pendulum - wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Pendulum 1/31

"Simplegravitypendulum"modelassumesnofrictionorairresistance.

Animationofapendulumshowingthevelocityandaccelerationvectors

PendulumFromWikipedia,thefreeencyclopedia

Apendulumisaweightsuspendedfromapivotsothatitcanswingfreely.[1]Whenapendulumisdisplacedsidewaysfromitsrestingequilibriumposition,itissubjecttoarestoringforceduetogravitythatwillaccelerateitbacktowardtheequilibriumposition.Whenreleased,therestoringforcecombinedwiththependulum'smasscausesittooscillateabouttheequilibriumposition,swingingbackandforth.Thetimeforonecompletecycle,aleftswingandarightswing,iscalledtheperiod.Theperioddependsonthelengthofthependulum,andalsotoaslightdegreeontheamplitude,thewidthofthependulum'sswing.

Fromitsexaminationinaround1602byGalileoGalilei,theregularmotionofpendulumswasusedfortimekeeping,andwastheworld'smostaccuratetimekeepingtechnologyuntilthe1930s.[2]Pendulumsareusedtoregulatependulumclocks,andareusedinscientificinstrumentssuchasaccelerometersandseismometers.Historicallytheywereusedasgravimeterstomeasuretheaccelerationofgravityingeophysicalsurveys,andevenasastandardoflength.Theword'pendulum'isnewLatin,fromtheLatinpendulus,meaning'hanging'.[3]

Thesimplegravitypendulum[4]isanidealizedmathematicalmodelofapendulum.[5][6][7]Thisisaweight(orbob)ontheendofamasslesscordsuspendedfromapivot,withoutfriction.Whengivenaninitialpush,itwillswingbackandforthataconstantamplitude.Realpendulumsaresubjecttofrictionandairdrag,sotheamplitudeoftheirswingsdeclines.

Contents

1Periodofoscillation2Compoundpendulum3History

3.11602:Galileo'sresearch3.21656:Thependulumclock3.31673:Huygens'HorologiumOscillatorium3.41721:Temperaturecompensatedpendulums3.51851:Foucaultpendulum


3.61930:Declineinuse4Usefortimemeasurement

4.1Clockpendulums4.2Temperaturecompensation

4.2.1Mercurypendulum4.2.2Gridironpendulum4.2.3Invarandfusedquartz

4.3Atmosphericpressure4.4Gravity

5Accuracyofpendulumsastimekeepers5.1Qfactor5.2Escapement5.3TheAirycondition

6Gravitymeasurement6.1Thesecondspendulum6.2Earlyobservations6.3Kater'spendulum6.4Laterpendulumgravimeters

7Standardoflength7.1Earlyproposals7.2Themetre7.3BritainandDenmark

8Otheruses8.1Seismometers8.2Schulertuning8.3Coupledpendulums8.4Religiouspractice8.5Execution

9Seealso10Notes11References12Furtherreading13Externallinks

Periodofoscillation

Theperiodofswingofasimplegravitypendulumdependsonitslength,thelocalstrengthofgravity,andtoasmallextentonthemaximumanglethatthependulumswingsawayfromvertical,0,calledtheamplitude.[8]Itis

independentofthemassofthebob.Iftheamplitudeislimitedtosmallswings,[Note1]theperiodTofasimple


Theperiodofapendulumgetslongerastheamplitude0(widthofswing)increases.

Thetrueperiodisshownasasolidpendulum,thesimpleperiodequationisshownasanemptypendulum

pendulum,thetimetakenforacompletecycle,is:[9]

whereListhelengthofthependulumandgisthelocalaccelerationofgravity.

Forsmallswingstheperiodofswingisapproximatelythesamefordifferentsizeswings:thatis,theperiodisindependentofamplitude.Thisproperty,calledisochronism,isthereasonpendulumsaresousefulfortimekeeping.[10]Successiveswingsofthependulum,evenifchanginginamplitude,takethesameamountoftime.

Forlargeramplitudes,theperiodincreasesgraduallywithamplitudesoitislongerthangivenbyequation(1).Forexample,atanamplitudeof0=23itis1%largerthangivenby(1).Theperiodincreasesasymptotically(toinfinity)as0approaches180,becausethevalue0=180isanunstableequilibriumpointforthependulum.Thetrueperiodofanidealsimplegravitypendulumcanbewritteninseveraldifferentforms(seePendulum(mathematics)),oneexamplebeingtheinfiniteseries:[11][12]

Thedifferencebetweenthistrueperiodandtheperiodforsmallswings(1)aboveiscalledthecircularerror.Inthecaseofatypicalgrandfatherclockwhosependulumhasaswingof6andthusanamplitudeof3(0.05radians),thedifferencebetweenthetrueperiodandthesmallangleapproximation(1)amountstoabout15secondsperday.

Forsmallswingsthependulumapproximatesaharmonicoscillator,anditsmotionasafunctionoftime,t,isapproximatelysimpleharmonicmotion:[5]


Forrealpendulums,correctionstotheperiodmaybeneededtotakeintoaccountthepresenceofair,themassofthestring,thesizeandshapeofthebobandhowitisattachedtothestring,flexibilityandstretchingofthestring,motionofthesupport,andlocalgravitationalgradients.[11][13][14][15]

Compoundpendulum

ThelengthLoftheidealsimplependulumdiscussedaboveisthedistancefromthepivotpointtothecenterofmassofthebob.Anyswingingrigidbodyfreetorotateaboutafixedhorizontalaxisiscalledacompoundpendulumorphysicalpendulum.TheappropriateequivalentlengthLforcalculatingtheperiodofanysuchpendulumisthedistancefromthepivottothecenterofoscillation.[16]Thispointislocatedunderthecenterofmassatadistancefromthepivottraditionallycalledtheradiusofoscillation,whichdependsonthemassdistributionofthependulum.Ifmostofthemassisconcentratedinarelativelysmallbobcomparedtothependulumlength,thecenterofoscillationisclosetothecenterofmass.[17]

TheradiusofoscillationorequivalentlengthLofanyphysicalpendulumcanbeshowntobe

whereIisthemomentofinertiaofthependulumaboutthepivotpoint,misthemassofthependulum,andRisthedistancebetweenthepivotpointandthecenterofmass.Substitutingthisexpressionin(1)above,theperiodTofacompoundpendulumisgivenby

forsufficientlysmalloscillations.[18]

ArigiduniformrodoflengthLpivotedabouteitherendhasmomentofinertiaI=(1/3)mL2.Thecenterofmassislocatedatthecenteroftherod,soR=L/2.SubstitutingthesevaluesintotheaboveequationgivesT=22L/3g.Thisshowsthatarigidrodpendulumhasthesameperiodasasimplependulumof2/3itslength.

ChristiaanHuygensprovedin1673thatthepivotpointandthecenterofoscillationareinterchangeable.[19]Thismeansifanypendulumisturnedupsidedownandswungfromapivotlocatedatitspreviouscenterofoscillation,itwillhavethesameperiodasbeforeandthenewcenterofoscillationwillbeattheoldpivotpoint.In1817HenryKaterusedthisideatoproduceatypeofreversiblependulum,nowknownasaKaterpendulum,forimprovedmeasurementsoftheaccelerationduetogravity.

History

Oneoftheearliestknownusesofapendulumwasa1stcenturyseismometerdeviceofHanDynastyChinesescientistZhangHeng.[20]Itsfunctionwastoswayandactivateoneofaseriesofleversafterbeingdisturbedbythetremorofanearthquakefaraway.[21]Releasedbyalever,asmallballwouldfalloutoftheurnshapeddeviceintooneofeightmetaltoad'smouthsbelow,attheeightpointsofthecompass,signifyingthedirectiontheearthquakewaslocated.[21]

Manysources[22][23][24][25]claimthatthe10thcenturyEgyptianastronomerIbnYunususedapendulumfortimemeasurement,butthiswasanerrorthatoriginatedin1684withtheBritishhistorianEdwardBernard.[26][27][28]


ReplicaofZhangHeng'sseismometer.Thependulumiscontainedinside.

Thefirstpendulumclock

DuringtheRenaissance,largependulumswereusedassourcesofpowerformanualreciprocatingmachinessuchassaws,bellows,andpumps.[29]LeonardodaVincimademanydrawingsofthemotionofpendulums,thoughwithoutrealizingitsvaluefortimekeeping.

