Measurement Analysis Measurement Uncertainty and Propagation You should read Sections - PDF document

MeasurementAnalysis1:MeasurementUncertaintyandPropagationYoushouldreadSections1.6and1.7onpp.12{16oftheSerwaytextbeforethisactivity.PleasenotethatwhileattendingtheMA1eveninglectureisoptional,theMA1assignmentisNOToptionalandmustbeturnedinbeforethedeadlineforyourdivisionforcredit.Thedeadlineforyourdivisionisspeci¯edinREADMEFIRST!atthefrontofthismanual.Attheendofthisactivity,youshould:1.Understandtheformofmeasurementsinthelaboratory,includingmeasuredvaluesanduncertainties.2.Knowhowtogetuncertaintiesformeasurementsmadeusinglaboratoryinstruments.3.Beabletodiscriminatebetweenmeasurementsthatagreeandthosethatarediscrepant.4.Understandthedi®erencebetweenprecisionandaccuracy.5.Beabletocombinemeasurementsandtheiruncertaintiesthroughaddition,subtrac-tion,multiplication,anddivision.6.Beabletoproperlyroundmeasurementsandtreatsigni¯cant¯gures.1Measurements1.1UncertaintyinmeasurementsInanidealworld,measurementsarealwaysperfect:there,woodenboardscanbecuttoexactlytwometersinlengthandablockofsteelcanhaveamassofexactlythreekilograms.However,weliveintherealworld,andheremeasurementsareperfect.Inourworld,measuringdeviceshavelimitations.Theimperfectioninherentinallmeasurementsiscalledanuncertainty.InthePhysics152laboratory,wewillwriteanuncertaintyalmosteverytimewemakeameasurement.Ournotationformeasurementsandtheiruncertaintiestakesthefollowingform:measuredvalueuncertaintyproperunitswheretheisread`plusorminus.' 9.8009.8029.8049.806 PurdueUniversityPhysics152LMeasurementAnalysis1Figure1:Measurementanduncertainty:(9Considerthemeasurement=(9003)m/s.Weinterpretthismeasurementasmeaningthattheexperimentallydeterminedvalueofcanlieanywherebetweenthevalues9801+0003mand9003m,or9798m804m.Asyoucansee,arealworldmeasurementisnotonesimplemeasuredvalue,butisactuallyarangeofpossiblevalues(seeFigure1).Thisrangeisdeterminedbytheuncertaintyinthemeasurement.Asuncertaintyisreduced,thisrangeisnarrowed.Herearetwoexamplesofmeasurements:=(4002)m/s=(6Lookoverthemeasurementsgivenabove,payingcloseattentiontothenumberofdecimalplacesinthemeasuredvaluesandtheuncertainties(whenthemeasurementisgoodtothethousandthsplace,soistheuncertainty;whenthemeasurementisgoodtothehundredthsplace,soistheuncertainty).Youshouldnoticethattheyalwaysagree,andthisismostimportant:|Inameasurement,themeasuredvalueanditsuncertaintymustalwayshavethesamenumberofdigitsafterthedecimalplace.Examplesofnonsensicalmeasurementsare(90001)mand(91)mwritingsuchnonsensicalmeasurementswillcausereaderstojudgeyouaseitherincompetentorsloppy.AvoidwritingimpropermeasurementsbyalwaysmakingsurethedecimalplacesSometimeswewanttotalkaboutmeasurementsmoregenerally,andsowewritethemwithoutactualnumbers.Inthesecases,weusethelowercaseGreekletter,orrepresenttheuncertaintyinthemeasurement.Examplesinclude:Althoughunitsarenotexplicitlywrittennexttothesemeasurements,theyareimplied.WewillusethesegeneralexpressionsformeasurementswhenwediscussthepropagationofuncertaintiesinSection4.1.2UncertaintiesinmeasurementsinlabInthelaboratoryyouwillbetakingrealworldmeasurements,andforsomemeasurementsyouwillrecordbothmeasuredvaluesanduncertainties.Gettingvaluesfrommeasuring 24681012 