experiment #2 motion in one dimension pre-lab questions实验# 2一维运动预实验问题 12页

Experiment#2MotioninOneDimensionPre-labQuestions**Disclaimer:Thispre-labisnottobecopied,inwholeorinpart,unlessaproperreferenceismadeastothesource.(Itisstronglyrecommendedthatyouusethisdocumentonlytogenerateideas,orasareferencetoexplaincomplexphysicsnecessaryforcompletionofyourwork.)Copyingofthecontentsofthiswebsiteandturninginthematerialas“originalmaterial”isplagiarismandwillresultinseriousconsequencesasdeterminedbyyourinstructor.Theseconsequencesmayincludeafailinggradefortheparticularpre-laborafailinggradefortheentiresemester,atthediscretionofyourinstructor.**·WhatarethetwoimportantlimitationstokeepinmindwhenusingtheMotionDetectors?Themotiondetectorscannotpickupaccuratelyanobjectlocatedwithinabout0.5metersfromthedetector.Thatis,anythingcloserthan0.5meterstothesensorwilllikelygivea“bad”reading.Second,allobjectsarepickedupbythesensor(includingstrayhair,clothing,arms,books,binders,cables,etc…).Likewise,thesensorpicksupsmallvariationsintheobjectit’smeasuring(likecurvesofthebody–whichcouldeffectivelyintroduceatilttothemeasurement).·Whatistheaccelerationofgravityontheearth(includeunits)?TheaccelerationofgravityontheEarthatzerolatitudeandsealevelisagiven“constant”.Thisisnamelygivenas:Notethatitisparticularlyimportanttocalloutthatthistheoreticalvalueisonly“constant”atzerolatitudeandsealevel.ThisiscausedbytheForceduetogravity(foratwobodysystem).Where“G”istheNewtonianGravitationalParameter,“M”isthemassoftheEarth(orthelargerofthetwomassivebodies),“m”isthemassoftheobjectbeingacteduponbytheEarth(orthesmallerofthetwomassivebodies),and“r”istherelativedistancebetweenthe“centerofmass”-esforthetwobodies.[NotethatthenegativeisprovidedduetothefactthatwearedealingwithamagnitudeofaForceVector.Byconvention,\nwetake“up”(thedirectionawayfromthecenterofmassoftheearth)tobethedefaultdirectionofapositivez-axis.Sincegravitytendstocreateaforceinthedirectionofthecenterofmassoftheearth,thedirectionoftheforceisthusinthenegativezdirection.Ifwewantedtobecompletelyexplicit,wewouldmaketheforcesintovectorsandspecificallycalloutthecoordinateframeanddirectionoftheforces,inadditiontotheirmagnitudes–becauseweallknowthatvectorsmusthaveboth:a)amagnitudeandb)adirection.](InthestandardSEZ[South-East-Zenith]CartesianCoordinateFrameonEarth)Howtocalculate“g”ontheEarth(oranyotherplanetforthatmatter):ByNewton’s2ndlaw:Thus,GravitationalParameterMassoftheEarth(orPlanet)RadiusoftheEarth(orPlanet)6.6726x10-11[m3/kgs2]5.9737x1024[kg]6374333[m]Noticeifweincrease“r”(i.e.,increaseouraltitudefromthe“sealevel”radiusoftheEarth[e.g.,climbupamountain])wedecreasethevalueof“g”.Likewise,wehaveassumedthattheEarthissphericalforallthesecalculations(inwhichcasethecalculationof“g”wouldbeindependentoflatitude.)However,fromverycarefulmeasurements,itisshownthattheEarthisactuallya“squished-sphere”andtheaveragedistancetothepolesislessthentheaveragedistancetotheequator[thisiscalledthe“oblateEarth”].Anexampleofthe“oblateEarth”modelisshowninthefollowingfigure.\nNotethatthepolarradiusisslightlylessthentheequatorialradiusoftheEarth.Thismeansthatasweincreaseordecreaselatitude,thedistancefromthecenterofmassoftheEarthtothecenterofmassofanobjectontheEarthwilllikewiseincreaseordecrease.Thisinturnwillchangethevalueof“g”.ThisvarianceisbeyondthescopeofPES115,butcanbefurtherinvestigatedinAstrophysicscoursesorAerospacecourses.Ifyouwerecurious,the“eccentricity”[squishiness]oftheEarthis:.