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Force

Forcesarealsodescribedasapushorpullonanobject.Theycanbeduetophenomenasuchasgravity,magnetism,oranythingthatmight

causeamasstoaccelerate.

Commonsymbols F,F

SIunit newton

InSIbaseunits 1kgm/s2

Derivationsfromotherquantities

F=ma

ForceFromWikipedia,thefreeencyclopedia

Inphysics,aforceisanyinteractionwhichtendstochangethemotionofanobject.[1]Inotherwords,aforcecancauseanobjectwithmasstochangeitsvelocity(whichincludestobeginmovingfromastateofrest),i.e.,toaccelerate.Forcecanalsobedescribedbyintuitiveconceptssuchasapushorapull.Aforcehasbothmagnitudeanddirection,makingitavectorquantity.ItismeasuredintheSIunitofnewtonsandrepresentedbythesymbolF.

TheoriginalformofNewton'ssecondlawstatesthatthenetforceactinguponanobjectisequaltotherateatwhichitsmomentumchangeswithtime.Ifthemassoftheobjectisconstant,thislawimpliesthattheaccelerationofanobjectisdirectlyproportionaltothenetforceactingontheobject,isinthedirectionofthenetforce,andisinverselyproportionaltothemassoftheobject.Asaformula,thisisexpressedas:

wherethearrowsimplyavectorquantitypossessingbothmagnitudeanddirection.

Relatedconceptstoforceinclude:thrust,whichincreasesthevelocityofanobjectdrag,whichdecreasesthevelocityofanobjectandtorquewhichproduceschangesinrotationalspeedofanobject.Inanextendedbody,eachpartusuallyappliesforcesontheadjacentpartsthedistributionofsuchforcesthroughthebodyisthesocalledmechanicalstress.Pressureisasimpletypeofstress.Stressusuallycausesdeformationofsolidmaterials,orflowinfluids.

Contents

1Developmentoftheconcept2PreNewtonianconcepts3Newtonianmechanics

3.1Firstlaw3.2Secondlaw3.3Thirdlaw

4Specialtheoryofrelativity5Descriptions

5.1Equilibrium5.1.1Static5.1.2Dynamic

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5.2ForcesinQuantumMechanics5.3Feynmandiagrams

6Fundamentalforces6.1Gravitational6.2Electromagnetic6.3Nuclear

7Nonfundamentalforces7.1Normalforce7.2Friction7.3Tension7.4Elasticforce7.5Continuummechanics7.6Fictitiousforces

8Rotationsandtorque8.1Centripetalforce

9Kinematicintegrals10Potentialenergy

10.1Conservativeforces10.2Nonconservativeforces

11Unitsofmeasurement12Forcemeasurement13Seealso14Notes15References16Furtherreading17Externallinks

Developmentoftheconcept

Philosophersinantiquityusedtheconceptofforceinthestudyofstationaryandmovingobjectsandsimplemachines,butthinkerssuchasAristotleandArchimedesretainedfundamentalerrorsinunderstandingforce.Inpartthiswasduetoanincompleteunderstandingofthesometimesnonobviousforceoffriction,andaconsequentlyinadequateviewofthenatureofnaturalmotion.[2]Afundamentalerrorwasthebeliefthataforceisrequiredtomaintainmotion,evenataconstantvelocity.MostofthepreviousmisunderstandingsaboutmotionandforcewereeventuallycorrectedbySirIsaacNewtonwithhismathematicalinsight,heformulatedlawsofmotionthatwerenotimprovedonfornearlythreehundredyears.[3]Bytheearly20thcentury,Einsteindevelopedatheoryofrelativitythatcorrectlypredictedtheactionofforcesonobjectswithincreasingmomentanearthespeedoflight,andalsoprovidedinsightintotheforcesproducedbygravitationandinertia.

Withmoderninsightsintoquantummechanicsandtechnologythatcanaccelerateparticlesclosetothespeedoflight,particlephysicshasdevisedaStandardModeltodescribeforcesbetweenparticlessmallerthanatoms.TheStandardModelpredictsthatexchangedparticlescalledgaugebosonsarethefundamentalmeansbywhich

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Aristotlefamouslydescribedaforceasanythingthatcausesanobjecttoundergo"unnaturalmotion"

forcesareemittedandabsorbed.Onlyfourmaininteractionsareknown:inorderofdecreasingstrength,theyare:strong,electromagnetic,weak,andgravitational.[4]:210[5]:79Highenergyparticlephysicsobservationsmadeduringthe1970sand1980sconfirmedthattheweakandelectromagneticforcesareexpressionsofamorefundamentalelectroweakinteraction.[6]

PreNewtonianconcepts

Sinceantiquitytheconceptofforcehasbeenrecognizedasintegraltothefunctioningofeachofthesimplemachines.Themechanicaladvantagegivenbyasimplemachineallowedforlessforcetobeusedinexchangeforthatforceactingoveragreaterdistanceforthesameamountofwork.AnalysisofthecharacteristicsofforcesultimatelyculminatedintheworkofArchimedeswhowasespeciallyfamousforformulatingatreatmentofbuoyantforcesinherentinfluids.[2]

AristotleprovidedaphilosophicaldiscussionoftheconceptofaforceasanintegralpartofAristoteliancosmology.InAristotle'sview,theterrestrialspherecontainedfourelementsthatcometorestatdifferent"naturalplaces"therein.AristotlebelievedthatmotionlessobjectsonEarth,thosecomposedmostlyoftheelementsearthandwater,tobeintheirnaturalplaceonthegroundandthattheywillstaythatwayifleftalone.Hedistinguishedbetweentheinnatetendencyofobjectstofindtheir"naturalplace"(e.g.,forheavybodiestofall),whichledto"naturalmotion",andunnaturalorforcedmotion,whichrequiredcontinuedapplicationofaforce.[7]Thistheory,basedontheeverydayexperienceofhowobjectsmove,suchastheconstantapplicationofaforceneededtokeepacartmoving,hadconceptualtroubleaccountingforthebehaviorofprojectiles,suchastheflightofarrows.Theplacewherethearchermovestheprojectilewasatthestartoftheflight,andwhiletheprojectilesailedthroughtheair,nodiscernibleefficientcauseactsonit.Aristotlewasawareofthisproblemandproposedthattheairdisplacedthroughtheprojectile'spathcarriestheprojectiletoitstarget.Thisexplanationdemandsacontinuumlikeairforchangeofplaceingeneral.[8]

AristotelianphysicsbeganfacingcriticisminMedievalscience,firstbyJohnPhiloponusinthe6thcentury.

TheshortcomingsofAristotelianphysicswouldnotbefullycorrecteduntilthe17thcenturyworkofGalileoGalilei,whowasinfluencedbythelateMedievalideathatobjectsinforcedmotioncarriedaninnateforceofimpetus.GalileoconstructedanexperimentinwhichstonesandcannonballswerebothrolleddownaninclinetodisprovetheAristoteliantheoryofmotionearlyinthe17thcentury.Heshowedthatthebodieswereacceleratedbygravitytoanextentwhichwasindependentoftheirmassandarguedthatobjectsretaintheirvelocityunlessactedonbyaforce,forexamplefriction.[9]

Newtonianmechanics

SirIsaacNewtonsoughttodescribethemotionofallobjectsusingtheconceptsofinertiaandforce,andindoingsohefoundthattheyobeycertainconservationlaws.In1687,NewtonwentontopublishhisthesisPhilosophiNaturalisPrincipiaMathematica.[3][10]InthisworkNewtonsetoutthreelawsofmotionthattothisdayarethewayforcesaredescribedinphysics.[10]

Firstlaw

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Newton'sFirstLawofMotionstatesthatobjectscontinuetomoveinastateofconstantvelocityunlessacteduponbyanexternalnetforceorresultantforce.[10]ThislawisanextensionofGalileo'sinsightthatconstantvelocitywasassociatedwithalackofnetforce(seeamoredetaileddescriptionofthisbelow).Newtonproposedthateveryobjectwithmasshasaninnateinertiathatfunctionsasthefundamentalequilibrium"naturalstate"inplaceoftheAristotelianideaofthe"naturalstateofrest".Thatis,thefirstlawcontradictstheintuitiveAristotelianbeliefthatanetforceisrequiredtokeepanobjectmovingwithconstantvelocity.Bymakingrestphysicallyindistinguishablefromnonzeroconstantvelocity,Newton'sFirstLawdirectlyconnectsinertiawiththeconceptofrelativevelocities.Specifically,insystemswhereobjectsaremovingwithdifferentvelocities,itisimpossibletodeterminewhichobjectis"inmotion"andwhichobjectis"atrest".Inotherwords,tophrasemattersmoretechnically,thelawsofphysicsarethesameineveryinertialframeofreference,thatis,inallframesrelatedbyaGalileantransformation.

Forinstance,whiletravelinginamovingvehicleataconstantvelocity,thelawsofphysicsdonotchangefrombeingatrest.Apersoncanthrowaballstraightupintheairandcatchitasitfallsdownwithoutworryingaboutapplyingaforceinthedirectionthevehicleismoving.Thisistrueeventhoughanotherpersonwhoisobservingthemovingvehiclepassbyalsoobservestheballfollowacurvingparabolicpathinthesamedirectionasthemotionofthevehicle.Itistheinertiaoftheballassociatedwithitsconstantvelocityinthedirectionofthevehicle'smotionthatensurestheballcontinuestomoveforwardevenasitisthrownupandfallsbackdown.Fromtheperspectiveofthepersoninthecar,thevehicleandeverythinginsideofitisatrest:Itistheoutsideworldthatismovingwithaconstantspeedintheoppositedirection.Sincethereisnoexperimentthatcandistinguishwhetheritisthevehiclethatisatrestortheoutsideworldthatisatrest,thetwosituationsareconsideredtobephysicallyindistinguishable.Inertiathereforeappliesequallywelltoconstantvelocitymotionasitdoestorest.