PurdueUniversityPhysics152LMeasurementAnalysis1equipmentisusuallyassimpleasreadingascaleoradigitalreadout.Determininguncer-taintiesisabitmorechallengingsinceyou|notthemeasuringdevice|mustdeterminethem.Whendetermininganuncertaintyfromameasuringdevice,youneedto¯rstdeter-minethesmallestquantitythatcanberesolvedonthedevice.Then,foryourworkinPHYS152L,theuncertaintyinthemeasurementistakentobethisvalue.Forexample,ifadigitalreadoutdisplays135g,thenyoushouldwritethatmeasurementas(101)g.Thesmallestdivisionyoucanclearlyreadisyouruncertainty.Ontheotherhand,readingascaleissomewhatsubjective.Supposeyouuseameterstickthatisdividedintocentimeterstodeterminethelength()ofarod,asillustratedinFigure2.First,youreadyourmeasuredvaluefromthisscaleand¯ndthattherodis6cm.Dependingonthesharpnessofyourvision,theclarityofthescale,andtheboundariesofthemeasuredobject,youmightreadtheuncertaintyas1cm,5cm,or2cm.Anuncertaintyof1cmorsmallerisdubiousbecausetheendsoftheobjectareroundedanditishardtoresolve1cm.Thus,youmightwanttorecordyourmeasurementas)=(61)cm)=(65)cmor()=(62)cmsinceallthreemeasurementswouldappearreasonable.Forthepurposesofdiscussionanduniformityinthislaboratorymanual,wewillusethelargestreasonableuncertainty.Forourexample,thisisFigure2:Ameasurementobtainedbyreadingascale.Acceptablemeasurementsrangefrom1cmto62cm,dependingonthesharpnessofyourvision,theclarityofthescale,andtheboundariesofthemeasuredobject.Examplesofunacceptablemeasurementsare62cmand601cm.1.3PercentageuncertaintyofmeasurementsWhenwespeakofameasurement,weoftenwanttoknowhowreliableitis.Weneedsomewayofjudgingtherelativeworthofameasurement,andthisisdoneby¯ndingthepercentageuncertaintyofameasurement.Wewillrefertothepercentageuncertaintyofameasurementastheratiobetweenthemeasurement'suncertaintyanditsmeasuredvaluemultipliedby100%.Youwilloftenhearthiskindofuncertaintyorsomethingcloselyrelatedusedwithmeasurements{ameterisgoodto3%offullscale,or1%ofthereading,orgoodtoonepartinamillion.percentageuncertaintyofameasurement()isde¯nedas Thinkaboutpercentageuncertaintyasawayoftellinghowmuchameasurementde-viatesfrom\perfection."Withthisideainmind,itmakessensethatastheuncertainty PurdueUniversityPhysics152LMeasurementAnalysis1forameasurementdecreases,thepercentageuncertainty 100%decreases,andsothemeasurementdeviateslessfromperfection.Forexample,ameasurementof(21)mhasapercentageuncertaintyof50%,oronepartintwo.Incontrast,ameasurementof01)mhasapercentageuncertaintyof0.5%(or1partin200)andisthereforethemoreprecisemeasurement.Ifthereweresomewaytomakethissamemeasurementwithzerouncertainty,thepercentageuncertaintywouldequal0%andtherewouldbenodeviationwhatsoeverfromthemeasuredvalue|wewouldhavea\perfect"measurement.Unfortunately,thisneverhappensintherealworld.1.4ImplieduncertaintiesWhenyoureadaphysicstextbook,youmaynoticethatalmostallthemeasurementsstatedaremissinguncertainties.Doesthismeanthattheauthorisabletomeasurethingsperfectly,withoutanyuncertainty?Notatall!Infact,itiscommonpracticeintextbooksnottowriteuncertaintieswithmeasurements,eventhoughtheyareactuallythere.Insuchcases,theuncertaintiesareimplied.Wetreattheseimplieduncertaintiesthesamewayaswedidwhentakingmeasurementsinlab:|Inameasurementwithanimplieduncertainty,theactualuncertaintyiswrit-tenas1inthesmallestplacevalueofthegivenmeasuredvalue.Forexample,ifyouread80146minatextbook,youknowthismeasuredvaluehasanimplieduncertaintyof000001m.Tobemorespeci¯c,youcouldthenwrite)=(900001)m1.5Decimalpoints|don'tlosethemIfadecimalpointgetslost,itcanhavedisastrousconsequences.Oneofthemostcommonplaceswhereadecimalpointgetslostisinfrontofanumber.Forexample,writing.52cmsometimesresultsinareadermissingthedecimalpoint,andreadingitas52cm|onehundredtimeslarger!Afterall,adecimalpointisonlyasimplesmalldot.However,writing0.52cmvirtuallyeliminatestheproblem,andwritingleadingzerosfordecimalnumbersisstandardscienti¯candengineeringpractice.2Agreement,Discrepancy,andDi®erenceInthelaboratory,youwillnotonlybetakingmeasurements,butalsocomparingthem.Youwillcompareyourexperimentalmeasurements(i.e.theonesyou¯ndinlab)tosometheoretical,predicted,orstandardmeasurements(i.e.thetypeyoucalculateorlookupinatextbook)aswellastoexperimentalmeasurementsyoumakeduringasecond(orthird...)datarun.Weneedamethodtodeterminehowcloselythesemeasurementscompare.Tosimplifythisprocess,weadoptthefollowingnotion:twomeasurements,whencom-pared,eitheragreewithinexperimentaluncertaintyortheyarediscrepant(thatis,theydo 9.7909.8009.810 g 9.7909.8009.810 g PurdueUniversityPhysics152LMeasurementAnalysis1notagree).Beforeweillustratehowthisclassi¯cationiscarriedout,youshould¯rstrecallthatameasurementinthelaboratoryisnotmadeupofonesinglevalue,butawholerangeofvalues.Withthisinmind,wecansay,Twomeasurementsareinagreementifthetwomeasurementssharevaluesincommon;thatis,theirrespectiveuncertaintyrangespartially(ortotally)overlap.Figure3:AgreementanddiscrepancyofgravitymeasurementsForexample,alaboratorymeasurementof()=(9004)mbeingcomparedtoascienti¯cstandardvalueof()=(90025)mAsillustratedinFigure3(a),weseethattherangesofthemeasurementspartiallyoverlap,andsoweconcludethatthetwomeasurementsagree.Rememberthatmeasurementsareeitherinagreementorarediscrepant.Itthenmakessensethat,Twomeasurementsarediscrepantifthetwomeasurementsdonotsharevaluesincommon;thatis,theirrespectiveuncertaintyrangesdonotoverlap.Supposeasanexamplethatalaboratorymeasurement()=(9004)misbeingcomparedtothevalueof()=(90025)mFromFigure3(b)wenoticethattherangesofthemeasurementsdonotoverlapatall,andsowesaythesemeasurementsarediscrepant. Precision & Accuracyprecise, but not accurate, but not (a)(b)PurdueUniversityPhysics152LMeasurementAnalysis1Whentwomeasurementsbeingcompareddonotagree,wewanttoknowbyhowmuchtheydonotagree.Wecallthisquantitythediscrepancybetweenmeasurements,andweusethefollowingformulatocomputeit:Thediscrepancybetweenanexperimentalmeasurement()andatheoreticalorstandardmeasurement()is:experimentalstandard standardAsanexample,takethetwodiscrepantmeasurements()and()fromthepreviousexample.Sincewefoundthatthesetwomeasurementsarediscrepant,wecancalculatethediscrepancybetweenthemas: 100%= Keepthefollowinginmindwhencomparingmeasurementsinthelaboratory:1.Ifyou¯ndthattwomeasurementsagree,statethisinyourreport.DoNOTcomputeadiscrepancy.2.Ifyou¯ndthattwomeasurementsarediscrepant,statethisinyourreportandthengoontocomputethediscrepancy.3PrecisionandaccuracyFigure4:Precisionandaccuracyintargetshooting.Ineverydaylanguage,thewordsprecisionandaccuracyareofteninterchangeable.Inthesciences,however,thetwotermshavedistinctmeanings: 123 a: neither accurate nor precise 123 123 PurdueUniversityPhysics152LMeasurementAnalysis1Precisiondescribesthedegreeofcertaintyonehasaboutameasurement.Accuracydescribeshowwellmeasurementsagreewithaknown,standardmea-surement.Let's¯rstexaminetheconceptofprecision.Figure4(a)showsaprecisetargetshooter,sincealltheshotsareclosetooneanother.Becausealltheshotsareclusteredaboutasinglepoint,thereisahighdegreeofcertaintyinwheretheshotshavegoneandsothereforetheshotsareprecise.InFigure5(b),themeasurementsontherulerareallclosetooneanother,andlikethetargetshots,theyarepreciseaswell.Accuracy,ontheotherhand,describeshowwellsomethingagreeswithastandard.InFigure4(b),the\standard"isthecenterofthetarget.Alltheshotsareclosetothiscenter,andsowewouldsaythatthetargetshooterisaccurate.However,theshotsarenotclosetooneanother,andsotheyarenotprecise.Hereweseethattheterms\precision"and\accuracy"arede¯nitelynotinterchangeable;onedoesnotimplytheother.Nevertheless,itispossibleforsomethingtobebothaccurateandprecise.InFigure5(c),themeasurementsareaccurate,sincetheyareallclosetothe\standard"measurementof1.5cm.Inaddition,themeasurementsareprecise,becausetheyareallclusteredaboutoneanother.Notethatitisalsopossibleforameasurementtobeneitherprecisenoraccurate.InFigure5(a),themeasurementsareneitherclosetooneanother(andthereforenotprecise),noraretheyclosetotheacceptedvalueof1.5cm(andhencenotaccurate).Figure5:Examplesofprecisionandaccuracyinlengthmeasurements.Herethehollowheadedarrowsindicatethe`actual'valueof1.5cm.Thesolidarrowsrepresentmeasurements.Youmayhavenoticedthatwehavealreadydevelopedtechniquestomeasureprecisionandaccuracy.InSection1.3,wecomparedtheuncertaintyofameasurementtoitsmeasuredvalueto¯ndthepercentageuncertainty.Thecalculationofpercentageuncertaintyisactuallyatesttodeterminehowcertainyouareaboutameasurement;inotherwords,howprecisethemeasurementis.InSection2,welearnedhowtocompareameasurementtoastandard PurdueUniversityPhysics152LMeasurementAnalysis1oracceptedvaluebycalculatingapercentdiscrepancy.Thiscomparisontoldyouhowcloseyourmeasurementwastothisstandardmeasurement,andso¯ndingpercentdiscrepancyisreallyatestforaccuracy.Itturnsoutthatinthelaboratory,precisionismucheasiertoachievethanaccuracy.Precisioncanbeachievedbycarefultechniquesandhandiwork,butaccuracyrequiresex-cellenceinexperimentaldesignandmeasurementanalysis.Duringthislaboratorycourse,youwillexaminebothaccuracyandprecisioninyourmeasurementsandsuggestmethodsofimprovingboth.4Propagationofuncertainty(worstcase)Inthelaboratory,wewillneedtocombinemeasurementsusingaddition,subtraction,mul-tiplication,anddivision.However,measurementsarecomposedoftwoparts|ameasuredvalueandanuncertainty|andsoanyalgebraiccombinationmustaccountforboth.Perform-ingtheseoperationsonthemeasuredvaluesiseasilyaccomplished;handlinguncertaintiesposesthechallenge.Wemakeuseofthepropagationofuncertaintytocombinemeasurementswiththeassumptionthatasmeasurementsarecombined,uncertaintyincreases|hencetheuncertaintypropagatesthroughthecalculation.Hereweshowhowtocombinetwomea-surementsandtheiruncertainties.Ofteninlabyouwillhavetokeepusingthepropagationformulaeoverandover,buildingupmoreandmoreuncertaintyasyoucombinethree,fouror¯vesetofnumbers.Whenaddingtwomeasurements,theuncertaintyinthe¯nalmeasurementisthesumoftheuncertaintiesintheoriginalmeasurements:)(1)Asanexample,letuscalculatethecombinedlength()oftwotableswhoselengthsare()=(304)mand()=(1001)m.Usingthisadditionrule,we¯ndthat)=(304)m+(1001)m=(1305)mWhensubtractingtwomeasurements,theuncertaintyinthe¯nalmeasurementisagainequaltothesumoftheuncertaintiesintheoriginalmeasurements:)(2)Forexample,thedi®erenceinlengthbetweenthetwotablesmentionedaboveis)=(1001)m04)m=[(1001+004)]m=(705)m PurdueUniversityPhysics152LMeasurementAnalysis1Becarefulnottosubtractuncertaintieswhensubtractingmeasurements|uncertaintyALWAYSgetsworseasmoremeasurementsarecombined.Whenmultiplyingtwomeasurements,theuncertaintyinthe¯nalmeasurementisfoundbysummingthepercentageuncertaintiesoftheoriginalmeasurementsandthenmultiplyingthatsumbytheproductofthemeasuredvalues: + Aquickderivationofthismultiplicationruleisgivenbelow.First,assumethatthemeasuredvaluesarelargecomparedtotheuncertainties;thatis,Then,usingthedistributivelawofmultiplication:A±BB±A)(4)Sincetheuncertaintiesaresmallcomparedtothemeasuredvalues,theproductoftwosmalluncertaintiesisanevensmallernumber,andsowediscardtheproduct).Withfurthersimpli¯cation,we¯nd: + Itshouldbenotedthattheaboveequationismathematicallyunde¯nedifeitherAorBiszero.Inthiscaseequation4isusedtoobtaintheuncertaintysinceitisvalidforallvaluesofAandB.Nowletususethemultiplicationruletodeterminetheareaofarectangularsheetwithlength()=(102)mandwidth()=(201)cm=(001)m.Thearea()isthen + =(1 1:50+0:01 §(0:0133+00500)]mm§0:0633]m=(002)m PurdueUniversityPhysics152LMeasurementAnalysis1Noticethatthe¯nalvaluesforuncertaintyintheabovecalculationweredeterminedbymultiplyingtheproduct()outsidethebracketbythesumofthetwopercentageuncertainties(±l=l±w=w)insidethebracket.Alwaysrememberthiscrucialstep!Also,noticehowthe¯nalmeasurementfortheareawasrounded.Thisroundingwasperformedbyfollowingtherulesofsigni¯cant¯gures,whichareexplainedindetaillaterinSection5.RecallourdiscussionofpercentageuncertaintyinSection1.3.Itisherethatweseethebene¯tsofusingsuchaquantity;speci¯cally,wecanuseittotellrightwhichofthetwooriginalmeasurementscontributedmosttothe¯nalareauncertainty.Intheaboveexample,weseethatthepercentageuncertaintyofthewidthmeasurement±w=w100%is5%,whichislargerthanthepercentageuncertainty(±l=l3%ofthelengthmeasurement.Hence,thewidthmeasurementcontributedmosttothe¯nalareauncertainty,andsoifwewantedtoimprovetheprecisionofourareameasurement,weshouldconcentrateonreducingwidthuncertainty(sinceitwouldhaveagreatere®ectonthetotaluncertainty)bychangingourmethodformeasuringWhendividingtwomeasurements,theuncertaintyinthe¯nalmeasurementisfoundbysummingthepercentageuncertaintiesoftheoriginalmeasurementsandthenmultiplyingthatsumbythequotientofthemeasuredvalues: (B§=µA B¶"1§Ã + Asanexample,let'scalculatetheaveragespeedofarunnerwhotravelsadistanceof2)min(912)susingtheequation D=t;where¹istheaveragespeed,isthedistancetraveled,andisthetimeittakestotravelthatdistance. §µD t¶"1§Ã + !