Force

== Examples of forces in classical mechanics <span class="anchor" id="Non-fundamental forces"></span> ==
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 a [[rigid body]].{{Citation needed|date=January 2024}}

=== Gravitational ===
{{main|Gravity}}
[[File:Falling ball.jpg|upright|thumb|Images of a freely falling basketball taken with a [[stroboscope]] at 20 flashes per second. The distance units on the right are multiples of about 12&nbsp;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 the [[acceleration]] of every object in [[free-fall]] was constant and independent of the mass of the object. Today, this [[Gravitational acceleration|acceleration due to gravity]] towards the surface of the Earth is usually designated as <math> \vec{g}</math> and has a magnitude of about 9.81 [[meter]]s per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth.<ref>{{cite journal |last=Cook |first=A. H. |journal=Nature |title=A New Absolute Determination of the Acceleration due to Gravity at the National Physical Laboratory |date=1965 |doi=10.1038/208279a0 |page=279 |volume=208 |bibcode=1965Natur.208..279C |issue=5007 |s2cid=4242827 |doi-access=free }}</ref> 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 <math>m</math> will experience a force:
<math display="block">\vec{F} = m\vec{g}.</math>

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.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

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 [[Newton's law of gravity|law of gravity]] that could account for the celestial motions that had been described earlier using [[Kepler's laws of planetary motion]].<ref name=uniphysics_ch4 />

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.<ref name=uniphysics_ch4 /> Combining these ideas gives a formula that relates the mass (<math> m_\oplus</math>) and the radius (<math> R_\oplus</math>) of the Earth to the gravitational acceleration:
<math display="block" qid=Q30006>\vec{g}=-\frac{Gm_\oplus}{{R_\oplus}^2} \hat{r},</math>
where the vector direction is given by <math>\hat{r}</math>, is the [[unit vector]] directed outward from the center of the Earth.<ref name="Principia"/>

In this equation, a dimensional constant <math>G</math> is used to describe the relative strength of gravity. This constant has come to be known as the [[Newtonian constant of gravitation]], though its value was unknown in Newton's lifetime. Not until 1798 was [[Henry Cavendish]] able to make the first measurement of <math>G</math> using a [[torsion balance]]; this was widely reported in the press as a measurement of the mass of the Earth since knowing <math>G</math> could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed the same [[Kepler's laws|laws 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 <math>m_1</math> due to the gravitational pull of mass <math>m_2</math> is
<math display="block" qid=Q11412>\vec{F}=-\frac{Gm_{1}m_{2}}{r^2} \hat{r},</math>
where <math>r</math> is the distance between the two objects' centers of mass and <math>\hat{r}</math> is the unit vector pointed in the direction away from the center of the first object toward the center of the second object.<ref name="Principia"/>

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]]<ref>{{cite web |last=Watkins |first=Thayer |title=Perturbation Analysis, Regular and Singular |work=Department of Economics |publisher=San José State University |url=http://www.sjsu.edu/faculty/watkins/perturb.htm |access-date=2008-01-05 |archive-date=2011-02-10 |archive-url=https://web.archive.org/web/20110210010802/http://www.sjsu.edu/faculty/watkins/perturb.htm |url-status=dead }}</ref> were invented to calculate the deviations of [[orbit]]s 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.<ref name='Neptdisc'>{{cite web |url=http://www.ucl.ac.uk/sts/nk/neptune/index.htm |title=Neptune's Discovery. The British Case for Co-Prediction. |access-date=2007-03-19 |last=Kollerstrom |first=Nick |year=2001 |publisher=University College London |archive-url= https://web.archive.org/web/20051111190351/http://www.ucl.ac.uk/sts/nk/neptune/index.htm |archive-date=2005-11-11}}</ref>