(Thismeansitisalmostperfectlyspherical–butnotquite.IftheEarthwereadiamond,Iimagineitwouldhavethefollowingcharacteristics:cut=VeryGood,color=D[rareblue/greendiamondJ],clarity=F,caratweight=2x1025[afterallitISaplanet],Symmetry=VeryGood[sinceitisoblate].So,needlesstosay,itwouldbeveryexpensive–andquiteanengagementring!Goodluckfindingafingertofitthaton!J)Youarehere.·Findtheaverageofthefollowingnumbers:10.02[m/s2],9.98[m/s2],9.91[m/s2],10.1[m/s2],9.87[m/s2]Theaverageiscalculatedusingthefollowingequation:\nWherexisthedatawearefindingtheaverageof,andxiaretheindividualcomponentsofthegivendata.Furthermore,Nisthetotalnumberofelementsinthegivendata.Forourcase,Nis5,andxisAcceleration;hence:Forthesakeofeducation,comparethisaveragetothevalueofg.Whattypeoferrorispresentinthedata?Randomerror,systematic.Explainyouranswer.Wewilltakethestandardacceptedvalueoftheaccelerationduetogravityatsea-levelandattheequatortobethetheoreticalvalueforg.Likewise,wewilltakethemeasuredvaluetobetheaveragecalculatedinthequestionabove.Hence:Usingthepercentdifference,wecancomparethe“measured”accelerationduetogravitywiththe“theoretical”accelerationduetogravity.Ifweconsiderthedataprovidedinthequestionabove,wecanseethatthedataisexhibitingthecharacteristicsofbothsystematicandrandomerror.Noticethatthedataappearstobe“high”foreverymeasuredvalue.Thatis–therearenotapproximatelythesamenumberofmeasuredvalueshigherthen9.81m/s2astherearenumberofmeasuredvalueslowerthen9.81m/s2.Allthevaluesappeartobe“shifted-off”tothehighendbyagivenamount.Thisissystematicerror.Furthermore,notallthevaluesarewithinasmalldeltarange.Theyrangefrom9.87m/s2to10.1m/s2.Thisisrandomerror.Wetooktheaverageofmanyvalues–thiswillcompensatefortherandomerror,becausethemoredatapointsavailable,themorelikelytheaverageofthevalueswillconvergetounity.\nThesystematicerrorwillalwaysbepresent,unlesswecanassuredlycalibrateourmeasurementtooltoawellknownsource.Byexaminingourresultofthepercentdifferencecalculation,wecanseethatwearenotfarofffromthetheoreticalvalueoftheaccelerationduetogravity.Thenegativemeansthatwewereslightlyhigher(byexactly1.5%)thetheoreticalvalue.ThiscouldbeexplainedbyColoradoSpringlatitudebeingabout40oNorthandapproximatelyanaltitudeof1.873km(6150ft)abovesealevel.TheactualgeodeticlocationofColoradoSpringsMunicipalisgiveninthefollowingtable.(ThisisavailableontheColoradoSpringsWebPageathttp://www.cospgs.com/weather.html).LATITUDELONGITUDEELEVATION(ft.)38°48'43"N104°42'40"W6145**Aside**Recently,scientistsbelievethattheremaybeprooffor“undergroundgravitationalanomalies”.Thesemassivebodiessuspendedinthemagma,andinadditionthefactthattheEarth’scoremaybeoff-center,couldleadtosmall,butdetectablevariancesinthemeasureoftheaccelerationduetogravityatspecific,“abnormallocales”.Whoknows…maybeoneofthese“undergroundgravitationalanomalies”couldexistbetweentheEarth’scenterofmassandtheexperimentalmassesofthePhysics115labsattheUniversityofColoradoatColoradoSprings–causingsystematic“erroneous”measurements.J**EndAside**·Calculateavelocityvs.timefromthefollowingpositionvs.timeasshowninthemanual.Time[sec]X-displacement[m]00.00000