#=µ100: 85s 100:0+0:12 =1015[1002000+001218)]ms=1015[101418)]m=(101439)m1)mInthisparticularexamplethe¯naluncertaintyresultsmainlyfromtheuncertaintyinthemeasurementof,whichisseenbycomparingthepercentageuncertaintiesofthetimeanddistancemeasurements,(±t=t22%and(±D=D20%,respectively.Therefore,toreducetheuncertaintyin( v§± ),wewouldwanttolook¯rstatchangingthewayismeasured.Specialcases|inversionandmultiplicationbyaconstant: PurdueUniversityPhysics152LMeasurementAnalysis1(a)Ifyouhaveaquantity,youcaninvertitandapplytheoriginalpercentageuncertainty: X§µ1 X¶"1§ (b)Tomultiplybyaconstant,t,kY§k±Y]Itisimportanttorealizethattheseformulasandtechniquesallowyoutoperformthefourbasicarithmeticoperations.Youcan(andwill)combinethembyrepetitionforthesumofthreemeasurements,orthecubeofameasurement.Normallyitisimpossibletousethesesimplerulesformorecomplicatedoperationssuchasasquarerootoralogarithm,butthetrigonometricfunctionssinandtanareexcep-tions.Becausethesefunctionsarede¯nedastheratiosbetweenlengths,wecanusethequotientruletoevaluatethem.Forexample,inarighttrianglewithoppositeside)andhypotenuse(),sin .Similarly,anyexpressionthatcanbebrokendownintoarithmeticstepsmaybeevaluatedwiththeseformulas;forexample,FindingtheuncertaintyofasquarerootThemethodforobtainingthesquarerootofameasurement.usessomealgebracoupledwiththemultiplicationrule.Let()and()betwomeasurements.Further,assumethatthesquarerootof()isequaltothemeasurement().Then, )(6)Squaringbothsides,weobtainUsingthemultiplicationruleon(,we¯nd B±BThus,B±B)whichmeans Aand B= p A: PurdueUniversityPhysics152LMeasurementAnalysis1MakingthissubstitutionintoEquation6,wearriveatthe¯nalresult (A§p A§ p Thistechniquefor¯ndingtheuncertaintyinasquarerootwillberequiredinE4|Anotherexample:acaseinvolvingatripleproduct.Theformulaforthevolumeofarodwithacircularcross{section()andlengthisgivenby.Giveninitialmeasurements()and(),deriveanexpressionfor(Notethatuncertainty.Usingthederivationoftheworstcasemultiplicationpropagationrule(Equation4)asaguide,westartwithandexpandthetermsinvolvingonthelefthandside.side.r2+r(§±r)+r(§±r)+(§±r)(§±r)](l§±l)Discardingtheterminvolvingtheproductofmeasurementuncertainties(),sinceitissmallcomparedtotheotherterms,weobtainobtainr2+2r(§±r)](l§±l)Nextwemultiplyoutthe¯nalproductontheleft.left.r2l+r2(§±l)+2rl(§±r)+2r(§±r)(§±l)]Againwediscardtermsinvolvingproductsofmeasurementuncertaintiessuchas(toobtainobtainr2l+r2(§±l)+2rl(§±r)]Finally,wecanfactorouttoobtain §2 Otheruncertaintypropagationtechniques.Theworstcaseuncertaintypropa-gationassumesthatallmeasurementuncertaintiesconspiretogivetheworstpossibleuncertaintyinyour¯nalresult.Fortunatelythisdoesnotusuallyhappeninnature,andtherearetechniquestotakethisintoaccount,thesimplestbeingtheadditionofuncertaintiesinquadratureandtakingthesquarerootofthesum.However,thesetechniquesaremorecomplexandinconsistentwiththemathematicalrequirementsforPHYS152,andwehaveavoidedthem.Agoodstartinlearningaboutthesemoresophisticatedtechniquesistoreadthereferenceslistedattheendofthischapter. PurdueUniversityPhysics152LMeasurementAnalysis15RoundingmeasurementsTheprevioussectionscontainthebulkofwhatyouneedtotakeandanalyzemeasurementsinthelaboratory.Nowitistimetodiscussthe¯nerdetailsofmeasurementanalysis.Thesubtletiesweareabouttopresentcauseaninordinateamountofconfusioninthelaboratory.Gettingcaughtupindetailsisafrustratingexperience,andthefollowingguidelinesshouldhelpalleviatetheseproblems.Anoften-askedquestionis,\HowshouldIroundmymeasurementsinthelaboratory?"Theansweristhatyoumustwatchsigni¯cant¯guresincalculationsandthensurethenumberofdecimalplacesofameasuredvalueanditsuncertaintyagree.Beforewegiveanexample,weshouldexplorethesetwoideasinsomedetail.5.1Treatingsigni¯cant¯guresThesimplestde¯nitionforasigni¯cant¯gureisadigit(0-9)thatactuallyrepresentssomequantity.Zerosthatareusedtolocateadecimalpointarenotconsideredsigni¯cant¯gures.Anymeasuredvalue,then,hasaspeci¯cnumberofsigni¯cant¯gures.SeeTable1forTherearetwomajorrulesforhandlingsigni¯cant¯guresincalculations.Oneappliesforadditionandsubtraction,theotherformultiplicationanddivision.Whenaddingorsubtractingquantities,thenumberofdecimalplacesintheresultshouldequalthesmallestnumberofdecimalplacesofanyterminthesum(or67=49771468=7133208+1872+0851=40Whenmultiplyingordividingquantities,thenumberofsigni¯cant¯guresinthe¯nalansweristhesameasthenumberofsigni¯cant¯guresintheleastaccurateofthequantitiesbeingmultiplied(ordivided).7=82425748=05.2Measuredvaluesanduncertainties:NumberofdecimalplacesAsmentionedearlierinSection1.1,welearnedthatforanymeasurement(),thenumberofdecimalplacesofthemeasuredvaluemustequalthoseofthecorrespondinguncertaintyBelowaresomeexamplesofcorrectlywrittenmeasurements.Noticehowthenumberofdecimalplacesofthemeasuredvalueanditscorrespondinguncertaintyagree.)=(3002)m()=(414)kg PurdueUniversityPhysics152LMeasurementAnalysis1MeasuredvalueNumberofsigni¯cant¯gures12331.2331.23040.0012330.0012304Table1:Examplesofsigni¯cant¯gures5.3RoundingSupposeweareaskedto¯ndthearea()ofarectanglewithlength(005)mandwidth()=(101)m.Beforepropagatingtheuncertaintiesbyusingthemultiplicationrule,weshould¯rst¯gureouthowmanysigni¯cant¯guresour¯nalmeasuredvaluemusthave.Inthiscase,,andsincehasfoursigni¯cant¯guresandhasthreesigni¯cant¯gures,islimitedtothreesigni¯cant¯gures.Rememberthisresult;wewillcomebacktoitinafewsteps.Wemaynowusethemultiplicationruletocalculatethearea: + =(2 2:708+0:01 =(2843)[1001846+0009524)]m843(1011370)m=(203232)mNoticethatintheintermediatestepdirectlyabove,weallowedeachnumberoneextrasigni¯cant¯gurebeyondwhatweknowour¯nalmeasuredvaluewillhave;thatis,weknowthe¯nalvaluewillhavethreesigni¯cant¯gures,butwehavewritteneachoftheseintermediatenumberswithfoursigni¯cant¯gures.Carryingtheextrasigni¯cant¯gureensuresthatwewillnotintroduceround-o®error.Wearejusttwostepsfromwritingour¯nalmeasurement.Steponeisrecallingtheresultwefoundearlier|thatour¯nalmeasuredvaluemusthavethreesigni¯cant¯gures.Thus,wewillround2843mto284m.Oncethisstepisaccomplished,weroundouruncertaintytomatchthenumberofdecimalplacesinthemeasuredvalue.Inthiscase,weround003233mto003m.Finally,wecanwrite)=(203)m PurdueUniversityPhysics152LMeasurementAnalysis16ReferencesThesebooksareonreserveinthePhysicsLibrary(PHYS291).Askforthembytheauthor'slastname.1.Bevington,P.R.,Datareductionanduncertaintyanalysisforthephysicalsciences(McGraw-Hill,NewYork,1969).2.Young,H.D.,Statisticaltreatmentofexperimentaldata(McGraw-Hill,NewYork, PurdueUniversityPhysics152LMeasurementAnalysis1Thispageisdeliberatelyleftblank. 23cm ABCD PurdueUniversityPhysics152LMeasurementAnalysis1MeasurementAnalysisProblemSetMA1 Labday/time Division GTA Writeyouranswersinthespaceprovided.Forfullcredit,showallessentialintermediatesteps,includeunitsandensurethatmeasuredvaluesanduncertaintiesagreeinthenumberofdecimalplaces.1.Usingtheabove¯gure,writethemeasuredvaluesanduncertaintiesforthefollowing:)cm.)cm.)cm.)cm.