=== Electromagnetic ===
{{main|Electromagnetic force}}
The [[electrostatic force]] was first described in 1784 by Coulomb as a force that existed intrinsically between two [[electric charge|charges]].<ref name=Cutnell/>{{rp|519}} The properties of the electrostatic force were that it varied as an [[inverse square law]] directed in the [[polar coordinates|radial direction]], was both attractive and repulsive (there was intrinsic [[Electrical polarity|polarity]]), was independent of the mass of the charged objects, and followed the [[superposition principle]]. [[Coulomb's law]] unifies all these observations into one succinct statement.<ref name="Coulomb">{{cite journal |first=Charles |last=Coulomb |journal=Histoire de l'Académie Royale des Sciences |year=1784 |title=Recherches théoriques et expérimentales sur la force de torsion et sur l'élasticité des fils de metal |pages=229–269}}</ref>

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.<ref name=FeynmanVol2/>{{rp|((4-6–4-8))}} Thus the electric field anywhere in space is defined as
<math display="block">\vec{E} = {\vec{F} \over{q}},</math>
where <math>q</math> 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|Lorentz force law]] gives the force upon a body with charge <math>q</math> due to electric and magnetic fields:
<math display="block" qid=Q849919>\vec{F} = q\left(\vec{E} + \vec{v} \times \vec{B}\right),</math>
where <math> \vec{F}</math> is the electromagnetic force, <math> \vec{E}</math> is the electric field at the body's location, <math>\vec{B}</math> is the magnetic field, and <math> \vec{v}</math> 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.<ref>{{Cite book|last=Tonnelat|first=Marie-Antoinette|url=https://www.worldcat.org/oclc/844001|title=The principles of electromagnetic theory and of relativity.|date=1966|publisher=D. Reidel|isbn=90-277-0107-5|location=Dordrecht|oclc=844001|author-link=Marie-Antoinette Tonnelat |page=85}}</ref><ref name="openstax-university-physics2">{{cite book|title=University Physics, Volume 2 |url=https://openstax.org/details/books/university-physics-volume-2 |publisher=[[OpenStax]] |year=2021 |first1=Samuel J. |last1=Ling |first2=Jeff |last2=Sanny |first3=William |last3=Moebs |isbn=978-1-947-17221-0}}</ref>{{rp|482}}

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]].<ref>{{cite book
|title=Polarized light in liquid crystals and polymers
|first1=Toralf
|last1=Scharf
|publisher=John Wiley and Sons
|year=2007
|isbn=978-0-471-74064-3
|page=19
|chapter=Chapter 2
|chapter-url=https://books.google.com/books?id=CQNE13opFucC&pg=PA19}}</ref> 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 a [[wave]] that traveled at a speed that he calculated to be the [[speed of light]]. This insight united the nascent fields of electromagnetic theory with [[optics]] and led directly to a complete description of the [[electromagnetic spectrum]].<ref>
{{cite book
 |first=William
 |last=Duffin
 |title=Electricity and Magnetism
 |publisher=McGraw-Hill
 |pages=[https://archive.org/details/electricitymagn00duff/page/364 364–383]
 |year=1980
 |edition=3rd
 |isbn=978-0-07-084111-6
 |url=https://archive.org/details/electricitymagn00duff/page/364
}}</ref>

=== Normal ===
[[File:Incline.svg|right|thumb|''F''<sub>N</sub> represents the [[normal force]] exerted on the object.]]
{{main|Normal force}}
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.<ref name=Cutnell>{{Cite book |last1=Cutnell |first1=John D. |title=Physics |last2=Johnson |first2=Kenneth W. |date=2004 |publisher=Wiley |isbn=978-0-471-44895-2 |edition=6th|location=Hoboken, NJ}}</ref>{{rp|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.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Friction ===
{{main|Friction}}
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]] and [[kinetic friction]].<ref name="openstax-university-physics"/>{{rp|267}}

The static friction force (<math>F_{\mathrm{sf}}</math>) will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the [[coefficient of static friction]] (<math>\mu_{\mathrm{sf}}</math>) multiplied by the normal force (<math>F_\text{N}</math>). In other words, the magnitude of the static friction force satisfies the inequality:
<math display="block">0 \le F_{\mathrm{sf}} \le \mu_{\mathrm{sf}} F_\mathrm{N}.</math>