1602:Galileo'sresearch

ItalianscientistGalileoGalileiwasthefirsttostudythepropertiesofpendulums,beginningaround1602.[30]TheearliestextantreportofhisresearchiscontainedinalettertoGuidoUbaldodalMonte,fromPadua,datedNovember29,1602.[31]Hisbiographerandstudent,VincenzoViviani,claimedhisinteresthadbeensparkedaround1582bytheswingingmotionofachandelierinthePisacathedral.[32][33]Galileodiscoveredthecrucialpropertythatmakespendulumsusefulastimekeepers,calledisochronismtheperiodofthependulumisapproximatelyindependentoftheamplitudeorwidthoftheswing.[34]Healsofoundthattheperiodisindependentofthemassofthebob,andproportionaltothesquarerootofthelengthofthependulum.Hefirstemployedfreeswingingpendulumsinsimpletimingapplications.Aphysicianfriendinventedadevicewhichmeasuredapatient'spulsebythelengthofapendulumthepulsilogium.[30]In1641GalileoconceivedanddictatedtohissonVincenzoadesignforapendulumclock[34]Vincenzobeganconstruction,buthadnotcompleteditwhenhediedin1649.[35]Thependulumwasthefirstharmonicoscillatorusedbyman.[34]

1656:Thependulumclock

In1656theDutchscientistChristiaanHuygensbuiltthefirstpendulumclock.[36]Thiswasagreatimprovementoverexistingmechanicalclockstheirbestaccuracywasincreasedfromaround15minutesdeviationadaytoaround15secondsaday.[37]PendulumsspreadoverEuropeasexistingclockswereretrofittedwiththem.[38]

TheEnglishscientistRobertHookestudiedtheconicalpendulumaround1666,consistingofapendulumthatisfreetoswingintwodimensions,withthebobrotatinginacircleorellipse.[39]Heusedthemotionsofthisdeviceasamodeltoanalyzetheorbitalmotionsoftheplanets.[40]HookesuggestedtoIsaacNewtonin1679thatthecomponentsoforbitalmotionconsistedofinertialmotionalongatangentdirectionplusanattractivemotionintheradialdirection.ThisplayedapartinNewton'sformulationofthelawofuniversalgravitation.[41][42]RobertHookewasalsoresponsibleforsuggestingasearlyas1666thatthependulumcouldbeusedtomeasuretheforceofgravity.[39]

DuringhisexpeditiontoCayenne,FrenchGuianain1671,JeanRicherfoundthatapendulumclockwas212minutesperdaysloweratCayennethanatParis.FromthishededucedthattheforceofgravitywasloweratCayenne.[43][44]In1687,IsaacNewtoninPrincipiaMathematicashowedthatthiswasbecausetheEarthwasnotatruespherebutslightlyoblate(flattenedatthepoles)fromtheeffectofcentrifugalforceduetoitsrotation,


causinggravitytoincreasewithlatitude.[45]Portablependulumsbegantobetakenonvoyagestodistantlands,asprecisiongravimeterstomeasuretheaccelerationofgravityatdifferentpointsonEarth,eventuallyresultinginaccuratemodelsoftheshapeoftheEarth.[46]

1673:Huygens'HorologiumOscillatorium

In1673,ChristiaanHuygenspublishedhistheoryofthependulum,HorologiumOscillatoriumsivedemotupendulorum.[47][48]MarinMersenneandRenDescarteshaddiscoveredaround1636thatthependulumwasnotquiteisochronousitsperiodincreasedsomewhatwithitsamplitude.[49]Huygensanalyzedthisproblembydeterminingwhatcurveanobjectmustfollowtodescendbygravitytothesamepointinthesametimeinterval,regardlessofstartingpointthesocalledtautochronecurve.Byacomplicatedmethodthatwasanearlyuseofcalculus,heshowedthiscurvewasacycloid,ratherthanthecirculararcofapendulum,[50]confirmingthatthependulumwasnotisochronousandGalileo'sobservationofisochronismwasaccurateonlyforsmallswings.[51]Huygensalsosolvedtheproblemofhowtocalculatetheperiodofanarbitrarilyshapedpendulum(calledacompoundpendulum),discoveringthecenterofoscillation,anditsinterchangeabilitywiththepivotpoint.[52]

Theexistingclockmovement,thevergeescapement,madependulumsswinginverywidearcsofabout100.[53]Huygensshowedthiswasasourceofinaccuracy,causingtheperiodtovarywithamplitudechangescausedbysmallunavoidablevariationsintheclock'sdriveforce.[54]Tomakeitsperiodisochronous,Huygensmountedcycloidalshapedmetal'chops'nexttothepivotsinhisclocks,thatconstrainedthesuspensioncordandforcedthependulumtofollowacycloidarc.[55]Thissolutiondidn'tproveaspracticalassimplylimitingthependulum'sswingtosmallanglesofafewdegrees.Therealizationthatonlysmallswingswereisochronousmotivatedthedevelopmentoftheanchorescapementaround1670,whichreducedthependulumswinginclocksto46.[53][56]

1721:Temperaturecompensatedpendulums

Duringthe18thand19thcentury,thependulumclock'sroleasthemostaccuratetimekeepermotivatedmuchpracticalresearchintoimprovingpendulums.Itwasfoundthatamajorsourceoferrorwasthatthependulumrodexpandedandcontractedwithchangesinambienttemperature,changingtheperiodofswing.[8][57]Thiswassolvedwiththeinventionoftemperaturecompensatedpendulums,themercurypendulumin1721[58]andthegridironpendulumin1726,reducingerrorsinprecisionpendulumclockstoafewsecondsperweek.[55]

Theaccuracyofgravitymeasurementsmadewithpendulumswaslimitedbythedifficultyoffindingthelocationoftheircenterofoscillation.Huygenshaddiscoveredin1673thatapendulumhasthesameperiodwhenhungfromitscenterofoscillationaswhenhungfromitspivot,[19]andthedistancebetweenthetwopointswasequaltothelengthofasimplegravitypendulumofthesameperiod.[16]In1818BritishCaptainHenryKaterinventedthereversibleKater'spendulum[59]whichusedthisprinciple,makingpossibleveryaccuratemeasurementsofgravity.Forthenextcenturythereversiblependulumwasthestandardmethodofmeasuringabsolutegravitationalacceleration.

1851:Foucaultpendulum

In1851,JeanBernardLonFoucaultshowedthattheplaneofoscillationofapendulum,likeagyroscope,tendstostayconstantregardlessofthemotionofthepivot,andthatthiscouldbeusedtodemonstratetherotationoftheEarth.Hesuspendedapendulumfreetoswingintwodimensions(laternamedtheFoucaultpendulum)fromthedomeofthePanthoninParis.Thelengthofthecordwas67m(220ft).Oncethependulumwassetinmotion,theplaneofswingwasobservedtoprecessorrotate360clockwiseinabout32hours.[60]Thiswasthe


TheFoucaultpendulumin1851wasthefirstdemonstrationoftheEarth'srotationthatdidnotinvolvecelestialobservations,anditcreateda"pendulummania".Inthisanimationtherateofprecessionisgreatlyexaggerated.

firstdemonstrationoftheEarth'srotationthatdidn'tdependoncelestialobservations,[61]anda"pendulummania"brokeout,asFoucaultpendulumsweredisplayedinmanycitiesandattractedlargecrowds.[62][63]

1930:Declineinuse

Around1900lowthermalexpansionmaterialsbegantobeusedforpendulumrodsinthehighestprecisionclocksandotherinstruments,firstinvar,anickelsteelalloy,andlaterfusedquartz,whichmadetemperaturecompensationtrivial.[64]Precisionpendulumswerehousedinlowpressuretanks,whichkepttheairpressureconstanttopreventchangesintheperiodduetochangesinbuoyancyofthependulumduetochangingatmosphericpressure.[64]Theaccuracyofthebestpendulumclockstoppedoutataroundasecondperyear.[65][66]

Thetimekeepingaccuracyofthependulumwasexceededbythequartzcrystaloscillator,inventedin1921,andquartzclocks,inventedin1927,replacedpendulumclocksastheworld'sbesttimekeepers.[2]PendulumclockswereusedastimestandardsuntilWorldWar2,althoughtheFrenchTimeServicecontinuedusingthemintheirofficialtimestandardensembleuntil1954.[67]Pendulumgravimetersweresupersededby"freefall"gravimetersinthe1950s,[68]butpenduluminstrumentscontinuedtobeusedintothe1970s.


Longcaseclock(Grandfatherclock)pendulum

OrnamentedpenduluminaFrenchComtoiseclock

Mercurypendulum Gridironpendulum

Ellicottpendulum,anothertemperaturecompensatedtype

InvarpenduluminlowpressuretankinRieflerregulatorclock,usedastheUStimestandardfrom1909to1929

Clockpendulums

Usefortimemeasurement

For300years,fromitsdiscoveryaround1581untildevelopmentofthequartzclockinthe1930s,thependulumwastheworld'sstandardforaccuratetimekeeping.[2][69]Inadditiontoclockpendulums,freeswingingsecondspendulumswerewidelyusedasprecisiontimersinscientificexperimentsinthe17thand18thcenturies.Pendulumsrequiregreatmechanicalstability:alengthchangeofonly0.02%,0.2mminagrandfatherclockpendulum,willcauseanerrorofaminuteperweek.[70]

Clockpendulums

Pendulumsinclocks(seeexampleatright)areusuallymadeofaweightorbob(b)suspendedbyarodofwoodormetal(a).[8][71]Toreduceairresistance(whichaccountsformostoftheenergylossinclocks)[72]thebobistraditionallyasmoothdiskwithalensshapedcrosssection,althoughinantiqueclocksitoftenhadcarvingsordecorationsspecifictothetypeofclock.Inqualityclocksthebobismadeasheavyasthesuspensioncansupportandthemovementcandrive,sincethisimprovestheregulationoftheclock(seeAccuracybelow).Acommonweightforsecondspendulumbobsis15pounds(6.8kg).[73]Insteadofhangingfromapivot,clockpendulumsareusuallysupportedbyashortstraightspring(d)offlexiblemetalribbon.Thisavoidsthefrictionand'play'causedbyapivot,andtheslightbendingforceofthespringmerelyaddstothependulum'srestoringforce.Afewprecisionclockshavepivotsof'knife'bladesrestingonagateplates.Theimpulsestokeepthependulumswingingareprovidedbyanarmhangingbehindthependulumcalledthecrutch,(e),whichendsinafork,(f)whoseprongsembracethependulumrod.Thecrutchispushedbackandforthbytheclock'sescapement,(g,h).


Pendulumandanchorescapementfromagrandfatherclock

Eachtimethependulumswingsthroughitscentreposition,itreleasesonetoothoftheescapewheel(g).Theforceoftheclock'smainspringoradrivingweighthangingfromapulley,transmittedthroughtheclock'sgeartrain,causesthewheeltoturn,andatoothpressesagainstoneofthepallets(h),givingthependulumashortpush.Theclock'swheels,gearedtotheescapewheel,moveforwardafixedamountwitheachpendulumswing,advancingtheclock'shandsatasteadyrate.

Thependulumalwayshasameansofadjustingtheperiod,usuallybyanadjustmentnut(c)underthebobwhichmovesitupordownontherod.[8][74]Movingthebobupdecreasesthependulum'slength,causingthependulumtoswingfasterandtheclocktogaintime.Someprecisionclockshaveasmallauxiliaryadjustmentweightonathreadedshaftonthebob,toallowfineradjustment.Sometowerclocksandprecisionclocksuseatrayattachedneartothemidpointofthependulumrod,towhichsmallweightscanbeaddedorremoved.Thiseffectivelyshiftsthecentreofoscillationandallowstheratetobeadjustedwithoutstoppingtheclock.[75][76]

Thependulummustbesuspendedfromarigidsupport.[8][77]Duringoperation,anyelasticitywillallowtinyimperceptibleswayingmotionsofthesupport,whichdisturbstheclock'speriod,resultinginerror.Pendulumclocksshouldbeattachedfirmlytoasturdywall.

Themostcommonpendulumlengthinqualityclocks,whichisalwaysusedingrandfatherclocks,isthesecondspendulum,about1metre(39inches)long.Inmantelclocks,halfsecondpendulums,25cm(9.8in)long,orshorter,areused.Onlyafewlargetowerclocksuselongerpendulums,the1.5secondpendulum,2.25m(7.4ft)long,oroccasionallythetwosecondpendulum,4m(13ft)[8][78]asisthecaseofBigBen.[79]

Temperaturecompensation

Thelargestsourceoferrorinearlypendulumswasslightchangesinlengthduetothermalexpansionandcontractionofthependulumrodwithchangesinambienttemperature.[80]Thiswasdiscoveredwhenpeoplenoticedthatpendulumclocksranslowerinsummer,byasmuchasaminuteperweek[57][81](oneofthefirstwasGodefroyWendelin,asreportedbyHuygensin1658).[82]ThermalexpansionofpendulumrodswasfirststudiedbyJeanPicardin1669.[83]Apendulumwithasteelrodwillexpandbyabout11.3partspermillion(ppm)witheachdegreeCelsiusincrease,causingittoloseabout0.27secondsperdayforeverydegreeCelsiusincreaseintemperature,or9secondsperdayfora33C(59F)change.Woodrodsexpandless,losingonlyabout6secondsperdayfora33C(59F)change,whichiswhyqualityclocksoftenhadwoodenpendulumrods.Thewoodhadtobevarnishedtopreventwatervaporfromgettingin,becausechangesinhumidityalsoaffectedthelength.

Mercurypendulum

Thefirstdevicetocompensateforthiserrorwasthemercurypendulum,inventedbyGeorgeGraham[58]in1721.[8][81]Theliquidmetalmercuryexpandsinvolumewithtemperature.Inamercurypendulum,thependulum'sweight(bob)isacontainerofmercury.Withatemperaturerise,thependulumrodgetslonger,butthemercuryalsoexpandsanditssurfacelevelrisesslightlyinthecontainer,movingitscentreofmassclosertothependulumpivot.Byusingthecorrectheightofmercuryinthecontainerthesetwoeffectswillcancel,leavingthependulum'scentreofmass,anditsperiod,unchangedwithtemperature.Itsmaindisadvantagewasthatwhenthetemperaturechanged,therodwouldcometothenewtemperaturequicklybutthemassofmercurymighttake


MercurypenduluminHowardastronomicalregulatorclock,1887

adayortwotoreachthenewtemperature,causingtheratetodeviateduringthattime.[84]Toimprovethermalaccommodationseveralthincontainerswereoftenused,madeofmetal.Mercurypendulumswerethestandardusedinprecisionregulatorclocksintothe20thcentury.[85]

Gridironpendulum

Themostwidelyusedcompensatedpendulumwasthegridironpendulum,inventedin1726byJohnHarrison.[8][81][84]Thisconsistsofalternatingrodsoftwodifferentmetals,onewithlowerthermalexpansion(CTE),steel,andonewithhigherthermalexpansion,zincorbrass.Therodsareconnectedbyaframe,asshowninthedrawingattheright,sothatanincreaseinlengthofthezincrodspushesthebobup,shorteningthependulum.Withatemperatureincrease,thelowexpansionsteelrodsmakethependulumlonger,whilethehighexpansionzincrodsmakeitshorter.Bymakingtherodsofthecorrectlengths,thegreaterexpansionofthezinccancelsouttheexpansionofthesteelrodswhichhaveagreatercombinedlength,andthependulumstaysthesamelengthwithtemperature.

Zincsteelgridironpendulumsaremadewith5rods,butthethermalexpansionofbrassisclosertosteel,sobrasssteelgridironsusuallyrequire9rods.Gridironpendulumsadjusttotemperaturechangesfasterthanmercurypendulums,butscientistsfoundthatfrictionoftherodsslidingintheirholesintheframecausedgridironpendulumstoadjustinaseriesoftinyjumps.[84]Inhighprecisionclocksthiscausedtheclock'sratetochangesuddenlywitheachjump.Lateritwasfoundthatzincissubjecttocreep.Forthesereasonsmercurypendulumswereusedinthehighestprecisionclocks,butgridironswereusedinqualityregulatorclocks.

Gridironpendulumsbecamesoassociatedwithgoodqualitythat,tothisday,manyordinaryclockpendulumshavedecorative'fake'gridironsthatdon'tactuallyhaveanytemperaturecompensationfunction.

Invarandfusedquartz

Around1900lowthermalexpansionmaterialsweredevelopedwhich,whenusedaspendulumrods,madeelaboratetemperaturecompensationunnecessary.[8][81]Thesewereonlyusedinafewofthehighestprecisionclocksbeforethependulumbecameobsoleteasatimestandard.In1896CharlesdouardGuillaumeinventedthenickelsteelalloyInvar.ThishasaCTEofaround0.5in/(inF),resultinginpendulumtemperatureerrorsover71Fofonly1.3secondsperday,andthisresidualerrorcouldbecompensatedtozerowithafewcentimetersofaluminiumunderthependulumbob[2][84](thiscanbeseenintheRieflerclockimageabove).Invarpendulumswerefirstusedin1898intheRieflerregulatorclock[86]whichachievedaccuracyof15millisecondsperday.SuspensionspringsofElinvarwereusedtoeliminatetemperaturevariationofthespring'srestoringforceonthependulum.LaterfusedquartzwasusedwhichhadevenlowerCTE.Thesematerialsarethechoiceformodernhighaccuracypendulums.[87]

Atmosphericpressure

Theeffectofthesurroundingaironamovingpendulumiscomplexandrequiresfluidmechanicstocalculateprecisely,butformostpurposesitsinfluenceontheperiodcanbeaccountedforbythreeeffects:[64][88]

ByArchimedes'principletheeffectiveweightofthebobisreducedbythebuoyancyoftheairitdisplaces,whilethemass(inertia)remainsthesame,reducingthependulum'saccelerationduringitsswingand


DiagramofagridironpendulumA:exteriorschematicB:normaltemperatureC:highertemperature

increasingtheperiod.Thisdependsontheairpressureandthedensityofthependulum,butnotitsshape.Thependulumcarriesanamountofairwithitasitswings,andthemassofthisairincreasestheinertiaofthependulum,againreducingtheaccelerationandincreasingtheperiod.Thisdependsonbothitsdensityandshape.Viscousairresistanceslowsthependulum'svelocity.Thishasanegligibleeffectontheperiod,butdissipatesenergy,reducingtheamplitude.Thisreducesthependulum'sQfactor,requiringastrongerdriveforcefromtheclock'smechanismtokeepitmoving,whichcausesincreaseddisturbancetotheperiod.

Soincreasesinbarometricpressureincreaseapendulum'speriodslightlyduetothefirsttwoeffects,byabout0.11secondsperdayperkilopascal(0.37secondsperdayperinchofmercuryor0.015secondsperdaypertorr).[64]Researchersusingpendulumstomeasuretheaccelerationofgravityhadtocorrecttheperiodfortheairpressureatthealtitudeofmeasurement,computingtheequivalentperiodofapendulumswinginginvacuum.ApendulumclockwasfirstoperatedinaconstantpressuretankbyFriedrichTiedein1865attheBerlinObservatory,[89][90]andby1900thehighestprecisionclocksweremountedintanksthatwerekeptataconstantpressuretoeliminatechangesinatmosphericpressure.Alternatively,insomeasmallaneroidbarometermechanismattachedtothependulumcompensatedforthiseffect.

Gravity

Pendulumsareaffectedbychangesingravitationalacceleration,whichvariesbyasmuchas0.5%atdifferentlocationsonEarth,sopendulumclockshavetoberecalibratedafteramove.Evenmovingapendulumclocktothetopofatallbuildingcancauseittolosemeasurabletimefromthereductioningravity.

Accuracyofpendulumsastimekeepers

Thetimekeepingelementsinallclocks,whichincludependulums,balancewheels,thequartzcrystalsusedinquartzwatches,andeventhevibratingatomsinatomicclocks,areinphysicscalledharmonicoscillators.Thereasonharmonicoscillatorsareusedinclocksisthattheyvibrateoroscillateataspecificresonantfrequencyorperiodandresistoscillatingatotherrates.However,theresonantfrequencyisnotinfinitely'sharp'.Aroundtheresonantfrequencythereisanarrownaturalbandoffrequencies(orperiods),calledtheresonancewidthorbandwidth,wheretheharmonicoscillatorwilloscillate.[91][92]Inaclock,theactualfrequencyofthependulummayvaryrandomlywithinthisbandwidthinresponsetodisturbances,butatfrequenciesoutsidethisband,theclockwillnotfunctionatall.

Qfactor

Themeasureofaharmonicoscillator'sresistancetodisturbancestoitsoscillationperiodisadimensionlessparametercalledtheQfactorequaltotheresonantfrequencydividedbythebandwidth.[92][93]ThehighertheQ,thesmallerthebandwidth,andthemoreconstantthefrequencyorperiodoftheoscillatorforagivendisturbance.[94]ThereciprocaloftheQisroughlyproportionaltothelimitingaccuracyachievablebyaharmonicoscillatorasatimestandard.[95]


AShorttSynchronomefreependulumclock,themostaccuratependulumclockevermade,attheNISTmuseum,Gaithersburg,MD,USA.Itkepttimewithtwosynchronizedpendulums.Themasterpenduluminthevacuumtank(left)swungfreeofvirtuallyanydisturbance,andcontrolledtheslavependulumintheclockcase(right)whichperformedtheimpulsingandtimekeepingtasks.Itsaccuracywasaboutasecondperyear.

TheQisrelatedtohowlongittakesfortheoscillationsofanoscillatortodieout.TheQofapendulumcanbemeasuredbycountingthenumberofoscillationsittakesfortheamplitudeofthependulum'sswingtodecayto1/e=36.8%ofitsinitialswing,andmultiplyingby2.

Inaclock,thependulummustreceivepushesfromtheclock'smovementtokeepitswinging,toreplacetheenergythependulumlosestofriction.Thesepushes,appliedbyamechanismcalledtheescapement,arethemainsourceofdisturbancetothependulum'smotion.TheQisequalto2timestheenergystoredinthependulum,dividedbytheenergylosttofrictionduringeachoscillationperiod,whichisthesameastheenergyaddedbytheescapementeachperiod.Itcanbeseenthatthesmallerthefractionofthependulum'senergythatislosttofriction,thelessenergyneedstobeadded,thelessthedisturbancefromtheescapement,themore'independent'thependulumisoftheclock'smechanism,andthemoreconstantitsperiodis.TheQofapendulumisgivenby:

whereMisthemassofthebob,=2/Tisthependulum'sradianfrequencyofoscillation,andisthefrictionaldampingforceonthependulumperunitvelocity.

isfixedbythependulum'speriod,andMislimitedbytheloadcapacityandrigidityofthesuspension.SotheQofclockpendulumsisincreasedbyminimizingfrictionallosses().Precisionpendulumsaresuspendedonlowfrictionpivotsconsistingoftriangularshaped'knife'edgesrestingonagateplates.Around99%oftheenergylossinafreeswingingpendulumisduetoairfriction,somountingapenduluminavacuumtankcanincreasetheQ,andthustheaccuracy,byafactorof100.[96]

TheQofpendulumsrangesfromseveralthousandinanordinaryclocktoseveralhundredthousandforprecisionregulatorpendulumsswinginginvacuum.[97]AqualityhomependulumclockmighthaveaQof10,000andanaccuracyof10secondspermonth.ThemostaccuratecommerciallyproducedpendulumclockwastheShorttSynchronomefreependulumclock,inventedin1921.[2][65][98][99][100]ItsInvarmasterpendulumswinginginavacuumtankhadaQof110,000[97]andanerrorrateofaroundasecondperyear.[65]

TheirQof103105isonereasonwhypendulumsaremoreaccuratetimekeepersthanthebalancewheelsinwatches,withQaround100300,butlessaccuratethanthequartzcrystalsinquartzclocks,withQof105106.[2][97]

Escapement

Pendulums(unlike,forexample,quartzcrystals)havealowenoughQthatthedisturbancecausedbytheimpulsestokeepthemmovingisgenerallythelimitingfactorontheirtimekeepingaccuracy.Thereforethedesignoftheescapement,themechanismthatprovidestheseimpulses,hasalargeeffectontheaccuracyofaclockpendulum.Iftheimpulsesgiventothependulumbytheescapementeachswingcouldbeexactlyidentical,theresponseofthependulumwouldbeidentical,anditsperiodwouldbeconstant.However,thisisnot


Thesecondspendulum,apendulumwithaperiodoftwosecondssoeachswingtakesonesecond

achievableunavoidablerandomfluctuationsintheforceduetofrictionoftheclock'spallets,lubricationvariations,andchangesinthetorqueprovidedbytheclock'spowersourceasitrunsdown,meanthattheforceoftheimpulseappliedbytheescapementvaries.

Ifthesevariationsintheescapement'sforcecausechangesinthependulum'swidthofswing(amplitude),thiswillcausecorrespondingslightchangesintheperiod,since(asdiscussedattop)apendulumwithafiniteswingisnotquiteisochronous.Therefore,thegoaloftraditionalescapementdesignistoapplytheforcewiththeproperprofile,andatthecorrectpointinthependulum'scycle,soforcevariationshavenoeffectonthependulum'samplitude.Thisiscalledanisochronousescapement.

TheAirycondition

In1826BritishastronomerGeorgeAiryprovedwhatclockmakershadknownforcenturiesthatthedisturbingeffectofadriveforceontheperiodofapendulumissmallestifgivenasashortimpulseasthependulumpassesthroughitsbottomequilibriumposition.[2]Specifically,heprovedthatifapendulumisdrivenbyanimpulsethatissymmetricalaboutitsbottomequilibriumposition,thependulum'samplitudewillbeunaffectedbychangesinthedriveforce.[101]Themostaccurateescapements,suchasthedeadbeat,approximatelysatisfythiscondition.[102]

Gravitymeasurement

Thepresenceoftheaccelerationofgravitygintheperiodicityequation(1)forapendulummeansthatthelocalgravitationalaccelerationoftheEarthcanbecalculatedfromtheperiodofapendulum.Apendulumcanthereforebeusedasagravimetertomeasurethelocalgravity,whichvariesbyover0.5%acrossthesurfaceoftheEarth.[103][Note2]Thependuluminaclockisdisturbedbythepushesitreceivesfromtheclockmovement,sofreeswingingpendulumswereused,andwerethestandardinstrumentsofgravimetryuptothe1930s.

Thedifferencebetweenclockpendulumsandgravimeterpendulumsisthattomeasuregravity,thependulum'slengthaswellasitsperiodhastobemeasured.Theperiodoffreeswingingpendulumscouldbefoundtogreatprecisionbycomparingtheirswingwithaprecisionclockthathadbeenadjustedtokeepcorrecttimebythepassageofstarsoverhead.Intheearlymeasurements,aweightonacordwassuspendedinfrontoftheclockpendulum,anditslengthadjusteduntilthetwopendulumsswunginexactsynchronism.Thenthelengthofthecordwasmeasured.Fromthelengthandtheperiod,gcouldbecalculatedfromequation(1).

Thesecondspendulum

Thesecondspendulum,apendulumwithaperiodoftwosecondssoeachswingtakesonesecond,waswidelyusedtomeasuregravity,becausemostprecisionclockshadsecondspendulums.Bythelate17thcentury,thelengthofthesecondspendulumbecamethestandardmeasureofthestrengthofgravitationalaccelerationatalocation.By1700itslengthhadbeenmeasuredwithsubmillimeteraccuracyatseveralcitiesinEurope.Forasecondspendulum,gisproportionaltoitslength:

Earlyobservations

1620:BritishscientistFrancisBaconwasoneofthefirsttoproposeusingapendulumtomeasuregravity,

suggestingtakingoneupamountaintoseeifgravityvarieswithaltitude.[104]


Borda&Cassini's1792measurementofthelengthofthesecondspendulum

1644:Evenbeforethependulumclock,FrenchpriestMarinMersennefirstdeterminedthelengthofthesecondspendulumwas39.1inches(990mm),bycomparingtheswingofapendulumtothetimeittookaweighttofallameasureddistance.1669:JeanPicarddeterminedthelengthofthesecondspendulumatParis,usinga1inch(25mm)copper

ballsuspendedbyanaloefiber,obtaining39.09inches(993mm).[105]

1672:ThefirstobservationthatgravityvariedatdifferentpointsonEarthwasmadein1672byJean

Richer,whotookapendulumclocktoCayenne,FrenchGuianaandfoundthatitlost212minutesperday

itssecondspendulumhadtobeshortenedby114lignes(2.6mm)shorterthanatParis,tokeepcorrect

time.[106][107]In1687IsaacNewtoninPrincipiaMathematicashowedthiswasbecausetheEarthhadaslightlyoblateshape(flattenedatthepoles)causedbythecentrifugalforceofitsrotation,sogravity

increasedwithlatitude.[107]Fromthistimeon,pendulumsbegantobetakentodistantlandstomeasuregravity,andtableswerecompiledofthelengthofthesecondspendulumatdifferentlocationsonEarth.In

1743AlexisClaudeClairautcreatedthefirsthydrostaticmodeloftheEarth,Clairaut'stheorem,[105]whichallowedtheellipticityoftheEarthtobecalculatedfromgravitymeasurements.ProgressivelymoreaccuratemodelsoftheshapeoftheEarthfollowed.1687:Newtonexperimentedwithpendulums(describedinPrincipia)andfoundthatequallengthpendulumswithbobsmadeofdifferentmaterialshadthesameperiod,provingthatthegravitationalforceondifferentsubstanceswasexactlyproportionaltotheirmass(inertia).

1737:FrenchmathematicianPierreBouguermadeasophisticatedseries

ofpendulumobservationsintheAndesmountains,Peru.[108]Heusedacopperpendulumbobintheshapeofadoublepointedconesuspendedbyathreadthebobcouldbereversedtoeliminatetheeffectsofnonuniformdensity.Hecalculatedthelengthtothecenterofoscillationofthreadandbobcombined,insteadofusingthecenterofthebob.Hecorrectedforthermalexpansionofthemeasuringrodandbarometricpressure,givinghisresultsforapendulumswinginginvacuum.Bouguerswungthesamependulumatthreedifferentelevations,fromsealeveltothetopofthehighPeruvianaltiplano.GravityshouldfallwiththeinversesquareofthedistancefromthecenteroftheEarth.Bouguerfoundthatitfelloffslower,andcorrectlyattributedthe'extra'gravitytothegravitationalfieldofthehugePeruvianplateau.Fromthedensityofrocksampleshecalculatedanestimateoftheeffectofthealtiplanoonthependulum,andcomparingthiswiththegravityoftheEarthwasabletomakethefirstroughestimateofthedensityoftheEarth.1747:DanielBernoullishowedhowtocorrectforthelengtheningoftheperiodduetoafiniteangleof

swing0byusingthefirstordercorrection02/16,givingtheperiodofapendulumwithaninfinitesimal

swing.[108]

1792:Todefineapendulumstandardoflengthforusewiththenewmetricsystem,in1792JeanCharles


AKater'spendulum

MeasuringgravitywithKater'sreversiblependulum,fromKater's1818paper

deBordaandJeanDominiqueCassinimadeaprecisemeasurementofthesecondspendulumatParis.

Theyuseda112inch(14mm)platinumballsuspendedbya12foot(3.7m)ironwire.Theirmain

innovationwasatechniquecalledthe"methodofcoincidences"whichallowedtheperiodofpendulumstobecomparedwithgreatprecision.(Bouguerhadalsousedthismethod).ThetimeintervalTbetweentherecurringinstantswhenthetwopendulumsswunginsynchronismwastimed.Fromthisthedifferencebetweentheperiodsofthependulums,T1andT2,couldbecalculated:

1821:FrancescoCarlinimadependulumobservationsontopofMountCenis,Italy,fromwhich,using

methodssimilartoBouguer's,hecalculatedthedensityoftheEarth.[109]Hecomparedhismeasurementstoanestimateofthegravityathislocationassumingthemountainwasn'tthere,calculatedfrompreviousnearbypendulummeasurementsatsealevel.Hismeasurementsshowed'excess'gravity,whichheallocatedtotheeffectofthemountain.Modelingthemountainasasegmentofasphere11miles(18km)indiameterand1mile(1.6km)high,fromrocksampleshecalculateditsgravitationalfield,andestimatedthedensityoftheEarthat4.39timesthatofwater.Laterrecalculationsbyothersgavevaluesof4.77and4.95,illustratingtheuncertaintiesinthesegeographicalmethods.

Kater'spendulum

Theprecisionoftheearlygravitymeasurementsabovewaslimitedbythedifficultyofmeasuringthelengthofthependulum,L.Lwasthelengthofanidealizedsimplegravitypendulum(describedattop),whichhasallitsmassconcentratedinapointattheendofthecord.In1673Huygenshadshownthattheperiodofarealpendulum(calledacompoundpendulum)wasequaltotheperiodofasimplependulumwithalengthequaltothedistancebetweenthepivotpointandapointcalledthecenterofoscillation,locatedunderthecenterofgravity,thatdependsonthemassdistributionalongthependulum.Buttherewasnoaccuratewayofdeterminingthecenterofoscillationinarealpendulum.

Togetaroundthisproblem,theearlyresearchersaboveapproximatedanidealsimplependulumascloselyaspossiblebyusingametalspheresuspendedbyalightwireorcord.Ifthewirewaslightenough,thecenterofoscillationwasclosetothecenterofgravityoftheball,atitsgeometriccenter.This"ballandwire"typeofpendulumwasn'tveryaccurate,becauseitdidn'tswingasarigidbody,andtheelasticityofthewirecauseditslengthtochangeslightlyasthependulumswung.


Measuringgravitywithaninvariablependulum,Madras,India,1821

HoweverHuygenshadalsoprovedthatinanypendulum,thepivotpointandthecenterofoscillationwereinterchangeable.[19]Thatis,ifapendulumwereturnedupsidedownandhungfromitscenterofoscillation,itwouldhavethesameperiodasitdidinthepreviousposition,andtheoldpivotpointwouldbethenewcenterofoscillation.

BritishphysicistandarmycaptainHenryKaterin1817realizedthatHuygens'principlecouldbeusedtofindthelengthofasimplependulumwiththesameperiodasarealpendulum.[59]Ifapendulumwasbuiltwithasecondadjustablepivotpointnearthebottomsoitcouldbehungupsidedown,andthesecondpivotwasadjusteduntiltheperiodswhenhungfrombothpivotswerethesame,thesecondpivotwouldbeatthecenterofoscillation,andthedistancebetweenthetwopivotswouldbethelengthofasimplependulumwiththesameperiod.

Katerbuiltareversiblependulum(shownatright)consistingofabrassbarwithtwoopposingpivotsmadeofshorttriangular"knife"blades(a)neareitherend.Itcouldbeswungfromeitherpivot,withtheknifebladessupportedonagateplates.Ratherthanmakeonepivotadjustable,heattachedthepivotsameterapartandinsteadadjustedtheperiodswithamoveableweightonthependulumrod(b,c).Inoperation,thependulumishunginfrontofaprecisionclock,andtheperiodtimed,thenturnedupsidedownandtheperiodtimedagain.Theweightisadjustedwiththeadjustmentscrewuntiltheperiodsareequal.Thenputtingthisperiodandthedistancebetweenthepivotsintoequation(1)givesthegravitationalaccelerationgveryaccurately.