(e)Howdidyoudeterminetheseuncertainties?2.Inthelaboratory,yourpartnerusesadigitalbalanceto¯ndthemassofasmallobject.He/shetellsyou(correctly)thatthedigitalreadoutshows43g.(a)Writethecorrectmassmeasurementanditsuncertainty.)g.(b)Whatisthepercentageuncertaintyofthismeasurement? PurdueUniversityPhysics152LMeasurementAnalysis13.Inlab,oneofyourpartnersdetermines(correctly)thatthesurfaceareaofanobjectis25.97cm.Allofthe¯guresinthismeasurementaresigni¯cant.(a)YourHP-9000calculatortellsyouthattheuncertaintyis0.04382361cm.Writetheappropriatemeasurementanditsuncertainty.)cm(b)Alternatively,yourHP-9000toldyouthattheuncertaintyis0.0012543797cmWritetheappropriatemeasurementanditsuncertainty.)cm(c)Finally,thecalculatortoldyouthattheuncertaintyis0.017386642cm.Writetheappropriatemeasurementanditsuncertainty.)cm(d)Brie°yexplainhowyoudeterminedthesethreenumericaluncertainties.4.Inthelaboratoryyoudeterminethegravitationalconstant()tobe006)m/s.Accordingtoageophysicalsurvey,theacceptedlocalvaluefor)is(9002)m/s(a)DrawadiagramlikeFigure3showingwhetheryourmeasurementagreeswiththeacceptedvaluewithinthelimitsofexperimentaluncertaintyornot.(b)Ifthesemeasuresdonotagree,whatistheactualdiscrepancy? PurdueUniversityPhysics152LMeasurementAnalysis15.Foralaboratoryexercise,youdeterminethemassesoftwoairtrackglidersas)=(4843)gand()=(3141)g.Determinethefollowingcombinationsofthemeasurements,andshowyourwork.(a)()g.(b)Determinetheprecisionsofthemeasurements()and(Whichcalculatedprecisionisthelarger?Usingthisinformation,determinewhichisthebettermeasurement.(c)()g.6.(a)Laboratorymeasurementsperformeduponarectangularsteelplateshowthelength()=(225)cmandthewidth()=(810)cm.Determinetheareaofthesteelplateandtheuncertaintyinthearea.)cm(b)AsDirectoroftheNationalScienceFoundation,youmustdecidewhatisthebestmeanstoimprovetheprecisionoftheareameasurementofthesteelplate.Youcanspendmoneyonaspacegizmotrontobettermeasurelength,oronasuperconductingwhizbangtobettermeasurewidth,butnotboth.Onwhichmeasurementshouldyouspendthemoney?Justifyyourdecisionwithnumbers. PurdueUniversityPhysics152LMeasurementAnalysis17.Duringanotherairtrackexperiment,youdeterminetheinitialpositionofaglidertobe()=(0001)mandits¯nalpositiontobe(003)m,withanelapsedtime()=(001)sduringthedisplacement.(a)Findthetotaldisplacementoftheglider()m.(b)Findtheaveragespeedoftheglider(¹gliderglidergliderglider)m/s.(c)Calculatethepercentageuncertaintiesforthetwoquantities()and(Basedontheseprecisions,determinewhichofthetwoquantitiescontributesmosttotheoveralluncertaintyglider(d)Ifwecouldchangetheapparatussoastomeasuretimetentimesmoreaccurately(butnotboth),whichshouldwechangeandwhy? PurdueUniversityPhysics152LMeasurementAnalysis18.Inthelaboratory,ameasurementfor()wastakenas()=(335)sWriteavalueforitsreciprocal.Showcalculations,andensurethatyouhandlesigni¯-cant¯guresproperly. (x§)1 9.Theformulaforthevolumeofaboxwithheight,base,andlengthisgivenby.Giveninitialmeasurements(),(),and(),deriveanexpressionfor().DoNOTusethemultiplicationrule(Equation3)inderivingthisequation.Hint:Usetheoftheworstcasemultiplicationpropagationrule(Equation4)asaguide.Startwith().Expand,showintermediatesteps,andregroupandsimplifyyoursolutionasmuchaspossible.Discardproductsofmeasurementuncertainties,suchas()and(