The kinetic friction force (<math>F_{\mathrm{kf}}</math>) is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:
<math display="block">F_{\mathrm{kf}} = \mu_{\mathrm{kf}} F_\mathrm{N},</math>

where <math>\mu_{\mathrm{kf}}</math> is the [[coefficient of kinetic friction]]. The coefficient of kinetic friction is normally less than the coefficient of static friction.<ref name="openstax-university-physics"/>{{rp|267–271}}

=== Tension ===
{{main|Tension (physics)}}
Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with ideal [[pulley]]s, 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.<ref>{{cite web
|title=Tension Force
|work=Non-Calculus Based Physics I
|url=http://www.mtsu.edu/~phys2010/Lectures/Part_2__L6_-_L11/Lecture_9/Tension_Force/tension_force.html
|access-date=2008-01-04
|archive-date=2007-12-27
|archive-url=https://web.archive.org/web/20071227065923/http://www.mtsu.edu/~phys2010/Lectures/Part_2__L6_-_L11/Lecture_9/Tension_Force/tension_force.html
|url-status=dead
}}</ref> 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 a [[mechanical advantage]] for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in the [[conservation of energy|conservation of mechanical energy]] since the [[#Kinematic integrals|work done on the load]] is the same no matter how complicated the machine.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner /><ref>{{cite web |last=Fitzpatrick |first=Richard |title=Strings, pulleys, and inclines |date=2006-02-02 |url=http://farside.ph.utexas.edu/teaching/301/lectures/node48.html |access-date=2008-01-04}}</ref>

=== Spring ===
{{main|Elasticity (physics)|Hooke's law}}
[[File:Mass-spring-system.png|upright|thumb|''F<sub>k</sub>'' is the force that responds to the load on the spring]]
A simple elastic force acts to return a [[Spring (device)|spring]] to its natural length. An [[ideal spring]] is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the [[displacement field (mechanics)|displacement]] of the spring from its equilibrium position.<ref>{{cite web |last=Nave |first=Carl Rod |title=Elasticity |work=HyperPhysics |publisher=University of Guelph |url=http://hyperphysics.phy-astr.gsu.edu/hbase/permot2.html |access-date=2013-10-28}}</ref> This linear relationship was described by [[Robert Hooke]] in 1676, for whom [[Hooke's law]] is named. If <math>\Delta x</math> is the displacement, the force exerted by an ideal spring equals:
<math display="block" qid=Q170282>\vec{F}=-k \Delta \vec{x},</math>
where <math>k</math> 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.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Centripetal ===
{{main|Centripetal force}}
For an object in [[uniform circular motion]], the net force acting on the object equals:<ref>{{cite web |last=Nave |first=Carl Rod |title=Centripetal Force |work=HyperPhysics |publisher=University of Guelph |url=http://hyperphysics.phy-astr.gsu.edu/hbase/cf.html |access-date=2013-10-28}}</ref>
<math display="block" qid=Q172881>\vec{F} = - \frac{mv^2 \hat{r}}{r},</math>
where <math>m</math> is the mass of the object, <math>v</math> is the velocity of the object and <math>r</math> is the distance to the center of the circular path and <math> \hat{r}</math> is the [[unit 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 the [[speed]] 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.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

=== Continuum mechanics ===
[[File:Stokes sphere.svg|thumb|upright|When the drag force (<math>F_\text{d}</math>) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (<math>F_\text{g}</math>), the object reaches a state of [[#Dynamic equilibrium|dynamic equilibrium]] at [[terminal velocity]].]]
{{main|Pressure|Drag (physics)|Stress (mechanics)}}
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized [[point particle]]s 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 of [[continuum mechanics]] describe the way forces affect the material. For example, in extended [[fluid mechanics|fluids]], differences in [[pressure]] result in forces being directed along the pressure [[gradient]]s as follows:
<math display="block">\frac{\vec{F}}{V} = - \vec{\nabla} P,</math>

where <math>V</math> is the volume of the object in the fluid and <math>P</math> is the [[scalar function]] that describes the pressure at all locations in space. Pressure gradients and differentials result in the [[buoyancy|buoyant force]] for fluids suspended in gravitational fields, winds in [[atmospheric science]], and the [[lift (physics)|lift]] associated with [[aerodynamics]] and [[flight]].<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