Theconceptofinertiacanbefurthergeneralizedtoexplainthetendencyofobjectstocontinueinmanydifferentformsofconstantmotion,eventhosethatarenotstrictlyconstantvelocity.TherotationalinertiaofplanetEarthiswhatfixestheconstancyofthelengthofadayandthelengthofayear.AlbertEinsteinextendedtheprincipleofinertiafurtherwhenheexplainedthatreferenceframessubjecttoconstantacceleration,suchasthosefreefallingtowardagravitatingobject,werephysicallyequivalenttoinertialreferenceframes.Thisiswhy,forexample,astronautsexperienceweightlessnesswheninfreefallorbitaroundtheEarth,andwhyNewton'sLawsofMotionaremoreeasilydiscernibleinsuchenvironments.Ifanastronautplacesanobjectwithmassinmidairnexttohimself,itwillremainstationarywithrespecttotheastronautduetoitsinertia.Thisisthesamethingthatwouldoccuriftheastronautandtheobjectwereinintergalacticspacewithnonetforceofgravityactingontheirsharedreferenceframe.Thisprincipleofequivalencewasoneofthefoundationalunderpinningsforthedevelopmentofthegeneraltheoryofrelativity.[11]

Secondlaw

AmodernstatementofNewton'sSecondLawisavectordifferentialequation:[Note1]

where isthemomentumofthesystem,andisthenet(vectorsum)force.Inequilibrium,thereiszeronetforcebydefinition,but(balanced)forcesmaybepresentnevertheless.Incontrast,thesecondlawstatesanunbalancedforceactingonanobjectwillresultintheobject'smomentumchangingovertime.[10]

Bythedefinitionofmomentum,

wheremisthemassandisthevelocity.[4]:91,92

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ThoughSirIsaacNewton'smostfamousequationis

,heactuallywrotedownadifferentformforhissecondlawofmotionthatdidnotusedifferentialcalculus.

Newton'ssecondlawappliesonlytoasystemofconstantmass,[Note2]andhencemmaybemovedoutsidethederivativeoperator.Theequationthenbecomes

Bysubstitutingthedefinitionofacceleration,thealgebraicversionofNewton'sSecondLawisderived:

Newtonneverexplicitlystatedtheformulainthereducedformabove.[12]

Newton'sSecondLawassertsthedirectproportionalityofaccelerationtoforceandtheinverseproportionalityofaccelerationtomass.Accelerationscanbedefinedthroughkinematicmeasurements.However,whilekinematicsarewelldescribedthroughreferenceframeanalysisinadvancedphysics,therearestilldeepquestionsthatremainastowhatistheproperdefinitionofmass.Generalrelativityoffersanequivalencebetweenspacetimeandmass,butlackingacoherenttheoryofquantumgravity,itisunclearastohoworwhetherthisconnectionisrelevantonmicroscales.Withsomejustification,Newton'ssecondlawcanbetakenasaquantitativedefinitionofmassbywritingthelawasanequalitytherelativeunitsofforceandmassthenarefixed.

TheuseofNewton'sSecondLawasadefinitionofforcehasbeendisparagedinsomeofthemorerigoroustextbooks,[4]:121[5]:59[13]becauseitisessentiallyamathematicaltruism.Notablephysicists,philosophersandmathematicianswhohavesoughtamoreexplicitdefinitionoftheconceptofforceincludeErnstMach,CliffordTruesdellandWalterNoll.[14][15]

Newton'sSecondLawcanbeusedtomeasurethestrengthofforces.Forinstance,knowledgeofthemassesofplanetsalongwiththeaccelerationsoftheirorbitsallowsscientiststocalculatethegravitationalforcesonplanets.

Thirdlaw

Newton'sThirdLawisaresultofapplyingsymmetrytosituationswhereforcescanbeattributedtothepresenceofdifferentobjects.Thethirdlawmeansthatallforcesareinteractionsbetweendifferentbodies,[16][Note3]andthusthatthereisnosuchthingasaunidirectionalforceoraforcethatactsononlyonebody.WheneverafirstbodyexertsaforceFonasecondbody,thesecondbodyexertsaforceFonthefirstbody.FandFareequalinmagnitudeandoppositeindirection.Thislawissometimesreferredtoastheactionreactionlaw,withFcalledthe"action"andFthe"reaction".Theactionandthereactionaresimultaneous:

Ifobject1andobject2areconsideredtobeinthesamesystem,thenthenetforceonthesystemduetotheinteractionsbetweenobjects1and2iszerosince

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Thismeansthatinaclosedsystemofparticles,therearenointernalforcesthatareunbalanced.Thatis,theactionreactionforcesharedbetweenanytwoobjectsinaclosedsystemwillnotcausethecenterofmassofthesystemtoaccelerate.Theconstituentobjectsonlyacceleratewithrespecttoeachother,thesystemitselfremainsunaccelerated.Alternatively,ifanexternalforceactsonthesystem,thenthecenterofmasswillexperienceanaccelerationproportionaltothemagnitudeoftheexternalforcedividedbythemassofthesystem.[4]:191[5]

CombiningNewton'sSecondandThirdLaws,itispossibletoshowthatthelinearmomentumofasystemisconserved.Using

andintegratingwithrespecttotime,theequation:

isobtained.Forasystemwhichincludesobjects1and2,

whichistheconservationoflinearmomentum.[17]Usingthesimilararguments,itispossibletogeneralizethistoasystemofanarbitrarynumberofparticles.Thisshowsthatexchangingmomentumbetweenconstituentobjectswillnotaffectthenetmomentumofasystem.Ingeneral,aslongasallforcesareduetotheinteractionofobjectswithmass,itispossibletodefineasystemsuchthatnetmomentumisneverlostnorgained.[4][5]

Specialtheoryofrelativity

Inthespecialtheoryofrelativity,massandenergyareequivalent(ascanbeseenbycalculatingtheworkrequiredtoaccelerateanobject).Whenanobject'svelocityincreases,sodoesitsenergyandhenceitsmassequivalent(inertia).Itthusrequiresmoreforcetoaccelerateitthesameamountthanitdidatalowervelocity.Newton'sSecondLaw

remainsvalidbecauseitisamathematicaldefinition.[18]:855876Butinordertobeconserved,relativisticmomentummustberedefinedas:

where

isthevelocityandisthespeedoflight

istherestmass.

Therelativisticexpressionrelatingforceandaccelerationforaparticlewithconstantnonzerorestmassmovinginthe directionis:

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Diagramsofablockonaflatsurfaceandaninclinedplane.Forcesareresolvedandaddedtogethertodeterminetheirmagnitudesandthenetforce.

wheretheLorentzfactor

[19]

Intheearlyhistoryofrelativity,theexpressions andwerecalledlongitudinalandtransversemass.Relativisticforcedoesnotproduceaconstantacceleration,butaneverdecreasingaccelerationastheobjectapproachesthespeedoflight.Notethatisundefinedforanobjectwithanonzerorestmassatthespeedoflight,andthetheoryyieldsnopredictionatthatspeed.

Ifisverysmallcomparedto ,thenisverycloseto1and

isacloseapproximation.Evenforuseinrelativity,however,onecanrestoretheformof

throughtheuseoffourvectors.Thisrelationiscorrectinrelativitywhen isthefourforce,istheinvariantmass,andisthefouracceleration.[20]

Descriptions

Sinceforcesareperceivedaspushesorpulls,thiscanprovideanintuitiveunderstandingfordescribingforces.[3]Aswithotherphysicalconcepts(e.g.temperature),theintuitiveunderstandingofforcesisquantifiedusingpreciseoperationaldefinitionsthatareconsistentwithdirectobservationsandcomparedtoastandardmeasurementscale.Throughexperimentation,itisdeterminedthatlaboratorymeasurementsofforcesarefullyconsistentwiththeconceptualdefinitionofforceofferedbyNewtonianmechanics.

Forcesactinaparticulardirectionandhavesizesdependentuponhowstrongthepushorpullis.Becauseofthesecharacteristics,forcesareclassifiedas"vectorquantities".Thismeansthatforcesfollowadifferentsetofmathematicalrulesthanphysicalquantitiesthatdonothavedirection(denotedscalarquantities).Forexample,whendeterminingwhathappenswhentwoforcesactonthesameobject,itisnecessarytoknowboththemagnitudeandthedirectionofbothforcestocalculatetheresult.Ifbothofthesepiecesofinformationarenotknownforeachforce,thesituationisambiguous.Forexample,ifyouknowthattwopeoplearepullingonthesameropewithknownmagnitudesofforcebutyoudonotknowwhichdirectioneitherpersonispulling,itisimpossibletodeterminewhattheaccelerationoftheropewillbe.Thetwopeoplecouldbepullingagainsteachotherasintugofwarorthetwopeoplecouldbepullinginthesamedirection.Inthissimpleonedimensionalexample,withoutknowingthedirectionoftheforcesitisimpossibletodecidewhetherthenetforceistheresultofaddingthetwoforcemagnitudesorsubtractingonefromtheother.Associatingforceswithvectorsavoidssuchproblems.

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Historically,forceswerefirstquantitativelyinvestigatedinconditionsofstaticequilibriumwhereseveralforcescanceledeachotherout.Suchexperimentsdemonstratethecrucialpropertiesthatforcesareadditivevectorquantities:theyhavemagnitudeanddirection.[3]Whentwoforcesactonapointparticle,theresultingforce,theresultant(alsocalledthenetforce),canbedeterminedbyfollowingtheparallelogramruleofvectoraddition:theadditionoftwovectorsrepresentedbysidesofaparallelogram,givesanequivalentresultantvectorwhichisequalinmagnitudeanddirectiontothetransversaloftheparallelogram.[4][5]Themagnitudeoftheresultantvariesfromthedifferenceofthemagnitudesofthetwoforcestotheirsum,dependingontheanglebetweentheirlinesofaction.However,iftheforcesareactingonanextendedbody,theirrespectivelinesofapplicationmustalsobespecifiedinordertoaccountfortheireffectsonthemotionofthebody.

Freebodydiagramscanbeusedasaconvenientwaytokeeptrackofforcesactingonasystem.Ideally,thesediagramsaredrawnwiththeanglesandrelativemagnitudesoftheforcevectorspreservedsothatgraphicalvectoradditioncanbedonetodeterminethenetforce.[21]

Aswellasbeingadded,forcescanalsoberesolvedintoindependentcomponentsatrightanglestoeachother.Ahorizontalforcepointingnortheastcanthereforebesplitintotwoforces,onepointingnorth,andonepointingeast.Summingthesecomponentforcesusingvectoradditionyieldstheoriginalforce.Resolvingforcevectorsintocomponentsofasetofbasisvectorsisoftenamoremathematicallycleanwaytodescribeforcesthanusingmagnitudesanddirections.[22]Thisisbecause,fororthogonalcomponents,thecomponentsofthevectorsumareuniquelydeterminedbythescalaradditionofthecomponentsoftheindividualvectors.Orthogonalcomponentsareindependentofeachotherbecauseforcesactingatninetydegreestoeachotherhavenoeffectonthemagnitudeordirectionoftheother.Choosingasetoforthogonalbasisvectorsisoftendonebyconsideringwhatsetofbasisvectorswillmakethemathematicsmostconvenient.Choosingabasisvectorthatisinthesamedirectionasoneoftheforcesisdesirable,sincethatforcewouldthenhaveonlyonenonzerocomponent.Orthogonalforcevectorscanbethreedimensionalwiththethirdcomponentbeingatrightanglestotheothertwo.[4][5]

Equilibrium

Equilibriumoccurswhentheresultantforceactingonapointparticleiszero(thatis,thevectorsumofallforcesiszero).Whendealingwithanextendedbody,itisalsonecessarythatthenettorqueinitis0.

Therearetwokindsofequilibrium:staticequilibriumanddynamicequilibrium.

Static

Staticequilibriumwasunderstoodwellbeforetheinventionofclassicalmechanics.Objectswhichareatresthavezeronetforceactingonthem.[23]

Thesimplestcaseofstaticequilibriumoccurswhentwoforcesareequalinmagnitudebutoppositeindirection.Forexample,anobjectonalevelsurfaceispulled(attracted)downwardtowardthecenteroftheEarthbytheforceofgravity.Atthesametime,surfaceforcesresistthedownwardforcewithequalupwardforce(calledthenormalforce).Thesituationisoneofzeronetforceandnoacceleration.[3]

Pushingagainstanobjectonafrictionalsurfacecanresultinasituationwheretheobjectdoesnotmovebecausetheappliedforceisopposedbystaticfriction,generatedbetweentheobjectandthetablesurface.Forasituationwithnomovement,thestaticfrictionforceexactlybalancestheappliedforceresultinginnoacceleration.Thestaticfrictionincreasesordecreasesinresponsetotheappliedforceuptoanupperlimitdeterminedbythecharacteristicsofthecontactbetweenthesurfaceandtheobject.[3]

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GalileoGalileiwasthefirsttopointouttheinherentcontradictionscontainedinAristotle'sdescriptionofforces.

Astaticequilibriumbetweentwoforcesisthemostusualwayofmeasuringforces,usingsimpledevicessuchasweighingscalesandspringbalances.Forexample,anobjectsuspendedonaverticalspringscaleexperiencestheforceofgravityactingontheobjectbalancedbyaforceappliedbythe"springreactionforce"whichequalstheobject'sweight.Usingsuchtools,somequantitativeforcelawswerediscovered:thattheforceofgravityisproportionaltovolumeforobjectsofconstantdensity(widelyexploitedformillenniatodefinestandardweights)Archimedes'principleforbuoyancyArchimedes'analysisoftheleverBoyle'slawforgaspressureandHooke'slawforsprings.ThesewereallformulatedandexperimentallyverifiedbeforeIsaacNewtonexpoundedhisThreeLawsofMotion.[3][4][5]

Dynamic

DynamicequilibriumwasfirstdescribedbyGalileowhonoticedthatcertainassumptionsofAristotelianphysicswerecontradictedbyobservationsandlogic.Galileorealizedthatsimplevelocityadditiondemandsthattheconceptofan"absoluterestframe"didnotexist.Galileoconcludedthatmotioninaconstantvelocitywascompletelyequivalenttorest.ThiswascontrarytoAristotle'snotionofa"naturalstate"ofrestthatobjectswithmassnaturallyapproached.SimpleexperimentsshowedthatGalileo'sunderstandingoftheequivalenceofconstantvelocityandrestwerecorrect.Forexample,ifamarinerdroppedacannonballfromthecrow'snestofashipmovingataconstantvelocity,Aristotelianphysicswouldhavethecannonballfallstraightdownwhiletheshipmovedbeneathit.Thus,inanAristotelianuniverse,thefallingcannonballwouldlandbehindthefootofthemastofamovingship.However,whenthisexperimentisactuallyconducted,thecannonballalwaysfallsatthefootofthemast,asifthecannonballknowstotravelwiththeshipdespitebeingseparatedfromit.Sincethereisnoforwardhorizontalforcebeingappliedonthecannonballasitfalls,theonlyconclusionleftisthatthecannonballcontinuestomovewiththesamevelocityastheboatasitfalls.Thus,noforceisrequiredtokeepthecannonballmovingattheconstantforwardvelocity.[9]

Moreover,anyobjecttravelingataconstantvelocitymustbesubjecttozeronetforce(resultantforce).Thisisthedefinitionofdynamicequilibrium:whenalltheforcesonanobjectbalancebutitstillmovesataconstantvelocity.

Asimplecaseofdynamicequilibriumoccursinconstantvelocitymotionacrossasurfacewithkineticfriction.Insuchasituation,aforceisappliedinthedirectionofmotionwhilethekineticfrictionforceexactlyopposestheappliedforce.Thisresultsinzeronetforce,butsincetheobjectstartedwithanonzerovelocity,itcontinuestomovewithanonzerovelocity.Aristotlemisinterpretedthismotionasbeingcausedbytheappliedforce.However,whenkineticfrictionistakenintoconsiderationitisclearthatthereisnonetforcecausingconstantvelocitymotion.[4][5]

ForcesinQuantumMechanics

Thenotion"force"keepsitsmeaninginquantummechanics,thoughoneisnowdealingwithoperatorsinsteadofclassicalvariablesandthoughthephysicsisnowdescribedbytheSchrdingerequationinsteadofNewtonianequations.Thishastheconsequencethattheresultsofameasurementarenowsometimes"quantized",i.e.theyappearindiscreteportions.Thisis,ofcourse,difficulttoimagineinthecontextof"forces".However,thepotentialsV(x,y,z)orfields,fromwhichtheforcesgenerallycanbederived,aretreatedsimilartoclassicalpositionvariables,i.e.,.

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Feynmandiagramforthedecayofaneutronintoaproton.TheWbosonisbetweentwoverticesindicatingarepulsion.

Thisbecomesdifferentonlyintheframeworkofquantumfieldtheory,wherethesefieldsarealsoquantized.

However,alreadyinquantummechanicsthereisone"caveat",namelytheparticlesactingontoeachotherdonotonlypossessthespatialvariable,butalsoadiscreteintrinsicangularmomentumlikevariablecalledthe"spin",andthereisthePauliprinciplerelatingthespaceandthespinvariables.Dependingonthevalueofthespin,identicalparticlessplitintotwodifferentclasses,fermionsandbosons.Iftwoidenticalfermions(e.g.electrons)haveasymmetricspinfunction(e.g.parallelspins)thespatialvariablesmustbeantisymmetric(i.e.theforcemustberepulsive),andviceversa,i.e.forantiparallelspinsthepositionvariablesmustbesymmetric(i.e.theforcemustbeattractive).Thusinthecaseoftwofermionsthereisastrictlynegativecorrelationbetweenspatialandspinvariables,whereasfortwobosons(e.g.quantaofelectromagneticwaves,photons)thecorrelationisstrictlypositive.

Thusthenotion"force"losesalreadypartofitsmeaning.