.052270.05000.086380.10000.11490.15000.13940.2000\nTocalculatethevelocity,weneedbothadistanceandatimemeasurement.Sinceourobjectisaccelerating,thevelocitywillbechangingoverthecourseofthemeasurements.Tocalculatethevelocities,weneedtotakeanaverageoftwosequentialtimesandconsiderthechangeofdisplacementoverthattime(rememberthattheslope[derivative]ofx-displacementversustimegraphisavelocity).Thefirstaveragevelocityisgivenas:Forthefirstandseconddatapoints(tofindthefirstaveragevelocity):Time[sec]X-displacement[m]00.00000.052270.0500Theaverageofthetwosequentialtimesis:\nForthesecondandthirddatapoints(tofindthesecondaveragevelocity):Time[sec]X-displacement[m]0.052270.05000.086380.1000Theaverageofthetwosequentialtimesis:Forthethirdandfourthdatapoints(tofindthethirdaveragevelocity):Time[sec]X-displacement[m]0.086380.10000.11490.1500Theaverageofthetwosequentialtimesis:\nForthefourthandfifthdatapoints(tofindthefourthaveragevelocity):Time[sec]X-displacement[m]0.11490.15000.13940.2000Theaverageofthetwosequentialtimesis:Ifweconsolidatetheabovefourcalculationsintoasingletable,wegetthefollowing:Time[sec]X-velocity[m/s]0.0261350.956570.0693251.465840.1006401.753160.1271502.04082\n**NotethatthisisthesamecalculationsperformedbytheLabProsoftware.****Alsonotethatifwefollowedthesameprocessyetagain,wecouldfindtheaverageaccelerations(andifwewantedtobesuperfancy,comparethatto“g”).J****Finally,notethevaluesoftheregressionforeachoftheabovegraphs;by“calculating”theaveragevelocities,thereisadegradationofthe“reliability”ofthedata.Hence,togetamoreaccuratevaluefortheinitialvelocity(aswedidforlab5),itwouldbemoreappropriatetousethefirstgraphofx-displacementversustime,ratherthenx-velocityversustime.**Ifwecomparethebestfitlinefromthex-displacementversustimegraphtotheequationforkinematicmotion:[m/s2][m/s][m]10.70580.6909-0.0002So,itisobviousthatforthisproblem,thex-directionispointingtowardthecenterofmassoftheearthinthiscoordinateframe.(Typically,we’vebeencallingthisvalue“y”;however,theaxisnamingisarbitrary–aslongasweareconsistentthroughouttheproblem.[Technically,wecouldhavecalledthisthe“nadir-axis”.])\n·F=3.5xisalinearrelationship.Whatdatawouldyouplotalongthey-axis,thex-axis?Sincethegivenequationisalinearrelationship,itmustbeoftheform:Whichweseeistrue!Alongthey-axis,Iwouldplotthemeasuredforce.Iwouldlabelthisaxis:“Force[N]”or“Force[kg-m/s2]”.Alongthex-axis,Iwouldplotthemeasureddisplacement.Iwouldlabelthisaxis:“Displacement[cm]”or“X[cm]”.·Whatwouldtheslopeofthisgraphrepresent?Whatshouldbethevalueofthey-intercept?Theslopeofthegraphforwouldbe3.5.ThisrepresentsaconstantofhowmuchtheForcewillchangewhenwechangethedisplacement.Ifweexaminethegivenequationandtheequationforalinearrelationship,wecaneasilydeterminethevalueofthey-intercept.They-interceptisthevaluebintheequationforalinearrelationship.Thisisthevalueofy,whenx=0.Thusforourgivenequation,byinspection,they-interceptis0.Bycalculation:·Matchthedescriptionswiththecorrectbuttons.\n1.Navigatethroughtheprogram’sdifferentpages.2.Startthecollectionofdata.3.Zoomin4.Tablesweredataisstoredinspreadsheetformat.5.Autoscaleagraph.6.Theplacewherethedataisdisplayedgraphically.7.Curvefitting\nThisisthesamesoftwarepackageusedinPES115.ThevariouscontrolsoftheLoggerProsoftwarepackageareshowninthefollowingfigure: