Experiment 1 Measuring “g” using the Kater Pendulum

1

Experiment1Measuring“g”usingtheKaterPendulum

Goal:Thefreefallacceleration,“g”,isaquantitythatvariesbylocationaroundthesurfaceoftheearth. While everywhereon the surface it is9.8m/s2 to two significantdigits, for someexperimentswewillneedtohavemoreprecision.Moreover,itwillbeimportanttoknowthe uncertainty in the local value of g. The purpose of this experiment is to perform acarefulmeasurementofthelocalvalueofgthatcanthenbeusedincalculationsinfutureexperiments.Background:The experiment will involve the use of a physical pendulum. Recall that the differencebetweenasimplependulumandaphysicalpendulumisthatasimplependulumhasallofits mass concentrated at the tip while a physical pendulum has mass distributedthroughoutitslength.Consider an arbitrary pendulum, as indicated above. The key characteristics are thedistancebetween the centerofmass and thepivotpoint (“d” in the abovediagram), themomentofinertia(I)aboutthepivotpoint,andthetotalmassofthependulum,M.IfyouconsultanintroductoryphysicstextbookorAnalyticalMechanicsbyFowlesandCassiday,youwillfindthattheperiodforsuchapendulumisgivenby

(1)

Thisequationappliesprovidedtheamplitudeofoscillationissufficientlysmall.

T = 2π IMgd

pivot

cmd

2

Notethatinthecaseofasimplependulum,thestringorrodistakentobemasslesssothatI=Md2,reducing(1)to

T = 2π dg (2)

Thiscommonlyappears in textbookswith“d”replacedby“L”. These twovariableshavethesamemeaninginthecontextofthesimplependulum.Aquickwaytodetermineg,thenistomeasuretheperiodofasimplependulumandrearrange(2)toobtain:

g = 4π 2 dT 2 (3)

Thismethodwillbeexploredbrieflyinthisexperiment,longenoughtoconvinceyouthattheKaterpendulumapproachisbetter.Returningtothephysicalpendulumasdescribedbyequation(1),themomentofinertiaofanobjectdependson the locationof theaxisof rotation (or thepivotpoint inourcase).TheParallelAxisTheoremsaysthatthemomentofinertia,I,aboutanaxisparalleltoonethatpassesthroughthecenterofmassofanobjectisgivenby

(4)whereIcmisthemomentofinertiaaboutthecenterofmassanddisthedistancebetweenthecenterofmassandtheaxisofrotation.Itisconvenienttowritethemomentofinertiaaboutthecenterofmassintermsofk,aquantityknownastheradiusofgyration:

(5)

Herekrepresentsthedistancefromtheaxisallthemasscouldbeconcentratedsuchthatthegivenmomentofinertiaisreproduced.Combiningthepreviousequations(1),(4),and(5)givesus

(6)

Now let us ask, are there two different pivot points we could use to produce the sameperiod?Whenwechangethepivotpoint,onlythevalueofdintheaboveequationchanges(kisdeterminedexclusivelybythemomentofinertiaaboutthecenterofmass). Wewillusesubscripts1and2torefertothetwodifferentpivotpoints:

I = Icm +Md2

Icm =Mk2

T = 2π k2 + d 2

gd

3

Alittlebitofalgebrareducesthistotheconstraint

(7)Wewillcallthiscommonvalueoftheperiodweobtainwhentheaboverelationissatisfied,T*.UsingtheequationforT1,wefind

Ofcourseourgoalinthisexperimentistofindg:

(8)

Thusourvalueofgwilldependon

• ourabilitytofindtwodifferentpivotsonthesamependulumthatproducethesameperiod,

• ourabilitytomeasurepreciselythedistancebetweenthesetwopivotpoints,• ourabilitytomeasurepreciselythependulumperiod.

Hadwe instead elected toworkwith the original equation for the period (1),wewouldhave been faced with the need to determine the position of the center of mass withprecisionaswellasthemomentofinertiaaboutthepivotpointwithprecision.Whilethesecalculations are easy for idealized situations, getting precise values for a real pendulum(with,forinstance,aholepunchedintoittoallowforapivotpoint)isnottrivial.

T1 = 2πk2 + d1

2

gd1T2 = 2π

k2 + d22

gd2

T1 = T2 ⇒k2 + d1

2

gd1=k2 + d2

2

gd2

k2 = d1d2

T*= 2π k2 + d12

gd1= 2π d1d2 + d1

2

gd1= 2π d1 + d2

g

g =4π 2 d1 + d2( )

T *( )2

4

Procedure:Part1:Theapparatusforthispartconsistsofastopwatch,arubberstopper,somethread,andatall support stand. If you are careful not to bump the apparatus and only make theadjustmentsaskedfor,andifyourpredecessorshavelefttheequipmentfullyaligned,youmaysaveyourselfsometimeinPart2,wherealignmentiscritical.1.Yourfirsttaskistomakea50cmlongsimplependulum.Ifthereisalreadyathreadoftherightlengthattachedtoastopper,skiptostep2.Cutapieceofthreadapproximately75cm long. Threadoneendof it throughthehole ina freerubberstopperandtie itoffbyloopingitbackabovethestopper.Tiealargeloopintheotherendofthethreadsothatthependulum length—asmeasured from the point atwhich the pendulumwill pivot to thecenterofmassofthestopper—isabout50cm.2.Loosenthesetscrewonthethinhorizontalrod(onthelefthandsideofthestand)andslide the rod forward so that the rubber tippedendextendspast the frontof the table’sedge.Suspendthependulumfromthelargestandandthenmeasurethedistancefromthepivotpointtothecenterofmassofthestopper,ascarefullyasyoucan,withameterstick.Estimate the uncertainty in thismeasured length, noting that the uncertainty associatedwiththeprocedureislikelygreaterthantheuncertaintyassociatedwiththeinstrument.Tohelpunderstandsourcesofuncertaintyandwaystoreducethem,youwillnextmeasuretheperiod in twodifferentways. Please followtheprocedureasdescribed,even thoughyoumayrecognizethefirstapproachascrude.3.ApproachA:Pullthependulumback10cmorso(thisdistanceisnotcritical)andreleaseit. Using a stopwatch, time one cycle of the oscillation. You can, for instance, start thestopwatchwhenthependulumisonthefarleftandstopitwhenitreturnstothatside.DONOTstartthestopwatchwhenthependulumisreleased.Youaremorelikelytointroducesystematic error that way (see the discussion in the Data Analysis section below). Atechniquethatisprobablymoreaccurateistotimethependulumasitpassesthroughthemidpointof theoscillation. Start thewatchwhen itpasses throughcoming from the left,and stop it when it comes back through the midpoint from the left again. Since thispendulumismovingfastthroughthecenter,youhaveashorter“triggering”eventthanifyouweretousetheendpointofthecycle(whenthependulummovesslowly)totimetheperiod. Makeandrecordatotalof10separate,singlecyclemeasurementsoftheperiod.Youdonotneedtorestarttheoscillationeachtime.4. Approach B: This process is the same as the previous one except you will time 10consecutive cycles. That is, start the stopwatchwhen the pendulumpasses through themidpointcomingfromtheleft. Thefirstcycleiscompletethenexttimeitcomesthroughfromtheleft,andsoon. After10cycleshavebeencompleted,stopthewatchandrecordthetotaltime.Repeatthis10times,foratotaloftenmeasurementsoftencycles.

5

5.Whenyouarefinished,removethependulumfromthethinrodandplaceitonthetableforthenextgroup.Thenforeyesafety,loosenthesetscrewonthethinrodandslidetherodbackawayfromtheedgeofthetable.Part2:Theapparatusforthispartconsistsofabrassrod(thependulum)withtwoholesdrilledinit(forthetwodifferentpivotpoints),aplasticslidethatcanbemovedupanddowntherod(allowing you to change the pendulum’s mass distribution), and a precision electronictimer.Iwillrefertotheholeneartheendoftherodasthe“endhole”andtheoneclosertothecenterasthe“midhole”.Therodhaslinesetchedinitat1cmintervals,exceptwherethelinewouldbeobscuredbyahole.Iwillrefertotheendoppositetotheendholeasthezeropointonthesliderpositionscale.Thediagrambelowshouldclarifythis:

1.Placetheslideratthe8cmmark.Thisisaprecisionexperiment.Donotplacetapeoranyother typeofmarkingson thebrasspendulum. Notethattherearenotmanyadjustmentstomakeinthisexperiment,buteachonemustbemadewithcareinordertoproduceaccurateresults.ALIGNMENT:Mountthependulumontherightsideofthestand,onthe“knife-edge”,whichisactuallyashortrodwithasquarecrosssection,orientedsothatonecornerofthesquareisatthetop.Startbymountingthependulumfromtheendhole. Donotmountthependulumall thewayatthetipoftheknifeedge;rathermoveittowardsthesupportrodbyacentimeterorso,becauselateryouwillneedsomeroomtoslide itout towards the tip.Make sure that the knife-edge is centered in the hole ascloselyaspossible.Thisturnsouttobeacriticalstepinordertogetreproducibleresults.You can use the providedmagnifying glass tomake sure the contact point between theknife-edgeandthetopoftheholeisinlinewiththeengraveddotsthatrundownthecenterofthependulum.Thiscenteringcheckneedstobeperformedeachtimeyouremountthependulum.2.Testtheswingforstabilitybysettingupaswingwitharoughly0.5cmamplitude(i.e.,atotalswingofabout1cm).Notethattheactualvalueoftheamplitudeisnotrelevant,butbeingrelativelycloseto0.5cmwillbenecessarytogetthephotogatetofunctionproperly.If the pendulum is mounted correctly, the oscillations should settle down into a planarmode.Iftherodneverstopstwistingorcircling,theneithertheknife-edgeisnotcenteredintheholeorthesupportmechanismisnotlevel(inthelattercase,youmayneedtoadjustthesetscrewsatthebaseofthetripod).

midhole endhole

01234….

6

3.Nextmount thephotogateso that the“U” isparallel to theplaneof the tableandsuchthatthependulumswingsthoughtheU,paralleltothesidesoftheUandperpendiculartothebaseoftheU.Letthependulumhangvertically,justbarelytouchingtheinneredgeofthefarsideoftheU.YouwillnoticethatontheinnersurfaceneareachtipoftheUthereisa small circular hole. These holes comprise the photogate. The photogate detectsinterruption of the light signal between these two points. Your goal is to line up thephotogatewiththepointmarkedAonthediagrambelow:

It seems tobeeasier tomake thisadjustmentwhen thependulum is restingagainstoneside of the U. Once you have the photogate correctly positioned, carefully slide thependulumalongtheknife-edgeawayfromthephotogatesothatwhenitswings,itwillnotrubthephotogate.Youwillneedabouta1cmgapbetweentherodandthephotogate,sothat later on in the experiment there is room for the slider at the photogate height. Asalways, make sure that the knife-edge remains centered in the pivot hole following theprocedureinstep1.Makeafinalchecktoensurethependulumswingsfreelyandwithoutanytwistingmotion.4.Thereshouldbeacablerunningfromthephotogatetothebackoftheelectronictimer,pluggedintotheΔTport.Turntheelectronictimeron.TheModeswitchshouldbeslidtotheright,sothattheLEDnexttothePeriodlabel(actuallyslightlyaboveit)islit.Oncethependulumissetinmotion,thereadoutwilldisplaytheperiod,updatingitaboutevery10s.Displace the pendulum about 0.5 cm to one side and release it. (Remember, 0.5 cm issmall—lookatametersticktorefreshyourmemory.) Ifthealignmentiscorrectandtheamplitude is correct, the pendulum will swing so that point B momentarily blocks thephotogate, but pointB does not continue onpast the photogate.When the swing settlesdown,youshouldgetaperiodverycloseto(i.e.,withinahundredthsofasecond),butnotexactlyequalto,onesecond.Troubleshootingtips:

• Thependulumcontinues towobble:Tryshifting toa slightlydifferentpositionontheknife-edge,checktomakesuretheknifeedgeiscenteredinthehole,makesureyouramplitudeisnottoolarge,orbemorecarefulhowyoureleasethependulum.