Feynmandiagrams

Inmodernparticlephysics,forcesandtheaccelerationofparticlesareexplainedasamathematicalbyproductofexchangeofmomentumcarryinggaugebosons.Withthedevelopmentofquantumfieldtheoryandgeneralrelativity,itwasrealizedthatforceisaredundantconceptarisingfromconservationofmomentum(4momentuminrelativityandmomentumofvirtualparticlesinquantumelectrodynamics).Theconservationofmomentumcanbedirectlyderivedfromthehomogeneityorsymmetryofspaceandsoisusuallyconsideredmorefundamentalthantheconceptofaforce.Thusthecurrentlyknownfundamentalforcesareconsideredmoreaccuratelytobe"fundamentalinteractions".[6]:199128WhenparticleAemits(creates)orabsorbs(annihilates)virtualparticleB,amomentumconservationresultsinrecoilofparticleAmakingimpressionofrepulsionorattractionbetweenparticlesAA'exchangingbyB.Thisdescriptionappliestoallforcesarisingfromfundamentalinteractions.Whilesophisticatedmathematicaldescriptionsareneededtopredict,infulldetail,theaccurateresultofsuchinteractions,thereisaconceptuallysimplewaytodescribesuchinteractionsthroughtheuseofFeynmandiagrams.InaFeynmandiagram,eachmatterparticleisrepresentedasastraightline(seeworldline)travelingthroughtimewhichnormallyincreasesuportotherightinthediagram.MatterandantimatterparticlesareidenticalexceptfortheirdirectionofpropagationthroughtheFeynmandiagram.Worldlinesofparticlesintersectatinteractionvertices,andtheFeynmandiagramrepresentsanyforcearisingfromaninteractionasoccurringatthevertexwithanassociatedinstantaneouschangeinthedirectionoftheparticleworldlines.Gaugebosonsareemittedawayfromthevertexaswavylinesand,inthecaseofvirtualparticleexchange,areabsorbedatanadjacentvertex.[24]

TheutilityofFeynmandiagramsisthatothertypesofphysicalphenomenathatarepartofthegeneralpictureoffundamentalinteractionsbutareconceptuallyseparatefromforcescanalsobedescribedusingthesamerules.Forexample,aFeynmandiagramcandescribeinsuccinctdetailhowaneutrondecaysintoanelectron,proton,andneutrino,aninteractionmediatedbythesamegaugebosonthatisresponsiblefortheweaknuclearforce.[24]

Fundamentalforces

Alloftheforcesintheuniversearebasedonfourfundamentalinteractions.Thestrongandweakforcesarenuclearforcesthatactonlyatveryshortdistances,andareresponsiblefortheinteractionsbetweensubatomicparticles,includingnucleonsandcompoundnuclei.Theelectromagneticforceactsbetweenelectriccharges,and

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thegravitationalforceactsbetweenmasses.Allotherforcesinnaturederivefromthesefourfundamentalinteractions.Forexample,frictionisamanifestationoftheelectromagneticforceactingbetweentheatomsoftwosurfaces,andthePauliexclusionprinciple,[25]whichdoesnotpermitatomstopassthrougheachother.Similarly,theforcesinsprings,modeledbyHooke'slaw,aretheresultofelectromagneticforcesandtheExclusionPrincipleactingtogethertoreturnanobjecttoitsequilibriumposition.Centrifugalforcesareaccelerationforceswhicharisesimplyfromtheaccelerationofrotatingframesofreference.[4]:1211[5]:359

Thedevelopmentoffundamentaltheoriesforforcesproceededalongthelinesofunificationofdisparateideas.Forexample,IsaacNewtonunifiedtheforceresponsibleforobjectsfallingatthesurfaceoftheEarthwiththeforceresponsiblefortheorbitsofcelestialmechanicsinhisuniversaltheoryofgravitation.MichaelFaradayandJamesClerkMaxwelldemonstratedthatelectricandmagneticforceswereunifiedthroughoneconsistenttheoryofelectromagnetism.Inthe20thcentury,thedevelopmentofquantummechanicsledtoamodernunderstandingthatthefirstthreefundamentalforces(allexceptgravity)aremanifestationsofmatter(fermions)interactingbyexchangingvirtualparticlescalledgaugebosons.[26]Thisstandardmodelofparticlephysicspositsasimilaritybetweentheforcesandledscientiststopredicttheunificationoftheweakandelectromagneticforcesinelectroweaktheorysubsequentlyconfirmedbyobservation.ThecompleteformulationofthestandardmodelpredictsanasyetunobservedHiggsmechanism,butobservationssuchasneutrinooscillationsindicatethatthestandardmodelisincomplete.AGrandUnifiedTheoryallowingforthecombinationoftheelectroweakinteractionwiththestrongforceisheldoutasapossibilitywithcandidatetheoriessuchassupersymmetryproposedtoaccommodatesomeoftheoutstandingunsolvedproblemsinphysics.Physicistsarestillattemptingtodevelopselfconsistentunificationmodelsthatwouldcombineallfourfundamentalinteractionsintoatheoryofeverything.Einsteintriedandfailedatthisendeavor,butcurrentlythemostpopularapproachtoansweringthisquestionisstringtheory.[6]:212219

Thefourfundamentalforcesofnature[27]

Property/Interaction GravitationWeak Electromagnetic Strong

(Electroweak) Fundamental Residual

Actson: MassEnergy Flavor Electriccharge ColorchargeAtomicnuclei

Particlesexperiencing: All Quarks,leptonsElectricallycharged

Quarks,Gluons Hadrons

Particlesmediating:Graviton(notyetobserved)

W+WZ0 Gluons Mesons

Strengthinthescaleofquarks: 10

41 104 1 60 NotapplicabletoquarksStrengthinthescaleofprotons/neutrons: 10

36 107 1 Notapplicabletohadrons 20

Gravitational

WhatwenowcallgravitywasnotidentifiedasauniversalforceuntiltheworkofIsaacNewton.BeforeNewton,thetendencyforobjectstofalltowardstheEarthwasnotunderstoodtoberelatedtothemotionsofcelestialobjects.Galileowasinstrumentalindescribingthecharacteristicsoffallingobjectsbydeterminingthattheaccelerationofeveryobjectinfreefallwasconstantandindependentofthemassoftheobject.Today,thisaccelerationduetogravitytowardsthesurfaceoftheEarthisusuallydesignatedasandhasamagnitudeofabout9.81meterspersecondsquared(thismeasurementistakenfromsealevelandmayvarydependingon

http://en.wikipedia.org/wiki/Force 12/25

Imagesofafreelyfallingbasketballtakenwithastroboscopeat20flashespersecond.Thedistanceunitsontherightaremultiplesofabout12millimetres.Thebasketballstartsatrest.Atthetimeofthefirstflash(distancezero)itisreleased,afterwhichthenumberofunitsfallenisequaltothesquareofthenumberofflashes.

location),andpointstowardthecenteroftheEarth.[28]ThisobservationmeansthattheforceofgravityonanobjectattheEarth'ssurfaceisdirectlyproportionaltotheobject'smass.Thusanobjectthathasamassofwillexperienceaforce:

Infreefall,thisforceisunopposedandthereforethenetforceontheobjectisitsweight.Forobjectsnotinfreefall,theforceofgravityisopposedbythereactionsoftheirsupports.Forexample,apersonstandingonthegroundexperienceszeronetforce,sincehisweightisbalancedbyanormalforceexertedbytheground.[4][5]

Newton'scontributiontogravitationaltheorywastounifythemotionsofheavenlybodies,whichAristotlehadassumedwereinanaturalstateofconstantmotion,withfallingmotionobservedontheEarth.HeproposedalawofgravitythatcouldaccountforthecelestialmotionsthathadbeendescribedearlierusingKepler'slawsofplanetarymotion.[29]

Newtoncametorealizethattheeffectsofgravitymightbeobservedindifferentwaysatlargerdistances.Inparticular,NewtondeterminedthattheaccelerationoftheMoonaroundtheEarthcouldbeascribedtothesameforceofgravityiftheaccelerationduetogravitydecreasedasaninversesquarelaw.Further,Newtonrealizedthattheaccelerationduetogravityisproportionaltothemassoftheattractingbody.[29]Combiningtheseideasgivesaformulathatrelatesthemass()andtheradius()oftheEarthtothegravitationalacceleration:

wherethevectordirectionisgivenby,theunitvectordirectedoutwardfromthecenteroftheEarth.[10]

Inthisequation,adimensionalconstantisusedtodescribetherelativestrengthofgravity.ThisconstanthascometobeknownasNewton'sUniversalGravitationConstant,[30]thoughitsvaluewasunknowninNewton'slifetime.Notuntil1798wasHenryCavendishabletomakethefirstmeasurementofusingatorsionbalancethiswaswidelyreportedinthepressasameasurementofthemassoftheEarthsinceknowingcouldallowonetosolvefortheEarth'smassgiventheaboveequation.Newton,however,realizedthatsinceallcelestialbodiesfollowedthesamelawsofmotion,hislawofgravityhadtobeuniversal.Succinctlystated,Newton'sLawofGravitationstatesthattheforceonasphericalobjectofmassduetothegravitationalpullofmassis

where isthedistancebetweenthetwoobjects'centersofmassandistheunitvectorpointedinthedirectionawayfromthecenterofthefirstobjecttowardthecenterofthesecondobject.[10]

Thisformulawaspowerfulenoughtostandasthebasisforallsubsequentdescriptionsofmotionwithinthesolarsystemuntilthe20thcentury.Duringthattime,sophisticatedmethodsofperturbationanalysis[31]wereinventedtocalculate

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thedeviationsoforbitsduetotheinfluenceofmultiplebodiesonaplanet,moon,comet,orasteroid.TheformalismwasexactenoughtoallowmathematicianstopredicttheexistenceoftheplanetNeptunebeforeitwasobserved.[32]

ItwasonlytheorbitoftheplanetMercurythatNewton'sLawofGravitationseemednottofullyexplain.Someastrophysicistspredictedtheexistenceofanotherplanet(Vulcan)thatwouldexplainthediscrepancieshowever,despitesomeearlyindications,nosuchplanetcouldbefound.WhenAlbertEinsteinformulatedhistheoryofgeneralrelativity(GR)heturnedhisattentiontotheproblemofMercury'sorbitandfoundthathistheoryaddedacorrectionwhichcouldaccountforthediscrepancy.ThiswasthefirsttimethatNewton'sTheoryofGravityhadbeenshowntobelesscorrectthananalternative.[33]

Sincethen,andsofar,generalrelativityhasbeenacknowledgedasthetheorywhichbestexplainsgravity.InGR,gravitationisnotviewedasaforce,butrather,objectsmovingfreelyingravitationalfieldstravelundertheirowninertiainstraightlinesthroughcurvedspacetimedefinedastheshortestspacetimepathbetweentwospacetimeevents.Fromtheperspectiveoftheobject,allmotionoccursasiftherewerenogravitationwhatsoever.Itisonlywhenobservingthemotioninaglobalsensethatthecurvatureofspacetimecanbeobservedandtheforceisinferredfromtheobject'scurvedpath.Thus,thestraightlinepathinspacetimeisseenasacurvedlineinspace,anditiscalledtheballistictrajectoryoftheobject.Forexample,abasketballthrownfromthegroundmovesinaparabola,asitisinauniformgravitationalfield.Itsspacetimetrajectory(whentheextractdimensionisadded)isalmostastraightline,slightlycurved(withtheradiusofcurvatureoftheorderoffewlightyears).Thetimederivativeofthechangingmomentumoftheobjectiswhatwelabelas"gravitationalforce".[5]

Electromagnetic

Theelectrostaticforcewasfirstdescribedin1784byCoulombasaforcewhichexistedintrinsicallybetweentwocharges.[18]:519Thepropertiesoftheelectrostaticforcewerethatitvariedasaninversesquarelawdirectedintheradialdirection,wasbothattractiveandrepulsive(therewasintrinsicpolarity),wasindependentofthemassofthechargedobjects,andfollowedthesuperpositionprinciple.Coulomb'slawunifiesalltheseobservationsintoonesuccinctstatement.[34]

Subsequentmathematiciansandphysicistsfoundtheconstructoftheelectricfieldtobeusefulfordeterminingtheelectrostaticforceonanelectricchargeatanypointinspace.Theelectricfieldwasbasedonusingahypothetical"testcharge"anywhereinspaceandthenusingCoulomb'sLawtodeterminetheelectrostaticforce.[35]:46to48Thustheelectricfieldanywhereinspaceisdefinedas

whereisthemagnitudeofthehypotheticaltestcharge.

Meanwhile,theLorentzforceofmagnetismwasdiscoveredtoexistbetweentwoelectriccurrents.IthasthesamemathematicalcharacterasCoulomb'sLawwiththeprovisothatlikecurrentsattractandunlikecurrentsrepel.Similartotheelectricfield,themagneticfieldcanbeusedtodeterminethemagneticforceonanelectriccurrentatanypointinspace.Inthiscase,themagnitudeofthemagneticfieldwasdeterminedtobe

whereisthemagnitudeofthehypotheticaltestcurrentand isthelengthofhypotheticalwirethroughwhichthetestcurrentflows.Themagneticfieldexertsaforceonallmagnetsincluding,forexample,thoseusedincompasses.ThefactthattheEarth'smagneticfieldisalignedcloselywiththeorientationoftheEarth'saxis

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causescompassmagnetstobecomeorientedbecauseofthemagneticforcepullingontheneedle.

Throughcombiningthedefinitionofelectriccurrentasthetimerateofchangeofelectriccharge,aruleofvectormultiplicationcalledLorentz'sLawdescribestheforceonachargemovinginamagneticfield.[35]Theconnectionbetweenelectricityandmagnetismallowsforthedescriptionofaunifiedelectromagneticforcethatactsonacharge.Thisforcecanbewrittenasasumoftheelectrostaticforce(duetotheelectricfield)andthemagneticforce(duetothemagneticfield).Fullystated,thisisthelaw:

whereistheelectromagneticforce,isthemagnitudeofthechargeoftheparticle, istheelectricfield,isthevelocityoftheparticlewhichiscrossedwiththemagneticfield().

Theoriginofelectricandmagneticfieldswouldnotbefullyexplaineduntil1864whenJamesClerkMaxwellunifiedanumberofearliertheoriesintoasetof20scalarequations,whichwerelaterreformulatedinto4vectorequationsbyOliverHeavisideandJosiahWillardGibbs.[36]These"MaxwellEquations"fullydescribedthesourcesofthefieldsasbeingstationaryandmovingcharges,andtheinteractionsofthefieldsthemselves.ThisledMaxwelltodiscoverthatelectricandmagneticfieldscouldbe"selfgenerating"throughawavethattraveledataspeedwhichhecalculatedtobethespeedoflight.Thisinsightunitedthenascentfieldsofelectromagnetictheorywithopticsandleddirectlytoacompletedescriptionoftheelectromagneticspectrum.[37]

However,attemptingtoreconcileelectromagnetictheorywithtwoobservations,thephotoelectriceffect,andthenonexistenceoftheultravioletcatastrophe,provedtroublesome.Throughtheworkofleadingtheoreticalphysicists,anewtheoryofelectromagnetismwasdevelopedusingquantummechanics.Thisfinalmodificationtoelectromagnetictheoryultimatelyledtoquantumelectrodynamics(orQED),whichfullydescribesallelectromagneticphenomenaasbeingmediatedbywaveparticlesknownasphotons.InQED,photonsarethefundamentalexchangeparticlewhichdescribedallinteractionsrelatingtoelectromagnetismincludingtheelectromagneticforce.[Note4]

Itisacommonmisconceptiontoascribethestiffnessandrigidityofsolidmattertotherepulsionoflikechargesundertheinfluenceoftheelectromagneticforce.However,thesecharacteristicsactuallyresultfromthePauliexclusionprinciple.Sinceelectronsarefermions,theycannotoccupythesamequantummechanicalstateasotherelectrons.Whentheelectronsinamaterialaredenselypackedtogether,therearenotenoughlowerenergyquantummechanicalstatesforthemall,sosomeofthemmustbeinhigherenergystates.Thismeansthatittakesenergytopackthemtogether.Whilethiseffectismanifestedmacroscopicallyasastructuralforce,itistechnicallyonlytheresultoftheexistenceofafinitesetofelectronstates.

Nuclear

Therearetwo"nuclearforces"whichtodayareusuallydescribedasinteractionsthattakeplaceinquantumtheoriesofparticlephysics.Thestrongnuclearforce[18]:940istheforceresponsibleforthestructuralintegrityofatomicnucleiwhiletheweaknuclearforce[18]:951isresponsibleforthedecayofcertainnucleonsintoleptonsandothertypesofhadrons.[4][5]

Thestrongforceistodayunderstoodtorepresenttheinteractionsbetweenquarksandgluonsasdetailedbythetheoryofquantumchromodynamics(QCD).[38]Thestrongforceisthefundamentalforcemediatedbygluons,actinguponquarks,antiquarks,andthegluonsthemselves.The(aptlynamed)stronginteractionisthe"strongest"ofthefourfundamentalforces.

http://en.wikipedia.org/wiki/Force 15/25

FNrepresentsthenormalforceexertedontheobject.

Thestrongforceonlyactsdirectlyuponelementaryparticles.However,aresidualoftheforceisobservedbetweenhadrons(thebestknownexamplebeingtheforcethatactsbetweennucleonsinatomicnuclei)asthenuclearforce.Herethestrongforceactsindirectly,transmittedasgluonswhichformpartofthevirtualpiandrhomesonswhichclassicallytransmitthenuclearforce(seethistopicformore).Thefailureofmanysearchesforfreequarkshasshownthattheelementaryparticlesaffectedarenotdirectlyobservable.Thisphenomenoniscalledcolorconfinement.

TheweakforceisduetotheexchangeoftheheavyWandZbosons.Itsmostfamiliareffectisbetadecay(ofneutronsinatomicnuclei)andtheassociatedradioactivity.Theword"weak"derivesfromthefactthatthefieldstrengthissome1013timeslessthanthatofthestrongforce.Still,itisstrongerthangravityovershortdistances.Aconsistentelectroweaktheoryhasalsobeendevelopedwhichshowsthatelectromagneticforcesandtheweakforceareindistinguishableatatemperaturesinexcessofapproximately1015kelvins.SuchtemperatureshavebeenprobedinmodernparticleacceleratorsandshowtheconditionsoftheuniverseintheearlymomentsoftheBigBang.

Nonfundamentalforces

Someforcesareconsequencesofthefundamentalones.Insuchsituations,idealizedmodelscanbeutilizedtogainphysicalinsight.

Normalforce

Thenormalforceisduetorepulsiveforcesofinteractionbetweenatomsatclosecontact.Whentheirelectroncloudsoverlap,Paulirepulsion(duetofermionicnatureofelectrons)followsresultingintheforcewhichactsinadirectionnormaltothesurfaceinterfacebetweentwoobjects.[18]:93Thenormalforce,forexample,isresponsibleforthestructuralintegrityoftablesandfloorsaswellasbeingtheforcethatrespondswheneveranexternalforcepushesonasolidobject.Anexampleofthenormalforceinactionistheimpactforceonanobjectcrashingintoanimmobilesurface.[4][5]

Friction

Frictionisasurfaceforcethatopposesrelativemotion.Thefrictionalforceisdirectlyrelatedtothenormalforcewhichactstokeeptwosolidobjectsseparatedatthepointofcontact.Therearetwobroadclassificationsoffrictionalforces:staticfrictionandkineticfriction.

Thestaticfrictionforce()willexactlyopposeforcesappliedtoanobjectparalleltoasurfacecontactuptothelimitspecifiedbythecoefficientofstaticfriction( )multipliedbythenormalforce().Inotherwordsthemagnitudeofthestaticfrictionforcesatisfiestheinequality:

.