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 "[[Drag (physics)#Very low Reynolds numbers – Stokes' drag|Stokes' drag]]" the force is approximately proportional to the velocity, but opposite in direction:
<math display="block" qid=Q824561>\vec{F}_\mathrm{d} = - b \vec{v}, </math>
where:
* <math>b</math> is a constant that depends on the properties of the fluid and the dimensions of the object (usually the [[Cross section (geometry)|cross-sectional area]]), and
* <math> \vec{v}</math> is the velocity of the object.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner />

More formally, forces in [[continuum mechanics]] are fully described by a [[Stress (mechanics)|stress]] [[tensor]] with terms that are roughly defined as
<math display="block" qid=Q206175>\sigma = \frac{F}{A},</math>
where <math>A</math> is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the [[matrix diagonal]]s of the tensor) as well as [[Shear stress|shear]] terms associated with forces that act [[Parallel (geometry)|parallel]] to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all [[strain (physics)|strains]] (deformations) including also [[tensile stress]]es and [[compression (physical)|compressions]].<ref name=uniphysics_ch2>{{cite book|title=University Physics |last1=Sears |first1=Francis W. |last2=Zemansky |first2=Mark W. |last3=Young |first3=Hugh D. |author-link1=Francis Sears |author-link2=Mark Zemansky |author-link3=Hugh D. Young |title-link=University Physics |pages=18–38 |publisher=Addison-Wesley |edition=6th |year=1982 |isbn=0-201-07199-1}}</ref><ref name=Kleppner>{{cite book |last1=Kleppner |first1=Daniel |last2=Kolenkow |first2=Robert J. |title=An Introduction to Mechanics|year=2014|publisher=Cambridge University Press|location=Cambridge|isbn=978-0521198110|edition=2nd|chapter=Chapter 3: Forces and equations of motion|chapter-url=https://archive.org/details/KleppnerD.KolenkowR.J.IntroductionToMechanics2014/page/n102}}</ref>{{rp|133–134}}<ref name=FeynmanVol2>{{cite book |last1=Feynman |first1=Richard P. |last2=Leighton |first2=Robert B. |last3=Sands |first3=Matthew |title=The Feynman lectures on physics. Vol. II: Mainly electromagnetism and matter |year=2010 |publisher=Basic Books |location=New York |isbn=978-0465024940 |edition=New millennium |title-link=The Feynman Lectures on Physics |author-link1=Richard Feynman |author-link2=Robert B. Leighton |author-link3=Matthew Sands}}</ref>{{rp|((38-1–38-11))}}

=== Fictitious ===
{{main|Fictitious forces}}
There are forces that are [[frame dependent]], meaning that they appear due to the adoption of non-Newtonian (that is, [[non-inertial frame|non-inertial]]) [[Frame of reference|reference frames]]. Such forces include the [[Centrifugal force (rotating reference frame)|centrifugal force]] and the [[Coriolis force]].<ref>{{cite web |last=Mallette |first=Vincent |title= The Coriolis Force |work=Publications in Science and Mathematics, Computing and the Humanities |publisher=Inwit Publishing, Inc. |date=1982–2008 |url=http://www.algorithm.com/inwit/writings/coriolisforce.html |access-date=2008-01-04}}</ref> These forces are considered fictitious because they do not exist in frames of reference that are not accelerating.<ref name=FeynmanVol1 />{{rp|at=ch.12}}<ref name=Kleppner /> Because these forces are not genuine they are also referred to as "pseudo forces".<ref name=FeynmanVol1 />{{rp|((12-11))}}

In [[general relativity]], [[gravity]] becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.<ref>{{Cite book |last=Choquet-Bruhat |first=Yvonne |url=https://www.worldcat.org/oclc/317496332 |title=General Relativity and the Einstein Equations |date=2009 |publisher=Oxford University Press |isbn=978-0-19-155226-7 |location=Oxford |oclc=317496332 |author-link=Yvonne Choquet-Bruhat |page=39}}</ref>