ofinertia and force. In 1687, Newton published his magnum opus,Philosophiæ Naturalis Principia Mathematica.[3][14] In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day.[14] The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.[15]
Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line.[14] The latter follows from the former because of theprinciple that the laws of physics are the same for allinertial observers, i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.[16]: 1–7
According to the first law, motion at constant speed in a straight line does not need a cause. It ischange in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion.Newton's second law states that the net force acting upon an object is equal to therate at which itsmomentum changes withtime. If the mass of the object is constant, this law implies that theacceleration of an object is directlyproportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to themass of the object.[17]: 204–207
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.[17]: 399 If a body is in equilibrium, there is zeronet force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is anunbalanced force acting on an object it will result in the object's momentum changing over time.[14]
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,wherem is themass and is thevelocity.[4]: 9-1,9-2 If Newton's second law is applied to a system ofconstant mass,m may be moved outside the derivative operator. The equation then becomesBy substituting the definition ofacceleration, the algebraic version ofNewton's second law is derived:
Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if is the force of body 1 on body 2 and that of body 2 on body 1, thenThis law is sometimes referred to as theaction-reaction law, with called theaction and thereaction.
Newton's third law is a result of applyingsymmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces areinteractions between different bodies.[18][19] and thus that there is no such thing as a unidirectional force or a force that acts on only one body.
In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:More generally, in aclosed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but thecenter 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.[4]: 19-1 [5]
Combining Newton's second and third laws, it is possible to show that thelinear momentum of a system is conserved in anyclosed system. In a system of two particles, if is the momentum of object 1 and the momentum of object 2, thenUsing 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.[4]: ch.12 [5]
Some textbooks use Newton's second law as adefinition of force.[20][21][22][23] However, for the equation for a constant mass to then have any predictive content, it must be combined with further information.[24][4]: 12-1 Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference.[5]: 59 The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways,[25][26]: vii which ultimately do not affect how the theory is used in practice.[25] Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force includeErnst Mach andWalter Noll.[27][28]
Forces act in a particulardirection and havesizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denotedscalar 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 theresult. If both of these pieces of information are not known for each force, the situation is ambiguous.[17]: 197
Historically, forces were first quantitatively investigated in conditions ofstatic equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additivevector quantities: they havemagnitude and direction.[3] When two forces act on apoint particle, the resulting force, theresultant (also called thenet force), can be determined by following theparallelogram rule ofvector 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.[4]: ch.12 [5]
Free body diagrams of a block on a flat surface and aninclined plane. Forces are resolved and added together to determine their magnitudes and the net force.
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 thatgraphical vector addition can be done to determine the net force.[29]
As well as being added, forces can also be resolved into independent components atright 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 ofbasis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions.[30] This is because, fororthogonal 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.[4]: ch.12 [5]
When all the forces that act upon an object are balanced, then the object is said to be in a state ofequilibrium.[17]: 566 Hence, equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extended body, it is also necessary that the net torque be zero. A body is instatic equilibrium with respect to a frame of reference if it at rest and not accelerating, whereas a body indynamic equilibrium is moving at a constant speed in a straight line, i.e., moving but not accelerating. What one observer sees as static equilibrium, another can see as dynamic equilibrium and vice versa.[17]: 566
Static equilibrium was understood well before the invention of classical mechanics. Objects that are not accelerating have zero net force acting on them.[31]
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 anormal force). The situation produces zero net force and hence no acceleration.[3]
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 bystatic friction, generated between the object and the table surface. For a situation with no movement, the static friction forceexactly 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.[3]
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such asweighing scales andspring balances. For example, an object suspended on a verticalspring 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 constantdensity (widely exploited for millennia to define standard weights);Archimedes' principle for buoyancy; Archimedes' analysis of thelever;Boyle's law for gas pressure; andHooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded histhree laws of motion.[3][4]: ch.12 [5]
Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's description of forces.
Dynamic equilibrium was first described byGalileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations andlogic. Galileo realized thatsimple velocity addition demands that the concept of an "absoluterest frame" did not exist. Galileo concluded that motion in a constantvelocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. When this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity.[11]
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 withkinetic 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.[4]: ch.12 [5]
Some forces are consequences of the fundamental ones. In such situations, idealized models can be used to gain physical insight. For example, each solid object is considered arigid body.[citation needed]
Images of a freely falling basketball taken with astroboscope at 20 flashes per second. The distance units on the right are multiples of about 12 millimeters. The basketball starts at rest. At the time of the first flash (distance zero) it is released, after which the number of units fallen is equal to the square of the number of flashes.
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 theacceleration of every object infree-fall was constant and independent of the mass of the object. Today, thisacceleration due to gravity towards the surface of the Earth is usually designated as and has a magnitude of about 9.81meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth.[32] 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 will experience a force:
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 anormal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.[4]: ch.12 [5]
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 alaw of gravity that could account for the celestial motions that had been described earlier usingKepler's laws of planetary motion.[33]
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 aninverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body.[33] Combining these ideas gives a formula that relates the mass () and the radius () of the Earth to the gravitational acceleration:where the vector direction is given by, is theunit vector directed outward from the center of the Earth.[14]
In this equation, a dimensional constant is used to describe the relative strength of gravity. This constant has come to be known as theNewtonian constant of gravitation, though its value was unknown in Newton's lifetime. Not until 1798 wasHenry Cavendish able to make the first measurement of using atorsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed the samelaws of motion, his law of gravity had to be universal. Succinctly stated,Newton's law of gravitation states that the force on a spherical object of mass due to the gravitational pull of mass iswhere 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.[14]
This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within theSolar System until the 20th century. During that time, sophisticated methods ofperturbation analysis[34] were invented to calculate the deviations oforbits due to the influence of multiple bodies on aplanet,moon,comet, orasteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planetNeptune before it was observed.[35]
Theelectrostatic force was first described in 1784 by Coulomb as a force that existed intrinsically between twocharges.[36]: 519 The properties of the electrostatic force were that it varied as aninverse square law directed in theradial direction, was both attractive and repulsive (there was intrinsicpolarity), was independent of the mass of the charged objects, and followed thesuperposition principle.Coulomb's law unifies all these observations into one succinct statement.[37]
Subsequent mathematicians and physicists found the construct of theelectric 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.[38]: 4-6–4-8 Thus the electric field anywhere in space is defined aswhere is the magnitude of the hypothetical test charge. Similarly, the idea of themagnetic field was introduced to express how magnets can influence one another at a distance. TheLorentz force law 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 thevelocity of the particle. The magnetic contribution to the Lorentz force is thecross product of the velocity vector with the magnetic field.[39][40]: 482
The origin of electric and magnetic fields would not be fully explained until 1864 whenJames Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations byOliver Heaviside andJosiah Willard Gibbs.[41] These "Maxwell's 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 awave that traveled at a speed that he calculated to be thespeed of light. This insight united the nascent fields of electromagnetic theory withoptics and led directly to a complete description of theelectromagnetic spectrum.[42]
When objects are in contact, the force directly between them is called the normal force, the component of the total force in the system exerted normal to the interface between the objects.[36]: 264 The normal force is closely related to Newton's third law. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.[4]: ch.12 [5]
Friction is a force that opposes relative motion of two bodies. At the macroscopic scale, the frictional force is directly related to the normal force at the point of contact. There are two broad classifications of frictional forces:static friction andkinetic friction.[17]: 267
The static friction force () will exactly oppose forces applied to an object parallel to a surface up to the limit specified by thecoefficient 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 thecoefficient of kinetic friction. The coefficient of kinetic friction is normally less than the coefficient of static friction.[17]: 267–271
Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with idealpulleys, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action–reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object.[43] By connecting the same string multiple times to the same object through the use of a configuration 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. Such machines allow amechanical advantage for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in theconservation of mechanical energy since thework done on the load is the same no matter how complicated the machine.[4]: ch.12 [5][44]
Fk is the force that responds to the load on the spring
A simple elastic force acts to return aspring to its natural length. Anideal 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 thedisplacement of the spring from its equilibrium position.[45] This linear relationship was described byRobert Hooke in 1676, for whomHooke's law is named. If is the displacement, the force exerted by an ideal spring equals:where is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to the applied load.[4]: ch.12 [5]
For an object inuniform circular motion, the net force acting on the object equals:[46]where is the mass of the object, is the velocity of the object and is the distance to the center of the circular path and is theunit vector pointing in the radial direction outwards from the center. This means that the net force felt by the 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 thespeed of the object (magnitude of the velocity), but only the direction of the velocity vector. More generally, the net 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.[4]: ch.12 [5]
When the drag force () associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (), the object reaches a state ofdynamic equilibrium atterminal velocity.
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealizedpoint particles rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories ofcontinuum mechanics describe the way forces affect the material. For example, in extendedfluids, differences inpressure result in forces being directed along the pressuregradients as follows:
where is the volume of the object in the fluid and is thescalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in thebuoyant force for fluids suspended in gravitational fields, winds inatmospheric science, and thelift associated withaerodynamics andflight.[4]: ch.12 [5]
A specific instance of such a force that is associated withdynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due toviscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:where:
is a constant that depends on the properties of the fluid and the dimensions of the object (usually thecross-sectional area), and
More formally, forces incontinuum mechanics are fully described by astresstensor with terms that are roughly defined aswhere 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 (thematrix diagonals of the tensor) as well asshear terms associated with forces that actparallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause allstrains (deformations) including alsotensile stresses andcompressions.[3][5]: 133–134 [38]: 38-1–38-11
There are forces that areframe dependent, meaning that they appear due to the adoption of non-Newtonian (that is,non-inertial)reference frames. Such forces include thecentrifugal force and theCoriolis force.[47] These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.[4]: ch.12 [5] Because these forces are not genuine they are also referred to as "pseudo forces".[4]: 12-11
Ingeneral relativity,gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.[48]
Forces that cause extended objects to rotate are associated withtorques. Mathematically, the torque of a force is defined relative to an arbitrary reference point as thecross product:where is theposition vector of the force application point relative to the reference point.[17]: 497
Torque is the rotation equivalent of force in the same way thatangle is the rotational equivalent forposition,angular velocity forvelocity, andangular momentum formomentum. As a consequence of Newton's first law of motion, there existsrotational 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 instantaneousangular 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 thetensor that, when properly analyzed, fully determines the characteristics of rotations includingprecession andnutation.[26]: 96–113
Equivalently, the differential form of Newton's second law provides an alternative definition of torque:[49]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,[50] and therefore also directly implies theconservation of angular momentum for closed systems that experience rotations andrevolutions through the action of internal torques.
Theyank is defined as the rate of change of force[51]: 131
The term is used in biomechanical analysis,[52] athletic assessment[53] and robotic control.[54] The second ("tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.[51]
Forces can be used to define a number of physical concepts byintegrating with respect tokinematic variables. For example, integrating with respect to time gives the definition ofimpulse:[55]which by Newton's second law must be equivalent to the change in momentum (yielding theImpulse momentum theorem).
Similarly, integrating with respect to position gives a definition for thework done by a force:[4]: 13-3 which is equivalent to changes inkinetic energy (yielding thework energy theorem).[4]: 13-3
PowerP is the rate of change dW/dt of the workW, as thetrajectory is extended by a position change in a time interval dt:[4]: 13-2 sowith thevelocity.
Instead of a force, often the mathematically related concept of apotential energy field is used. For instance, the gravitational force acting upon an object can be seen as the action of thegravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition ofwork), a potentialscalar field is defined as that field whosegradient is equal and opposite to the force produced at every point:
Forces can be classified asconservative or nonconservative. Conservative forces are equivalent to the gradient of apotential while nonconservative forces are not.[4]: ch.12 [5]
A conservative force that acts on aclosed system has an associated mechanical work that allows energy to convert only betweenkinetic orpotential forms. This means that for a closed system, the netmechanical 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,[56] 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 thecontour map of the elevation of an area.[4]: ch.12 [5]
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 ofmicrostates. For example, static friction is caused by the gradients of numerous electrostatic potentials between theatoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include othercontact forces,tension,compression, anddrag. 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.[4]: ch.12 [5]
The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment withstatistical mechanics. In macroscopic closed systems, nonconservative forces act to change theinternal energies of the system, and are often associated with the transfer of heat. According to theSecond law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions asentropy increases.[4]: ch.12 [5]
TheSI unit of force is thenewton (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.The correspondingCGS unit is thedyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s−2. A newton is thus equal to 100,000 dynes.[58]
The gravitationalfoot-pound-secondEnglish unit of force is thepound-force (lbf), defined as the force exerted by gravity on apound-mass in thestandard gravitational field of 9.80665 m·s−2.[58] The pound-force provides an alternative unit of mass: oneslug is the mass that will accelerate by one foot per second squared when acted on by one pound-force.[58] An alternative unit of force in a different foot–pound–second system, the absolute fps system, is thepoundal, defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared.[58]
The pound-force has a metric counterpart, less commonly used than the newton: thekilogram-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: themetric 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.[58]
At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly.
In thespecial theory of relativity, mass andenergy 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,remains valid because it is a mathematical definition.[36]: 855–876 But for momentum to be conserved at relativistic relative velocity,, momentum must be redefined as:where is therest mass and thespeed of light.
The expression relating force and acceleration for a particle with constant non-zerorest mass moving in the direction at velocity is:[59]: 216 whereis called theLorentz 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.[59]: 26 [4]: §15–8 If is very small compared to, then is very close to 1 andis a close approximation. Even for use in relativity, one can restore the form ofthrough the use offour-vectors. This relation is correct in relativity when is thefour-force, is theinvariant mass, and is thefour-acceleration.[60]
Thegeneral 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 fullybelow.
Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an objecthas, one considers what result mightappear when ameasurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.[61][62] Theexpectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.[63]
In quantum mechanics, interactions are typically described in terms of energy rather than force. TheEhrenfest 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, theBorn 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.[64][65]
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, theuncertainty principle limits the minimum extent of an electron probability distribution[66] and thePauli exclusion principle prevents electrons from sharing the same probability distribution.[67] This gives rise to an emergent pressure known asdegeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solidsstability.[68]
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 ofFeynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (seeworld 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 interactionvertices, 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.[69] The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture offundamental 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 aneutrondecays into anelectron,proton, andantineutrino, an interaction mediated by the same gauge boson that is responsible for theweak nuclear force.[69]
The fundamental theories for forces developed from theunification of different ideas. For example, Newton's universal theory ofgravitation showed that the force responsible for objects falling near the surface of theEarth is also the force responsible for the falling of celestial bodies about the Earth (theMoon) and around the Sun (the planets).Michael Faraday andJames Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development ofquantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchangingvirtual particles calledgauge bosons.[70] ThisStandard Model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces inelectroweak theory, which was subsequently confirmed by observation.[71]
Instruments like GRAVITY provide a powerful probe for gravity force detection.[73]
Newton's law of gravitation is an example ofaction at a distance: one body, like the Sun, exerts an influence upon any other body, like the Earth, no matter how far apart they are. Moreover, this action at a distance isinstantaneous. According to Newton's theory, the one body shifting position changes the gravitational pulls felt by all other bodies, all at the same instant of time.Albert Einstein recognized that this was inconsistent with special relativity and its prediction that influences cannot travel faster than thespeed of light. So, he sought a new theory of gravitation that would be relativistically consistent.[74][75]Mercury's orbit did not match that predicted by Newton's law of gravitation. Some astrophysicists predicted the existence of an undiscovered planet (Vulcan) that could explain the discrepancies. When Einstein formulated his theory ofgeneral relativity (GR) he focused on Mercury's problematic orbit and found that his theory addeda correction, which could account for the discrepancy. This was the first time that Newton's theory of gravity had been shown to be inexact.[76]
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 instraight lines throughcurved 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 theballistictrajectory of the object. For example, abasketball thrown from the ground moves in aparabola, as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with theradius of curvature of the order of fewlight-years). The time derivative of the changing momentum of the object is what we label as "gravitational force".[5]
Maxwell's equations and the set of techniques built around them adequately describe a wide range of physics involving force in electricity and magnetism. This classical theory already includes relativity effects.[77] Understanding quantized electromagnetic interactions between elementary particles requiresquantum electrodynamics (QED). In QED, photons are fundamental exchange particles, describing all interactions relating to electromagnetism including the electromagnetic force.[78]
There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. Thestrong nuclear force is the force responsible for the structural integrity ofatomic nuclei, and gains its name from its ability to overpower the electromagnetic repulsion between protons.[36]: 940 [79]
The strong force is today understood to represent theinteractions betweenquarks andgluons as detailed by the theory ofquantum chromodynamics (QCD).[80] The strong force is thefundamental force mediated by gluons, acting upon quarks,antiquarks, and the gluons themselves. The strong force only actsdirectly upon elementary particles. A residual is observed betweenhadrons (notably, thenucleons in atomic nuclei), known as thenuclear force. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rhomesons, the classical transmitters of the nuclear force. The failure of many searches forfree quarks has shown that the elementary particles affected are not directly observable. This phenomenon is calledcolor confinement.[81]: 232
Unique among the fundamental interactions, the weak nuclear force creates no bound states.[82] The weak force is due to the exchange of the heavyW and Z bosons. Since the weak force is mediated by two types of bosons, it can be divided into two types of interaction or "vertices" —charged current, involving the electrically charged W+ and W− bosons, andneutral current, involving electrically neutral Z0 bosons. The most familiar effect of weak interaction isbeta decay (of neutrons in atomic nuclei) and the associatedradioactivity.[36]: 951 This is a type of charged-current interaction. The word "weak" derives from the fact that the field strength is some 1013 times less than that of thestrong 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 approximately1015K.[83] Such temperatures occurred in the plasma collisions in the early moments of theBig Bang.[82]: 201
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