Katertimedtheswingofhispendulumusingthe"methodofcoincidences"andmeasuredthedistancebetweenthetwopivotswithamicroscope.Afterapplyingcorrectionsforthefiniteamplitudeofswing,thebuoyancyofthebob,thebarometricpressureandaltitude,andtemperature,heobtainedavalueof39.13929inchesforthesecondspendulumatLondon,invacuum,atsealevel,at62F.Thelargestvariationfromthemeanofhis12observationswas0.00028in.[110]representingaprecisionofgravitymeasurementof7106(7mGalor70m/s2).Kater'smeasurementwasusedasBritain'sofficialstandardoflength(seebelow)from1824to1855.

Reversiblependulums(knowntechnicallyas"convertible"pendulums)employingKater'sprinciplewereusedforabsolutegravitymeasurementsintothe1930s.

Laterpendulumgravimeters

TheincreasedaccuracymadepossiblebyKater'spendulumhelpedmakegravimetryastandardpartofgeodesy.Sincetheexactlocation(latitudeandlongitude)ofthe'station'wherethegravitymeasurementwasmadewasnecessary,gravitymeasurementsbecamepartofsurveying,andpendulumsweretakenonthegreatgeodeticsurveysofthe18thcentury,particularlytheGreatTrigonometricSurveyofIndia.

Invariablependulums:Katerintroducedtheideaofrelativegravitymeasurements,tosupplementtheabsolute

measurementsmadebyaKater'spendulum.[111]ComparingthegravityattwodifferentpointswasaneasierprocessthanmeasuringitabsolutelybytheKatermethod.Allthatwasnecessarywastotimetheperiodofanordinary(singlepivot)pendulumatthefirstpoint,thentransportthependulumtotheotherpointandtimeitsperiodthere.Sincethependulum'slengthwasconstant,from(1)theratioofthegravitationalaccelerationswasequaltotheinverseoftheratiooftheperiodssquared,andnoprecisionlengthmeasurementswerenecessary.Sooncethegravityhadbeenmeasuredabsolutelyatsomecentralstation,bytheKaterorotheraccuratemethod,thegravityatother


Repsoldpendulum,1864

pointscouldbefoundbyswingingpendulumsatthecentralstationandthentakingthemtothenearbypoint.Katermadeupasetof"invariable"pendulums,withonlyoneknifeedgepivot,whichweretakentomanycountriesafterfirstbeingswungatacentralstationatKewObservatory,UK.Airy'scoalpitexperiments:Startingin1826,usingmethodssimilartoBouguer,BritishastronomerGeorgeAiryattemptedtodeterminethedensityoftheEarthbypendulumgravitymeasurementsatthetop

andbottomofacoalmine.[112][113]ThegravitationalforcebelowthesurfaceoftheEarthdecreasesratherthanincreasingwithdepth,becausebyGauss'slawthemassofthesphericalshellofcrustabovethesubsurfacepointdoesnotcontributetothegravity.The1826experimentwasabortedbythefloodingofthemine,butin1854heconductedanimprovedexperimentattheHartoncoalmine,usingsecondspendulumsswingingonagateplates,timedbyprecisionchronometerssynchronizedbyanelectricalcircuit.Hefoundthelowerpendulumwasslowerby2.24secondsperday.Thismeantthatthegravitationalaccelerationatthebottomofthemine,1250ftbelowthesurface,was1/14,000lessthanitshouldhavebeenfromtheinversesquarelawthatistheattractionofthesphericalshellwas1/14,000oftheattractionoftheEarth.Fromsamplesofsurfacerockheestimatedthemassofthesphericalshellofcrust,andfromthisestimatedthatthedensityoftheEarthwas6.565timesthatofwater.VonSterneckattemptedtorepeattheexperimentin1882butfoundinconsistentresults.

RepsoldBesselpendulum:ItwastimeconsuminganderrorpronetorepeatedlyswingtheKater'spendulumandadjusttheweightsuntiltheperiodswereequal.FriedrichBessel

showedin1835thatthiswasunnecessary.[114]Aslongastheperiodswereclosetogether,thegravitycouldbecalculatedfromthetwoperiodsandthecenterofgravityofthe

pendulum.[115]Sothereversiblependulumdidn'tneedtobeadjustable,itcouldjustbeabarwithtwopivots.Besselalsoshowedthatifthependulumwasmadesymmetricalinformaboutitscenter,butwasweightedinternallyatoneend,theerrorsduetoairdragwouldcancelout.Further,anothererrorduetothefinitediameteroftheknifeedgescouldbemadetocanceloutiftheywereinterchangedbetweenmeasurements.Besseldidn'tconstructsuchapendulum,butin1864AdolfRepsold,undercontractbytheSwissGeodeticCommissionmadeapendulumalongtheselines.TheRepsoldpendulumwasabout56cmlongandhada

periodofabout34second.ItwasusedextensivelybyEuropeangeodeticagencies,andwith

theKaterpendulumintheSurveyofIndia.SimilarpendulumsofthistypeweredesignedbyCharlesPierceandC.Defforges.

VonSterneckandMendenhallgravimeters:In1887AustroHungarianscientistRobertvonSterneckdevelopedasmallgravimeterpendulummountedinatemperaturecontrolledvacuumtanktoeliminatetheeffectsoftemperatureandairpressure.Theseused"halfsecondpendulums,"havingaperiodclosetoonesecond,andwereabout25cmlong.Theywerenonreversible,soitwasusedforrelativegravitymeasurements,buttheirsmallsizemadethemsmallandportable.Theperiodofthependulumwaspickedoffbyreflectingtheimageofanelectricsparkcreatedbyaprecisionchronometeroffamirrormountedatthetopofthependulumrod.TheVonSterneckinstrument,andasimilarinstrumentdevelopedbyThomas

C.MendenhalloftheUSCoastandGeodeticSurveyin1890,[116]wereusedextensivelyforsurveysinto


PendulumsusedinMendenhallgravimeter,1890

QuartzpendulumsusedinGulfgravimeter,1929

the1920s.

TheMendenhallpendulumwasactuallyamoreaccuratetimekeeperthanthehighestprecisionclocksofthetime,andasthe'world'sbestclock'itwasusedbyAlbertA.Michelsoninhis1924measurementsof

thespeedoflightonMt.Wilson,California.[116]

Doublependulumgravimeters:Startingin1875,theincreasingaccuracyofpendulummeasurementsrevealedanothersourceoferrorinexistinginstruments:theswingofthependulumcausedaslightswayingofthetripodstandusedtosupportportablependulums,introducingerror.In1875CharlesSPeircecalculatedthatmeasurementsofthelengthofthesecondspendulummadewiththeRepsoldinstrumentrequireda

correctionof0.2mmduetothiserror.[117]In1880C.DefforgesusedaMichelsoninterferometertomeasuretheswayofthestanddynamically,andinterferometerswereaddedtothestandardMendenhallapparatustocalculatesway

corrections.[118]Amethodofpreventingthiserrorwasfirstsuggestedin1877byHervFayeandadvocatedbyPeirce,CellrierandFurtwangler:mounttwoidenticalpendulumsonthesamesupport,swingingwiththesameamplitude,180outofphase.Theoppositemotionofthependulumswouldcanceloutanysidewaysforcesonthesupport.Theideawasopposedduetoitscomplexity,butbythestartofthe20thcenturytheVonSterneckdeviceandotherinstrumentsweremodifiedtoswingmultiplependulumssimultaneously.

Gulfgravimeter:Oneofthelastandmostaccuratependulumgravimeterswastheapparatusdevelopedin1929bytheGulf

ResearchandDevelopmentCo.[119][120]Itusedtwopendulumsmadeoffusedquartz,each10.7inches(270mm)inlengthwithaperiodof0.89second,swingingonpyrexknifeedgepivots,180outofphase.Theyweremountedinapermanentlysealedtemperatureandhumiditycontrolledvacuumchamber.Strayelectrostaticchargesonthequartzpendulumshadtobedischargedbyexposingthemtoaradioactivesaltbeforeuse.Theperiodwasdetectedbyreflectingalightbeamfromamirroratthetopofthependulum,recordedbyachartrecorderandcomparedtoaprecisioncrystaloscillatorcalibratedagainsttheWWVradiotimesignal.Thisinstrumentwasaccurateto

within(0.30.5)107(3050microgalsor35nm/s2).[119]Itwasusedintothe1960s.


RelativependulumgravimetersweresupersededbythesimplerLaCostezerolengthspringgravimeter,inventedin1934byLucienLaCoste.[116]Absolute(reversible)pendulumgravimeterswerereplacedinthe1950sbyfreefallgravimeters,inwhichaweightisallowedtofallinavacuumtankanditsaccelerationismeasuredbyanopticalinterferometer.[68]

Standardoflength

BecausetheaccelerationofgravityisconstantatagivenpointonEarth,theperiodofasimplependulumatagivenlocationdependsonlyonitslength.Additionally,gravityvariesonlyslightlyatdifferentlocations.Almostfromthependulum'sdiscoveryuntiltheearly19thcentury,thispropertyledscientiststosuggestusingapendulumofagivenperiodasastandardoflength.

Untilthe19thcentury,countriesbasedtheirsystemsoflengthmeasurementonprototypes,metalbarprimarystandards,suchasthestandardyardinBritainkeptattheHousesofParliament,andthestandardtoiseinFrance,keptatParis.Thesewerevulnerabletodamageordestructionovertheyears,andbecauseofthedifficultyofcomparingprototypes,thesameunitoftenhaddifferentlengthsindistanttowns,creatingopportunitiesforfraud.[121]Enlightenmentscientistsarguedforalengthstandardthatwasbasedonsomepropertyofnaturethatcouldbedeterminedbymeasurement,creatinganindestructible,universalstandard.Theperiodofpendulumscouldbemeasuredverypreciselybytimingthemwithclocksthatweresetbythestars.ApendulumstandardamountedtodefiningtheunitoflengthbythegravitationalforceoftheEarth,forallintentsconstant,andthesecond,whichwasdefinedbytherotationrateoftheEarth,alsoconstant.Theideawasthatanyone,anywhereonEarth,couldrecreatethestandardbyconstructingapendulumthatswungwiththedefinedperiodandmeasuringitslength.

Virtuallyallproposalswerebasedonthesecondspendulum,inwhicheachswing(ahalfperiod)takesonesecond,whichisaboutameter(39inches)long,becausebythelate17thcenturyithadbecomeastandardformeasuringgravity(seeprevioussection).Bythe18thcenturyitslengthhadbeenmeasuredwithsubmillimeteraccuracyatanumberofcitiesinEuropeandaroundtheworld.

Theinitialattractionofthependulumlengthstandardwasthatitwasbelieved(byearlyscientistssuchasHuygensandWren)thatgravitywasconstantovertheEarth'ssurface,soagivenpendulumhadthesameperiodatanypointonEarth.[121]Sothelengthofthestandardpendulumcouldbemeasuredatanylocation,andwouldnotbetiedtoanygivennationorregionitwouldbeatrulydemocratic,worldwidestandard.AlthoughRicherfoundin1672thatgravityvariesatdifferentpointsontheglobe,theideaofapendulumlengthstandardremainedpopular,becauseitwasfoundthatgravityonlyvarieswithlatitude.Gravitationalaccelerationincreasessmoothlyfromtheequatortothepoles,duetotheoblateshapeoftheEarth.Soatanygivenlatitude(eastwestline),gravitywasconstantenoughthatthelengthofasecondspendulumwasthesamewithinthemeasurementcapabilityofthe18thcentury.Sotheunitoflengthcouldbedefinedatagivenlatitudeandmeasuredatanypointatthatlatitude.Forexample,apendulumstandarddefinedat45northlatitude,apopularchoice,couldbemeasuredinpartsofFrance,Italy,Croatia,Serbia,Romania,Russia,Kazakhstan,China,Mongolia,theUnitedStatesandCanada.Inaddition,itcouldberecreatedatanylocationatwhichthegravitationalaccelerationhadbeenaccuratelymeasured.

Bythemid19thcentury,increasinglyaccuratependulummeasurementsbyEdwardSabineandThomasYoungrevealedthatgravity,andthusthelengthofanypendulumstandard,variedmeasurablywithlocalgeologicfeaturessuchasmountainsanddensesubsurfacerocks.[122]SoapendulumlengthstandardhadtobedefinedatasinglepointonEarthandcouldonlybemeasuredthere.Thistookmuchoftheappealfromtheconcept,andeffortstoadoptpendulumstandardswereabandoned.

Earlyproposals


OneofthefirsttosuggestdefininglengthwithapendulumwasFlemishscientistIsaacBeeckman[123]whoin1631recommendedmakingthesecondspendulum"theinvariablemeasureforallpeopleatalltimesinallplaces".[124]MarinMersenne,whofirstmeasuredthesecondspendulumin1644,alsosuggestedit.ThefirstofficialproposalforapendulumstandardwasmadebytheBritishRoyalSocietyin1660,advocatedbyChristiaanHuygensandOleRmer,basingitonMersenne'swork,[125]andHuygensinHorologiumOscillatoriumproposeda"horaryfoot"definedas1/3ofthesecondspendulum.ChristopherWrenwasanotherearlysupporter.Theideaofapendulumstandardoflengthmusthavebeenfamiliartopeopleasearlyas1663,becauseSamuelButlersatirizesitinHudibras:[126]

UponthebenchIwillsohandleemThatthevibrationofthispendulumShallmakealltaylorsyardsofoneUnanimousopinion

In1671JeanPicardproposedapendulumdefined'universalfoot'inhisinfluentialMesuredelaTerre.[127]GabrielMoutonaround1670suggesteddefiningthetoiseeitherbyasecondspendulumoraminuteofterrestrialdegree.Aplanforacompletesystemofunitsbasedonthependulumwasadvancedin1675byItalianpolymathTitoLivioBurratini.InFrancein1747,geographerCharlesMariedelaCondamineproposeddefininglengthbyasecondspendulumattheequatorsinceatthislocationapendulum'sswingwouldn'tbedistortedbytheEarth'srotation.BritishpoliticiansJamesSteuart(1780)andGeorgeSkeneKeithwerealsosupporters.

Bytheendofthe18thcentury,whenmanynationswerereformingtheirweightandmeasuresystems,thesecondspendulumwastheleadingchoiceforanewdefinitionoflength,advocatedbyprominentscientistsinseveralmajornations.In1790,thenUSSecretaryofStateThomasJeffersonproposedtoCongressacomprehensivedecimalizedUS'metricsystem'basedonthesecondspendulumat38Northlatitude,themeanlatitudeoftheUnitedStates.[128]Noactionwastakenonthisproposal.InBritaintheleadingadvocateofthependulumwaspoliticianJohnRiggsMiller.[129]WhenhiseffortstopromoteajointBritishFrenchAmericanmetricsystemfellthroughin1790,heproposedaBritishsystembasedonthelengthofthesecondspendulumatLondon.Thisstandardwasadoptedin1824(below).

Themetre

InthediscussionsleadinguptotheFrenchadoptionofthemetricsystemin1791,theleadingcandidateforthedefinitionofthenewunitoflength,themetre,wasthesecondspendulumat45Northlatitude.ItwasadvocatedbyagroupledbyFrenchpoliticianTalleyrandandmathematicianAntoineNicolasCaritatdeCondorcet.ThiswasoneofthethreefinaloptionsconsideredbytheFrenchAcademyofSciencescommittee.However,onMarch19,1791thecommitteeinsteadchosetobasethemetreonthelengthofthemeridianthroughParis.Apendulumdefinitionwasrejectedbecauseofitsvariabilityatdifferentlocations,andbecauseitdefinedlengthbyaunitoftime.(However,since1983themetrehasbeenofficiallydefinedintermsofthelengthofthesecondandthespeedoflight.)ApossibleadditionalreasonisthattheradicalFrenchAcademydidn'twanttobasetheirnewsystemonthesecond,atraditionalandnondecimalunitfromtheancienregime.

Althoughnotdefinedbythependulum,thefinallengthchosenforthemetre,107ofthepoletoequatormeridianarc,wasveryclosetothelengthofthesecondspendulum(0.9937m),within0.63%.Althoughnoreasonforthisparticularchoicewasgivenatthetime,itwasprobablytofacilitatetheuseofthesecondspendulumasasecondarystandard,aswasproposedintheofficialdocument.Sothemodernworld'sstandardunitoflengthiscertainlycloselylinkedhistoricallywiththesecondspendulum.

BritainandDenmark


BritainandDenmarkappeartobetheonlynationsthat(forashorttime)basedtheirunitsoflengthonthependulum.In1821theDanishinchwasdefinedas1/38ofthelengthofthemeansolarsecondspendulumat45latitudeatthemeridianofSkagen,atsealevel,invacuum.[130][131]TheBritishparliamentpassedtheImperialWeightsandMeasuresActin1824,areformoftheBritishstandardsystemwhichdeclaredthatiftheprototypestandardyardwasdestroyed,itwouldberecoveredbydefiningtheinchsothatthelengthofthesolarsecondspendulumatLondon,atsealevel,inavacuum,at62Fwas39.1393inches.[132]ThisalsobecametheUSstandard,sinceatthetimetheUSusedBritishmeasures.However,whentheprototypeyardwaslostinthe1834HousesofParliamentfire,itprovedimpossibletorecreateitaccuratelyfromthependulumdefinition,andin1855Britainrepealedthependulumstandardandreturnedtoprototypestandards.

Otheruses

Seismometers

Apenduluminwhichtherodisnotverticalbutalmosthorizontalwasusedinearlyseismometersformeasuringearthtremors.Thebobofthependulumdoesnotmovewhenitsmountingdoes,andthedifferenceinthemovementsisrecordedonadrumchart.

Schulertuning

AsfirstexplainedbyMaximilianSchulerina1923paper,apendulumwhoseperiodexactlyequalstheorbitalperiodofahypotheticalsatelliteorbitingjustabovethesurfaceoftheearth(about84minutes)willtendtoremainpointingatthecenteroftheearthwhenitssupportissuddenlydisplaced.Thisprinciple,calledSchulertuning,isusedininertialguidancesystemsinshipsandaircraftthatoperateonthesurfaceoftheEarth.Nophysicalpendulumisused,butthecontrolsystemthatkeepstheinertialplatformcontainingthegyroscopesstableismodifiedsothedeviceactsasthoughitisattachedtosuchapendulum,keepingtheplatformalwaysfacingdownasthevehiclemovesonthecurvedsurfaceoftheEarth.

Coupledpendulums

In1665Huygensmadeacuriousobservationaboutpendulumclocks.Twoclockshadbeenplacedonhismantlepiece,andhenotedthattheyhadacquiredanopposingmotion.Thatis,theirpendulumswerebeatinginunisonbutintheoppositedirection180outofphase.Regardlessofhowthetwoclockswerestarted,hefoundthattheywouldeventuallyreturntothisstate,thusmakingthefirstrecordedobservationofacoupledoscillator.[133]

Thecauseofthisbehaviorwasthatthetwopendulumswereaffectingeachotherthroughslightmotionsofthesupportingmantlepiece.Thisprocessiscalledentrainmentormodelockinginphysicsandisobservedinothercoupledoscillators.Synchronizedpendulumshavebeenusedinclocksandwerewidelyusedingravimetersintheearly20thcentury.AlthoughHuygensonlyobservedoutofphasesynchronization,recentinvestigationshaveshowntheexistenceofinphasesynchronization,aswellas"death"stateswhereinoneorbothclocksstops.[134][135]

Religiouspractice

Pendulummotionappearsinreligiousceremoniesaswell.Theswingingincenseburnercalledacenser,alsoknownasathurible,isanexampleofapendulum.[136]PendulumsarealsoseenatmanygatheringsineasternMexicowheretheymarktheturningofthetidesonthedaywhichthetidesareattheirhighestpoint.Seealsopendulumsfordivinationanddowsing.


PendulumintheMetropolitanCathedral,MexicoCity.

Execution

DuringtheMiddleAges,pendulumswereusedasamethodoftorturebytheSpanishInquisition.Usingthebasicprincipleofthependulum,theweight(bob)isreplacedbyanaxehead.Thevictimisstrappedtoatablebelow,thedeviceisactivated,andtheaxebeginstoswingbackandforththroughtheair.Witheachpass,orreturn,thependulumislowered,graduallycomingclosertothevictim'storso,untilfinallycleaved.Becauseofthetimerequiredbeforethemortalactionoftheaxeiscomplete,thependulumisconsideredamethodoftorturingthevictimbeforehisorherdemise.[137]

Seealso

Notes

1. ^A"small"swingisoneinwhichtheangleissmallenoughthatsin()canbeapproximatedbywhenismeasuredinradians

2. ^Thevalueof"g"(accelerationduetogravity)attheequatoris9.780m/s2andatthepolesis9.832m/s2,adifferenceof0.53%.

Barton'sPendulumsBlackburnpendulumConicalpendulumDoubochinski'spendulumDoublependulumDoubleinvertedpendulumFoucaultpendulumFurutapendulumGridironpendulumInertiawheelpendulumInvertedpendulumHarmonograph(a.k.a."Lissajouspendulum")Kapitza'spendulumKater'spendulumMetronome

Npendulum[138]

Pendulum(mathematics)PendulumclockPendulumrocketfallacySecondspendulumSimpleharmonicmotionSphericalpendulumTorsionalpendulum


Thevalueofgreflectedbytheperiodofapendulumvariesfromplacetoplace.ThegravitationalforcevarieswithdistancefromthecenteroftheEarth,i.e.withaltitudeorbecausetheEarth'sshapeisoblate,gvarieswithlatitude.Amoreimportantcauseofthisreductioningattheequatorisbecausetheequatorisspinningatonerevolutionperday,reducingthegravitationalforcethere.

References

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id=dg0RAAAAIAAJ&pg=PA23).London:CharlesGriffin.pp.2224.110. ^Cox,John(1904).Mechanics(http://books.google.com/?id=EXQLAAAAMAAJ&pg=PA311).Cambridge,UK:

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Furtherreading

G.L.BakerandJ.A.Blackburn(2009).ThePendulum:ACaseStudyinPhysics(OxfordUniversityPress).M.Gitterman(2010).TheChaoticPendulum(WorldScientific).MichaelR.Matthews,ArthurStinner,ColinF.Gauld(2005)ThePendulum:Scientific,Historical,PhilosophicalandEducationalPerspectives,SpringerMichaelR.Matthews,ColinGauldandArthurStinner(2005)ThePendulum:ItsPlaceinScience,CultureandPedagogy.Science&Education,13,261277.SchlomoSilbermann,(2014)"PendulumFundamentalThePathOfNowhere"(Book)Matthys,RobertJ.(2004).AccuratePendulumClocks(http://books.google.com/?id=Lx0v2dhnZo8C&pg=PA27).UK:OxfordUniv.Press.ISBN0198529716.Nelson,RobertM.G.Olsson(February1986)."ThependulumRichphysicsfromasimplesystem"(http://fy.chalmers.se/~f7xiz/TIF080/pendulum.pdf).AmericanJournalofPhysics54(2):112121.Bibcode:1986AmJPh..54..112N(http://adsabs.harvard.edu/abs/1986AmJPh..54..112N).doi:10.1119/1.14703(https://dx.doi.org/10.1119%2F1.14703).Retrieved20081029.L.P.Pook(2011).UnderstandingPendulums:ABriefIntroduction(Springer).

Externallinks

NAWCCNationalAssociationofWatch&ClockCollectorsMuseum(http://www.nawcc.org/)Graphicalderivationofthetimeperiodforasimplependulum(http://www.sque.co.uk/physics/simplependulum/)Amoregeneralexplanationofpendula(http://scienceworld.wolfram.com/physics/Pendulum.html)Webbasedcalculatorofpendulumpropertiesfromnumericalinputs

132. ^Trautwine,JohnCresson(1907).TheCivilEngineer'sPocketBook,18thEd.(http://books.google.com/?id=qg41AAAAMAAJ&pg=PA216)NewYork:JohnWiley.p.216.

133. ^Toon,John(September8,2000)."OutofTime:ResearchersRecreate1665ClockExperimenttoGainInsightsintoModernSynchronizedOscillators"(http://gtresearchnews.gatech.edu/newsrelease/PENDULUM.html).GeorgiaTech.Retrieved20070531.

134. Â.L.FradkovandB.Andrievsky,"Synchronizationandphaserelationsinthemotionoftwopendulumsystem",InternationalJournalofNonlinearMechanics,vol.42(2007),pp.895901.

135. Î.I.Blekhman,"Synchronizationinscienceandtechnology",ASMEPress,NewYork,1988,(TranslatedfromRussianintoEnglish)

136. Âninterestingsimulationofthuriblemotioncanbefoundatthissite(http://www.sciences.univnantes.fr/physique/perso/gtulloue/Meca/Oscillateurs/botafumeiro.html#manip).

137. ^R.D.Melville(1905),"TheUseandFormsofJudicialTortureinEnglandandScotland",TheScottishHistoricalReview,vol.2,p.228GeoffreyAbbott(2006)Execution:theguillotine,thePendulum,theThousandCuts,theSpanishDonkey,and66OtherWaysofPuttingSomeonetoDeath,MacMillan,ISBN0312352220,p.213.Bothrefertotheuseofthependulum(pendola)byinquisitorialtribunals.Melville,however,refersonlytoitsuseasatorturemethod,whileAbbottsuggeststhatthedevicewaspurposelyallowedtokillthevictimifherefusedtoconfess.

138. ^YurchenkoD.,AlevrasP.DynamicsoftheNpendulumanditsapplicationtoawaveenergyconverterconcept.InternationalJournalofDynamicsandControl,1,4,2013.


(http://www.calctool.org/CALC/phys/newtonian/pendulum)AnanimatedandinteractiverigidpendulummodelinMSExcel(http://excelunusual.com/archive/2010/09/pendulum/)

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