Thekineticfrictionforce()isindependentofboththeforcesappliedandthemovementoftheobject.Thus,themagnitudeoftheforceequals:

,

whereisthecoefficientofkineticfriction.Formostsurfaceinterfaces,thecoefficientofkineticfrictionislessthanthecoefficientofstaticfriction.

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Fkistheforcethatrespondstotheloadonthespring

Tension

Tensionforcescanbemodeledusingidealstringswhicharemassless,frictionless,unbreakable,andunstretchable.Theycanbecombinedwithidealpulleyswhichallowidealstringstoswitchphysicaldirection.Idealstringstransmittensionforcesinstantaneouslyinactionreactionpairssothatiftwoobjectsareconnectedbyanidealstring,anyforcedirectedalongthestringbythefirstobjectisaccompaniedbyaforcedirectedalongthestringintheoppositedirectionbythesecondobject.[39]Byconnectingthesamestringmultipletimestothesameobjectthroughtheuseofasetupthatusesmovablepulleys,thetensionforceonaloadcanbemultiplied.Foreverystringthatactsonaload,anotherfactorofthetensionforceinthestringactsontheload.However,eventhoughsuchmachinesallowforanincreaseinforce,thereisacorrespondingincreaseinthelengthofstringthatmustbedisplacedinordertomovetheload.Thesetandemeffectsresultultimatelyintheconservationofmechanicalenergysincetheworkdoneontheloadisthesamenomatterhowcomplicatedthemachine.[4][5][40]

Elasticforce

Anelasticforceactstoreturnaspringtoitsnaturallength.Anidealspringistakentobemassless,frictionless,unbreakable,andinfinitelystretchable.Suchspringsexertforcesthatpushwhencontracted,orpullwhenextended,inproportiontothedisplacementofthespringfromitsequilibriumposition.[41]ThislinearrelationshipwasdescribedbyRobertHookein1676,forwhomHooke'slawisnamed.Ifisthedisplacement,theforceexertedbyanidealspringequals:

whereisthespringconstant(orforceconstant),whichisparticulartothespring.Theminussignaccountsforthetendencyoftheforcetoactinoppositiontotheappliedload.[4][5]

Continuummechanics

Newton'slawsandNewtonianmechanicsingeneralwerefirstdevelopedtodescribehowforcesaffectidealizedpointparticlesratherthanthreedimensionalobjects.However,inreallife,matterhasextendedstructureandforcesthatactononepartofanobjectmightaffectotherpartsofanobject.Forsituationswherelatticeholdingtogethertheatomsinanobjectisabletoflow,contract,expand,orotherwisechangeshape,thetheoriesofcontinuummechanicsdescribethewayforcesaffectthematerial.Forexample,inextendedfluids,differencesinpressureresultinforcesbeingdirectedalongthepressuregradientsasfollows:

whereisthevolumeoftheobjectinthefluidand isthescalarfunctionthatdescribesthepressureatalllocationsinspace.Pressuregradientsanddifferentialsresultinthebuoyantforceforfluidssuspendedingravitationalfields,windsinatmosphericscience,andtheliftassociatedwithaerodynamicsandflight.[4][5]

Aspecificinstanceofsuchaforcethatisassociatedwithdynamicpressureisfluidresistance:abodyforcethatresiststhemotionofanobjectthroughafluidduetoviscosity.Forsocalled"Stokes'drag"theforceisapproximatelyproportionaltothevelocity,butoppositeindirection:

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Whenthedragforce( )associatedwithairresistancebecomesequalinmagnitudetotheforceofgravityonafallingobject( ),theobjectreachesastateofdynamicequilibriumatterminalvelocity.

Relationshipbetweenforce(F),torque(),andmomentumvectors(pandL)inarotatingsystem.

where:

isaconstantthatdependsonthepropertiesofthefluidandthedimensionsoftheobject(usuallythecrosssectionalarea),and

isthevelocityoftheobject.[4][5]

Moreformally,forcesincontinuummechanicsarefullydescribedbyastresstensorwithtermsthatareroughlydefinedas

whereistherelevantcrosssectionalareaforthevolumeforwhichthestresstensorisbeingcalculated.Thisformalismincludespressuretermsassociatedwithforcesthatactnormaltothecrosssectionalarea(thematrixdiagonalsofthetensor)aswellassheartermsassociatedwithforcesthatactparalleltothecrosssectionalarea(theoffdiagonalelements).Thestresstensoraccountsforforcesthatcauseallstrains(deformations)includingalsotensilestressesandcompressions.[3][5]:133134[35]:3813811

Fictitiousforces

Thereareforceswhichareframedependent,meaningthattheyappearduetotheadoptionofnonNewtonian(thatis,noninertial)referenceframes.SuchforcesincludethecentrifugalforceandtheCoriolisforce.[42]Theseforcesareconsideredfictitiousbecausetheydonotexistinframesofreferencethatarenotaccelerating.[4][5]

Becausetheseforcesarenotgenuinetheyarealsoreferredtoas"pseudoforces".[4]:1211

Ingeneralrelativity,gravitybecomesafictitiousforcethatarisesinsituationswherespacetimedeviatesfromaflatgeometry.Asanextension,KaluzaKleintheoryandstringtheoryascribeelectromagnetismandtheotherfundamentalforcesrespectivelytothecurvatureofdifferentlyscaleddimensions,whichwouldultimatelyimplythatallforcesarefictitious.

Rotationsandtorque

Forcesthatcauseextendedobjectstorotateareassociatedwithtorques.Mathematically,thetorqueofaforceisdefinedrelativetoanarbitraryreferencepointasthecrossproduct:

where

isthepositionvectoroftheforceapplicationpointrelativetothereferencepoint.

Torqueistherotationequivalentofforceinthesamewaythatangleistherotationalequivalentforposition,angularvelocityforvelocity,andangularmomentumformomentum.AsaconsequenceofNewton'sFirstLawofMotion,thereexistsrotational

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inertiathatensuresthatallbodiesmaintaintheirangularmomentumunlessacteduponbyanunbalancedtorque.Likewise,Newton'sSecondLawofMotioncanbeusedtoderiveananalogousequationfortheinstantaneousangularaccelerationoftherigidbody:

where

isthemomentofinertiaofthebodyistheangularaccelerationofthebody.

Thisprovidesadefinitionforthemomentofinertiawhichistherotationalequivalentformass.Inmoreadvancedtreatmentsofmechanics,wheretherotationoveratimeintervalisdescribed,themomentofinertiamustbesubstitutedbythetensorthat,whenproperlyanalyzed,fullydeterminesthecharacteristicsofrotationsincludingprecessionandnutation.

Equivalently,thedifferentialformofNewton'sSecondLawprovidesanalternativedefinitionoftorque:

[43]where istheangularmomentumoftheparticle.

Newton'sThirdLawofMotionrequiresthatallobjectsexertingtorquesthemselvesexperienceequalandoppositetorques,[44]andthereforealsodirectlyimpliestheconservationofangularmomentumforclosedsystemsthatexperiencerotationsandrevolutionsthroughtheactionofinternaltorques.

Centripetalforce

Foranobjectacceleratingincircularmotion,theunbalancedforceactingontheobjectequals:[45]

whereisthemassoftheobject,isthevelocityoftheobjectand isthedistancetothecenterofthecircularpathandistheunitvectorpointingintheradialdirectionoutwardsfromthecenter.Thismeansthattheunbalancedcentripetalforcefeltbyanyobjectisalwaysdirectedtowardthecenterofthecurvingpath.Suchforcesactperpendiculartothevelocityvectorassociatedwiththemotionofanobject,andthereforedonotchangethespeedoftheobject(magnitudeofthevelocity),butonlythedirectionofthevelocityvector.Theunbalancedforcethatacceleratesanobjectcanberesolvedintoacomponentthatisperpendiculartothepath,andonethatistangentialtothepath.Thisyieldsboththetangentialforcewhichacceleratestheobjectbyeitherslowingitdownorspeedingitupandtheradial(centripetal)forcewhichchangesitsdirection.[4][5]

Kinematicintegrals

Forcescanbeusedtodefineanumberofphysicalconceptsbyintegratingwithrespecttokinematicvariables.Forexample,integratingwithrespecttotimegivesthedefinitionofimpulse:[46]

which,byNewton'sSecondLaw,mustbeequivalenttothechangeinmomentum(yieldingtheImpulsemomentumtheorem).

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Similarly,integratingwithrespecttopositiongivesadefinitionfortheworkdonebyaforce:[4]:133

whichisequivalenttochangesinkineticenergy(yieldingtheworkenergytheorem).[4]:133

PowerPistherateofchangedW/dtoftheworkW,asthetrajectoryisextendedbyapositionchangeinatimeintervaldt:[4]:132

with thevelocity.

Potentialenergy

Insteadofaforce,oftenthemathematicallyrelatedconceptofapotentialenergyfieldcanbeusedforconvenience.Forinstance,thegravitationalforceactinguponanobjectcanbeseenastheactionofthegravitationalfieldthatispresentattheobject'slocation.Restatingmathematicallythedefinitionofenergy(viathedefinitionofwork),apotentialscalarfieldisdefinedasthatfieldwhosegradientisequalandoppositetotheforceproducedateverypoint:

Forcescanbeclassifiedasconservativeornonconservative.Conservativeforcesareequivalenttothegradientofapotentialwhilenonconservativeforcesarenot.[4][5]

Conservativeforces

Aconservativeforcethatactsonaclosedsystemhasanassociatedmechanicalworkthatallowsenergytoconvertonlybetweenkineticorpotentialforms.Thismeansthatforaclosedsystem,thenetmechanicalenergyisconservedwheneveraconservativeforceactsonthesystem.Theforce,therefore,isrelateddirectlytothedifferenceinpotentialenergybetweentwodifferentlocationsinspace,[47]andcanbeconsideredtobeanartifactofthepotentialfieldinthesamewaythatthedirectionandamountofaflowofwatercanbeconsideredtobeanartifactofthecontourmapoftheelevationofanarea.[4][5]

Conservativeforcesincludegravity,theelectromagneticforce,andthespringforce.Eachoftheseforceshasmodelswhicharedependentonapositionoftengivenasaradialvectoremanatingfromsphericallysymmetricpotentials.[48]Examplesofthisfollow:

Forgravity:

whereisthegravitationalconstant,andisthemassofobjectn.

Forelectrostaticforces:

where iselectricpermittivityoffreespace,andistheelectricchargeofobjectn.

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Forspringforces:

whereisthespringconstant.[4][5]

Nonconservativeforces

Forcertainphysicalscenarios,itisimpossibletomodelforcesasbeingduetogradientofpotentials.Thisisoftenduetomacrophysicalconsiderationswhichyieldforcesasarisingfromamacroscopicstatisticalaverageofmicrostates.Forexample,frictioniscausedbythegradientsofnumerouselectrostaticpotentialsbetweentheatoms,butmanifestsasaforcemodelwhichisindependentofanymacroscalepositionvector.Nonconservativeforcesotherthanfrictionincludeothercontactforces,tension,compression,anddrag.However,foranysufficientlydetaileddescription,alltheseforcesaretheresultsofconservativeonessinceeachofthesemacroscopicforcesarethenetresultsofthegradientsofmicroscopicpotentials.[4][5]

Theconnectionbetweenmacroscopicnonconservativeforcesandmicroscopicconservativeforcesisdescribedbydetailedtreatmentwithstatisticalmechanics.Inmacroscopicclosedsystems,nonconservativeforcesacttochangetheinternalenergiesofthesystem,andareoftenassociatedwiththetransferofheat.AccordingtotheSecondlawofthermodynamics,nonconservativeforcesnecessarilyresultinenergytransformationswithinclosedsystemsfromorderedtomorerandomconditionsasentropyincreases.[4][5]

Unitsofmeasurement

TheSIunitofforceisthenewton(symbolN),whichistheforcerequiredtoaccelerateaonekilogrammassatarateofonemeterpersecondsquared,orkgms2.[49]ThecorrespondingCGSunitisthedyne,theforcerequiredtoaccelerateaonegrammassbyonecentimeterpersecondsquared,orgcms2.Anewtonisthusequalto100,000dynes.

ThegravitationalfootpoundsecondEnglishunitofforceisthepoundforce(lbf),definedastheforceexertedbygravityonapoundmassinthestandardgravitationalfieldof9.80665ms2.[49]Thepoundforceprovidesanalternativeunitofmass:oneslugisthemassthatwillacceleratebyonefootpersecondsquaredwhenactedonbyonepoundforce.[49]

Analternativeunitofforceinadifferentfootpoundsecondsystem,theabsolutefpssystem,isthepoundal,definedastheforcerequiredtoaccelerateaonepoundmassatarateofonefootpersecondsquared.[49]TheunitsofslugandpoundalaredesignedtoavoidaconstantofproportionalityinNewton'sSecondLaw.

Thepoundforcehasametriccounterpart,lesscommonlyusedthanthenewton:thekilogramforce(kgf)(sometimeskilopond),istheforceexertedbystandardgravityononekilogramofmass.[49]Thekilogramforceleadstoanalternate,butrarelyusedunitofmass:themetricslug(sometimesmugorhyl)isthatmasswhichacceleratesat1ms2whensubjectedtoaforceof1kgf.ThekilogramforceisnotapartofthemodernSIsystem,andisgenerallydeprecatedhoweveritstillseesuseforsomepurposesasexpressingaircraftweight,jetthrust,bicyclespoketension,torquewrenchsettingsandengineoutputtorque.Otherarcaneunitsofforceincludethesthnewhichisequivalentto1000Nandthekipwhichisequivalentto1000lbf.

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Unitsofforcenewton(SIunit)

dyne kilogramforce,kilopond poundforce poundal

1N 1kgm/s2 =105dyn 0.10197kp 0.22481lbF 7.2330pdl

1dyn =105N 1gcm/s2 1.0197106kp 2.2481106lbF 7.2330105pdl

1kp =9.80665N =980665dyn gn(1kg) 2.2046lbF 70.932pdl

1lbF 4.448222N 444822dyn 0.45359kp gn(1lb) 32.174pdl

1pdl 0.138255N 13825dyn 0.014098kp 0.031081lbF 1lbft/s2

Thevalueofgnasusedintheofficialdefinitionofthekilogramforceisusedhereforallgravitationalunits.

SeealsoTonforce.

Forcemeasurement

Seeforcegauge,springscale,loadcell

Seealso

Ordersofmagnitude(force)

Notes

1. ^Newton'sPrincipiaMathematicaactuallyusedafinitedifferenceversionofthisequationbaseduponimpulse.SeeImpulse.

2. ^"ItisimportanttonotethatwecannotderiveageneralexpressionforNewton'ssecondlawforvariablemasssystemsbytreatingthemassinF=dP/dt=d(Mv)asavariable.[...]WecanuseF=dP/dttoanalyzevariablemasssystemsonlyifweapplyittoanentiresystemofconstantmasshavingpartsamongwhichthereisaninterchangeofmass."[Emphasisasintheoriginal](Halliday,Resnick&Krane2001,p.199)

3. ^"Anysingleforceisonlyoneaspectofamutualinteractionbetweentwobodies."(Halliday,Resnick&Krane2001,pp.7879)

4. ^ForacompletelibraryonquantummechanicsseeQuantummechanicsReferences

References

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2. ^abHeath,T.L."TheWorksofArchimedes(1897).TheunabridgedworkinPDFform(19MB)"(http://www.archive.org/details/worksofarchimede029517mbp).InternetArchive.Retrieved20071014.

3. ^abcdefghUniversityPhysics,Sears,Young&Zemansky,pp.1838

4. ^abcdefghijklmnopqrstuvwxyzaaabFeynmanvolume1

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6. ^abcWeinberg,S.(1994).DreamsofaFinalTheory.VintageBooksUSA.ISBN0679744088.7. ^Lang,HelenS.(1998).TheorderofnatureinAristotle'sphysics:placeandtheelements(1.publ.ed.).Cambridge:

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CambridgeUniv.Press.ISBN9780521624534.8. ^Hetherington,NorrissS.(1993).Cosmology:Historical,Literary,Philosophical,Religious,andScientific

Perspectives.GarlandReferenceLibraryoftheHumanities.p.100.ISBN0815310854.

9. ^abDrake,Stillman(1978).GalileoAtWork.Chicago:UniversityofChicagoPress.ISBN0226162265

10. ^abcdefNewton,Isaac(1999).ThePrincipiaMathematicalPrinciplesofNaturalPhilosophy.Berkeley:UniversityofCaliforniaPress.ISBN0520088174.ThisisarecenttranslationintoEnglishbyI.BernardCohenandAnneWhitman,withhelpfromJuliaBudenz.

11. ^DiSalle,Robert(20020330)."SpaceandTime:InertialFrames"(http://plato.stanford.edu/entries/spacetimeiframes/).StanfordEncyclopediaofPhilosophy.Retrieved20080324.

12. ^Howland,R.A.(2006).Intermediatedynamicsalinearalgebraicapproach(OnlineAusg.ed.).NewYork:Springer.pp.255256.ISBN9780387280592.

13. ^Oneexceptiontothisruleis:Landau,L.D.Akhiezer,A.I.Lifshitz,A.M.(196).GeneralPhysicsmechanicsandmolecularphysics(FirstEnglished.).Oxford:PergamonPress.ISBN0080033040.Translatedby:J.B.Sykes,A.D.Petford,andC.L.Petford.LibraryofCongressCatalogNumber6730260.Insection7,pages1214,thisbookdefinesforceasdp/dt.

14. ^Jammer,Max(1999).Conceptsofforce:astudyinthefoundationsofdynamics(Facsim.ed.).Mineola,N.Y.:DoverPublications.pp.220222.ISBN9780486406893.

15. ^Noll,Walter(April2007)."OntheConceptofForce"(http://www.math.cmu.edu/~wn0g/Force.pdf)(pdf).CarnegieMellonUniversity.Retrieved28October2013.

16. ^C.Hellingman(1992)."Newton'sthirdlawrevisited".Phys.Educ.27(2):112115.Bibcode:1992PhyEd..27..112H(http://adsabs.harvard.edu/abs/1992PhyEd..27..112H).doi:10.1088/00319120/27/2/011(https://dx.doi.org/10.1088%2F00319120%2F27%2F2%2F011)."QuotingNewtoninthePrincipia:ItisnotoneactionbywhichtheSunattractsJupiter,andanotherbywhichJupiterattractstheSunbutitisoneactionbywhichtheSunandJupitermutuallyendeavourtocomenearertogether."

17. ^Dr.Nikitin(2007)."Dynamicsoftranslationalmotion"(http://physicshelp.info/physicsguide/mechanics/translational_dynamics.shtml).Retrieved20080104.

18. ^abcdeCutnell&Johnson200319. ^"Seminar:VisualizingSpecialRelativity"(http://www.anu.edu.au/Physics/Searle/Obsolete/Seminar.html).The

RelativisticRaytracer.Retrieved20080104.20. ^Wilson,JohnB."FourVectors(4Vectors)ofSpecialRelativity:AStudyofElegantPhysics"

(http://SciRealm.com/4Vectors.html).TheScienceRealm:John'sVirtualSciTechUniverse.Archived(http://web.archive.org/web/20090626152836/http://www.austininc.com/SciRealm/4Vectors.html)fromtheoriginalon26June2009.Retrieved20080104.

21. ^"IntroductiontoFreeBodyDiagrams"(http://eta.physics.uoguelph.ca/tutorials/fbd/intro.html).PhysicsTutorialMenu.UniversityofGuelph.Retrieved20080102.

22. ^Henderson,Tom(2004)."ThePhysicsClassroom"(http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/vectors/u3l1b.html).ThePhysicsClassroomandMathsoftEngineering&Education,Inc.Retrieved20080102.

23. ^"StaticEquilibrium"(http://web.archive.org/web/20071019054156/http://www.uvi.edu/Physics/SCI3xxWeb/Structure/StaticEq.html).PhysicsStaticEquilibrium(forcesandtorques).UniversityoftheVirginIslands.Archivedfromtheoriginal(http://www.uvi.edu/Physics/SCI3xxWeb/Structure/StaticEq.html)onOctober19,2007.Retrieved20080102.

24. ^abShifman,Mikhail(1999).ITEPlecturesonparticlephysicsandfieldtheory.WorldScientific.ISBN981022639X.

25. ^Nave,CarlRod."PauliExclusionPrinciple"(http://hyperphysics.phyastr.gsu.edu/hbase/pauli.html).HyperPhysics.

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UniversityofGuelph.Retrieved20131028.26. ^"Fermions&Bosons"(http://particleadventure.org/frameless/fermibos.html).TheParticleAdventure.Retrieved

20080104.27. ^http://www.pha.jhu.edu/~dfehling/particle.gif28. ^Cook,A.H.(161601965)."ANewAbsoluteDeterminationoftheAccelerationduetoGravityattheNational

PhysicalLaboratory"(http://www.nature.com/nature/journal/v208/n5007/abs/208279a0.html).Nature208(5007):279.Bibcode:1965Natur.208..279C(http://adsabs.harvard.edu/abs/1965Natur.208..279C).doi:10.1038/208279a0(https://dx.doi.org/10.1038%2F208279a0).Retrieved20080104.Checkdatevaluesin:|date=(help)

29. ^abUniversityPhysics,Sears,Young&Zemansky,pp598230. ^"SirIsaacNewton:TheUniversalLawofGravitation"

(http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html).Astronomy161TheSolarSystem.Retrieved20080104.

31. ^Watkins,Thayer."PerturbationAnalysis,RegularandSingular"(http://www.sjsu.edu/faculty/watkins/perturb.htm).DepartmentofEconomics.SanJosStateUniversity.

32. ^Kollerstrom,Nick(2001)."Neptune'sDiscovery.TheBritishCaseforCoPrediction."(http://web.archive.org/web/20051111190351/http://www.ucl.ac.uk/sts/nk/neptune/index.htm).UniversityCollegeLondon.Archivedfromtheoriginal(http://www.ucl.ac.uk/sts/nk/neptune/index.htm)on20051111.Retrieved20070319.

33. ^Einstein,Albert(1916)."TheFoundationoftheGeneralTheoryofRelativity"(http://www.alberteinstein.info/gallery/gtext3.html)(PDF).AnnalenderPhysik49(7):769822.Bibcode:1916AnP...354..769E(http://adsabs.harvard.edu/abs/1916AnP...354..769E).doi:10.1002/andp.19163540702(https://dx.doi.org/10.1002%2Fandp.19163540702).Retrieved20060903.

34. ^Coulomb,Charles(1784)."Recherchesthoriquesetexprimentalessurlaforcedetorsionetsurl'lasticitdesfilsdemetal".Histoiredel'AcadmieRoyaledesSciences:229269.

35. ^abcFeynmanvolume236. ^Scharf,Toralf(2007).Polarizedlightinliquidcrystalsandpolymers(http://books.google.com/?

id=CQNE13opFucC).JohnWileyandSons.p.19.ISBN0471740640.,Chapter2,p.19(http://books.google.com/books?id=CQNE13opFucC&pg=PA19)

37. ^Duffin,William(1980).ElectricityandMagnetism,3rdEd.McGrawHill.pp.364383.ISBN007084111X.38. ^Stevens,Tab(10/07/2003)."QuantumChromodynamics:ADefinitionScienceArticles"

(http://web.archive.org/web/20111016103116/http://www.physicspost.com/sciencearticle168.html).Archivedfromtheoriginal(http://www.physicspost.com/sciencearticle168.html)on20111016.Retrieved20080104.Checkdatevaluesin:|date=(help)

39. ^"TensionForce"(http://www.mtsu.edu/~phys2010/Lectures/Part_2__L6__L11/Lecture_9/Tension_Force/tension_force.html).NonCalculusBasedPhysicsI.Retrieved20080104.

40. ^Fitzpatrick,Richard(20060202)."Strings,pulleys,andinclines"(http://farside.ph.utexas.edu/teaching/301/lectures/node48.html).Retrieved20080104.

41. ^Nave,CarlRod."Elasticity"(http://hyperphysics.phyastr.gsu.edu/hbase/permot2.html).HyperPhysics.UniversityofGuelph.Retrieved20131028.

42. ^Mallette,Vincent(19822008)."InwitPublishing,Inc.andInwit,LLCWritings,LinksandSoftwareDistributionsTheCoriolisForce"(http://www.algorithm.com/inwit/writings/coriolisforce.html).PublicationsinScienceandMathematics,ComputingandtheHumanities.InwitPublishing,Inc.Retrieved20080104.

43. ^Nave,CarlRod."Newton's2ndLaw:Rotation"(http://hyperphysics.phyastr.gsu.edu/HBASE/n2r.html).HyperPhysics.UniversityofGuelph.Retrieved20131028.

44. ^Fitzpatrick,Richard(20070107)."Newton'sthirdlawofmotion"(http://farside.ph.utexas.edu/teaching/336k/lectures/node26.html).Retrieved20080104.

http://en.wikipedia.org/wiki/Force 24/25

WikimediaCommonshasmediarelatedtoForces.

LookupforceinWiktionary,thefreedictionary.

Furtherreading

Corben,H.C.PhilipStehle(1994).ClassicalMechanics.NewYork:Doverpublications.pp.2831.ISBN0486680630.Cutnell,JohnD.Johnson,KennethW.(2003).Physics,SixthEdition.Hoboken,NewJersey:JohnWiley&SonsInc.ISBN0471151831.Feynman,RichardP.LeightonSands,Matthew(2010).TheFeynmanlecturesonphysics.Vol.I:Mainlymechanics,radiationandheat(Newmillenniumed.).NewYork:BasicBooks.ISBN9780465024933.Feynman,RichardP.Leighton,RobertB.Sands,Matthew(2010).TheFeynmanlecturesonphysics.Vol.II:Mainlyelectromagnetismandmatter(Newmillenniumed.).NewYork:BasicBooks.ISBN9780465024940.Halliday,DavidResnick,RobertKrane,KennethS.(2001).Physicsv.1.NewYork:JohnWiley&Sons.ISBN0471320579.Kleppner,DanielKolenkow,RobertJ.(2010).Anintroductiontomechanics(3.printed.).Cambridge:CambridgeUniversityPress.ISBN0521198216.Parker,Sybil(1993)."force".EncyclopediaofPhysics.Ohio:McGrawHill.p.107,.ISBN0070514003.SearsF.,ZemanskyM.&YoungH.(1982).UniversityPhysics.Reading,Massachusetts:AddisonWesley.ISBN0201071991.Serway,RaymondA.(2003).PhysicsforScientistsandEngineers.Philadelphia:SaundersCollegePublishing.ISBN0534408427.Tipler,Paul(2004).PhysicsforScientistsandEngineers:Mechanics,OscillationsandWaves,Thermodynamics(5thed.).W.H.Freeman.ISBN0716708094.Verma,H.C.(2004).ConceptsofPhysicsVol1.(2004Reprinted.).BhartiBhavan.ISBN8177091875.

Externallinks

VideolectureonNewton'sthreelaws(http://ocw.mit.edu/OcwWeb/Physics/801PhysicsIFall1999/VideoLectures/detail/VideoSegmentIndexforL6.htm)byWalterLewinfromMITOpenCourseWareAJavasimulationonvectoradditionofforces(http://phy.hk/wiki/englishhtm/Vector.htm)Forcedemonstratedasanyinfluenceonanobjectthatchangestheobject'sshapeormotion(video)

(http://farside.ph.utexas.edu/teaching/336k/lectures/node26.html).Retrieved20080104.45. ^Nave,CarlRod."CentripetalForce"(http://hyperphysics.phyastr.gsu.edu/hbase/cf.html).HyperPhysics.University

ofGuelph.Retrieved20131028.46. ^Hibbeler,RussellC.(2010).EngineeringMechanics,12thedition.PearsonPrenticeHall.p.222.ISBN013

6077919.47. ^Singh,SunilKumar(20070825)."Conservativeforce"(http://cnx.org/content/m14104/latest/).Connexions.

Retrieved20080104.48. ^Davis,Doug."ConservationofEnergy"(http://www.ux1.eiu.edu/~cfadd/1350/08PotEng/ConsF.html).General

physics.Retrieved20080104.

49. ^abcdeWandmacher,CorneliusJohnson,Arnold(1995).MetricUnitsinEngineering.ASCEPublications.p.15.ISBN0784400709.

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Categories: Naturalphilosophy Classicalmechanics Conceptsinphysics Force Physicalquantities

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