A B

midholeçClosedendofU

7

• Theperiodreadscloseto2sratherthan1s.Mostlikely,eithertheamplitudeistoo

smallortheelectriceyehasbeenlinedupwiththeholeinsteadofatpointA.

• The period is too small: Try using a smaller amplitude or waiting longer for thewobblestodieout.

• Oscillationsdieouttorapidly: thiscanhappenif there is toomuchtwisting inthe

initialswingofthependulum—tryamorecarefulrelease;ifthatdoesnotwork,tryusingadifferentspotontheknifeedge

• The timewillnot settledown:view the swing fromdifferent angles tomake sure

thereisnotwistingandtomakesurethependulumtipisnotmovinginanellipse;sometimes,though,youjustneedtobepatientandtryagainseveraltimes

• Thetimeisnotupdatingeverytenseconds(i.e.,aboutevery10cycles):oftenthisis

becausethephotogateisnotproperlyaligned(seestep3)If the period is right around one second and you have a nice stable swing, then thealignmentiscomplete.Remainingadjustmentsshouldbeeasyandminimal,providedyoudonotbumptheapparatus.5.Totakedata, letthependulumswinguntiltheperiodsettlesdown.Avoidtouchingthetableduringthistime.WatchthePeriodreadoutas itupdatesevery10seconds. If thereare four consecutive readings all within 0.0004 s of each other, then the oscillation issufficientlystable,soyoucanrecordtheaverageofthose4times(tothenearest0.0001s)andthevariation(i.e.,uncertainty).Youdonotneedtorecordallfourindividualtimes.Ihave found that this stability criterion is sufficient to generate reproducible results onalmostall ofmy trials.Patience is required—evenon thebestof swings Iusuallyhad towait for about a dozen period updates before I had reached this level of stability. Thisperiodyouhavejustmeasuredcorrespondstopivotingabouttheendholewiththeslideratthe8cmposition.6.Nowcarefullyremovethependulumfromtheknife-edgeandflipitover,remountingitinthemidhole.Donotrepositiontheslider.Checkforcenteringfollowingtheprocedureinstep1.Measuretheperiodasyoudidinstep5.Thiswillbetheperiodfortheslideratthe8cmpositionwiththepivotatthemidhole.7.Movetheslidertothe10cmmark,returnthependulumtotheknife-edge,andmeasuretheperiodsonceagainforbothmountingholes.Repeatforthesliderat12cm,14cm,and16cm,so thatyouwillhave fivepositionsaltogether,eachwithperiodmeasurementsatbothsuspensionholes.Thelasttwosliderpositionsareontheothersideofthemidhole.Rememberthatoneofthemarksontherodismissing,duetothepresenceofthemidhole(i.e.,thatholecountsasaline). Eachoftheperiodsshouldbedifferent,butallshouldbe

8

within0.01sof1.0000s.Inall,youshouldhaverecordedthevalueof10differentperiods.Becarefulwhenthesliderisatthe12cmmarkthatitdoesnotrubintothephotogate(thisaddednoisetomydatauntilIspottedtheproblem).8.When you have completed yourmeasurements, take the pendulum off the knife-edgeandplaceitonthetablewhereitisvisibleforthenextgroupbutwhereitwillnoteasilybeknockedoff. Makesuretoturnoff thetimer. Theonlydatayouneedtoacquire for thispartoftheexperimentaretheseperiods(twosetsofperiods,eachwithfivetimes).

9

DataAnalysis:Part1ApproachA:Determinetheaveragevalueofthetensingle-cyclemeasurements,andthendetermine themaximumvalue and theminimumvalue. Whilewith large data sets it isappropriatetocalculatetheuncertaintyusingthestandarddeviationorthesquarerootofthe variance, it is not clear that this approach is appropriate when a small number ofmeasurementsaremade.Formostexperimentsinthiscourse,wewillnotberepeatingourmeasurements often enough to make it worthwhile calculating the standard deviation(althoughyoushouldfeelfreetodosoifyouwish).Instead,wewilluseacruderapproachthatgetstheorderofmagnituderight.Theuncertainty,δT,inourmeasuredvalueofTisthesmallestnumbersuchthatT+δT incorporatesallof thedata. In the followingsimplecase,Trial 1 2 3 4 5Time(s) 1.10 1.13 1.04 1.25 1.08wewouldcalculateT=1.12+/-0.13ssincetheaveragetimeis1.12sandallofthedataliesin the interval of (0.99s, 1.25s). Therewill never be any case in this coursewhere youshouldusemorethan2significantdigitsinyouruncertainty,andusually1significantdigitis fine. Onceyouhavedeterminedthenumberofsignificantdigitsyouwillreport in theuncertainty, theprecisionof thecomputedvaluewouldberoundedoff tothesamelevel.Thatis,wecouldalsoreporttheaboveresultas1.1+/-0.1s(thefinalresultgiventothenearesttenthofasecondinthiscase).ApproachB:Inthisapproachyoumeasuredatimeintervalfor10fullcycles.DenotethisasT10. Divideeachof thevaluesofT10by10 toget the time foroneperiod. Thenanalyzethose times as you did above. You should notice that the uncertainty is smaller in thisapproachbyaboutafactorof10.Infact,ifyoureviewyourcalculations,youwillseehowtheuncertaintyeffectivelygetsreducedbyafactorof10.Noticethatwhenwemeasurethetimefortencyclestothenearesthundredthofasecondandthendividebytentogettheperiod,thisgivesusinformationabouttheperiodpotentiallytothenearestthousandthofasecond. Keep this thousandths digit in your calculations until you see if, based on youruncertainty,itissignificant.Why is there uncertainty in this data? The primary reason is that your timing intervaldependsonyourobservingthestartingpointsandstoppingpointscorrectlyandonyourability tophysically respond toyourobservations. We loosely refer to thisasa reactiontimeissue. Ifyourreactiontimewerealwaysthesame,thenitwoulddelaythestartandstopofthewatchinthesameway,resultinginnoerrorintroducedintothetimeinterval.Itis the fact that the reaction time varies some that introduces some uncertainty into thedata.

10

Systematicerrorvs.randomerror:Itisnothardtoimaginethepossibilitythatthereactiontimeforstartingthewatchandstoppingthewatchcouldbeslightlydifferent,dependingonhowtheexperimentisperformed.Ifthereactiontimeforstartingwerealwayslongerthanforstopping,thatwouldleadtoasystematicerror,inthiscaseallofyourmeasuredtimesbeing too short. This systematic error would not show up in the uncertainty analysisabove.Thatanalysisassumedthatthetimingerrorwasuniformlydistributedaroundtheactual time. A careful experimental procedure will minimize systematic error (seeProcedurePart1,Step3). Insomecases,theremayberemaining,identifiablesystematicerror. For instance, equation (3),which isused toanalyze thisdata, isbasedona smallangleapproximation. Theapproximationusedthereforeintroducessystematicerrorintocalculatedresults. Ideally,anexperimentalistcan identifyall sourcesofsystematicerroranddemonstratethatanyresultingcorrectionswouldbesmallerthanthoseintroducedbyrandomerrors,andhencetheydonotneedtobe includedintheanalysis. Thereality is,though,thatsystematicerroroftencreepsinthroughpoorexperimentaldesignandhencemayonlybespottedbyanoutsideobserver.Thebottomlineisthatyourgoalshouldbetoreduceanysourcesofsystematicerrortoalevellessthanthatoftherandomerror,sothatsystematicsourcesnolongerbecomerelevant.Sowewillassumethatthetiminguncertaintyisalltheproductofrandom,notsystematic,error.Nextweturntothelengthmeasurement(“d”inequation3).Therearetwoissuestoconsiderhere.First,howpreciselycanyoumeasurewithameterstick?Thesmallestunitmarked is1mm. Somewouldargue thatyoucan thereforeonlymeasure to thenearestmm,otherswillarguethatyoucanvisuallydetecttothenearest0.5mm.Myperspectiveisthateitherpositionisjustifiableanddependsonthepersonmakingthemeasurement.Thisillustratesthepointthatthereisaroleforscientificjudgmentinperformingandanalyzingexperiments.Wearemorethanmachinesfollowingrules.Thesecondissueindeterminingtheuncertaintyindisourabilitytoaccuratelylocatethepivot point and the center of mass of the stopper. The latter is generally based on aneducatedguess.Itseemslikelythattheuncertaintyinitspositionexceedstheuncertaintyintroduced due to the measuring limit of the meter stick. I would estimate thiscontributiontotheuncertaintytobe3-5mm.Insum,itwouldbereasonabletoattributeanuncertaintyof5mmtothelengthmeasurement.Notice that in both the time and distance measurements, the uncertainty for the datasignificantlyexceeded themeasurement limitationof thedevice. Thatwillbea commonoccurrenceinthiscourse.

11

CarryinguncertaintyanalysisthroughinyourcalculationsA word of caution: make sure to retain enough significant digits in your intermediatecalculationstosupportyourfinalresult.Forinstance,inPart2ofthisexperiment,youwillbecalculatinggtothreeorfoursignificantdigits,soyourintermediatecalculationsmustbegoodtoatleastfourdigits.Thismeans,forinstance,youcannotusethevalueof3.14forpi.Returningtoequation(3):

g = 4π 2 dT 2 ,

wehaveseenthereisuncertaintyinvaluesforbothdandT.Oncetheaverage(best)valuesandtheuncertaintieshavebeenestablished(asabove),then

gbest = 4π2 dbestTbest( )2

gmin = 4π2 dbest −δdTbest +δT( )2

gmax = 4π2 dbest +δdTbest −δT( )2

Noticethattogetthemaximumvalueofg,wemaximizethenumeratorandminimizethedenominator. Likewise, to get the minimum value of g possible, we minimize thenumeratorandmaximizethedenominator.Yourresultisthenreportedasg+/-δg,wheregis thebestvalueandδg is taken just largeenoughto incorporateboth theminimumandmaximumvaluesofg.Thiscrudeapproachtocalculatinguncertaintiesusuallyoverstatestheuncertainty and thus canbe viewed as a cautious estimate of the actual uncertainty.Makesuretofollowthisanalysisthroughforbothapproachesinpart1.Here isanalternateapproach tocalculating theuncertainty thatyou learned inPHY223andyouarewelcometousehere.Forafunction,f,oftheform

,wherexandyhaveuncertaintiesδxandδyrespectively,theuncertaintyinfisgivenby

. (9)

Applyingthistoourequationforg,

δg = g δdd

⎛

⎝⎜

⎞

⎠⎟2

+ −2δTT

⎛

⎝⎜

⎞

⎠⎟2⎡

⎣⎢⎢

⎤

⎦⎥⎥

12

= g δdd

⎛

⎝⎜

⎞

⎠⎟2

+ 2δTT

⎛

⎝⎜

⎞

⎠⎟2⎡

⎣⎢⎢

⎤

⎦⎥⎥

12

12

You should be careful in the future when using this technique since it only applies tofunctionsoftheform ; not all equations in this course will have that simple form.EVERY MEASURED VALUE IN THIS COURSE SHOULD HAVE AN UNCERTAINTYASSOCIATEDWITH IT. EVERY CALCULATEDVALUE IN THIS COURSE SHOULDHAVEACALCULATEDUNCERTAINTY.Itissurprisinghowmanystudentsseemtoforgetthisfactbetweenthefirstandsecondexperiments,resultinginthelossofmanypoints.Part2Everytimeyoumovedtheslider,youchangedthemassdistributionandeffectivelycreatedanewpendulum. Thequestionis,whichofthosependulumshadidenticalperiodswhenpivotedatbothpoints?Whenyouidentifythatpendulum,thenyoucanapplyequation(8)toit.Using Excel or a similar program, plot the period (vertical axis) as a function of sliderposition(horizontalaxis)foreachofthetwopivotpoints,onthesamegraph. Haveeachdatasetfittoapolynomialofdegree2anddisplaytheequationonyourplot. Makesureyouraxesareappropriately labeledandthat there isa legendthatmakessense(i.e.,onethatconveysmoreinformationthanthedefault“Series1”).Where those two linescross,youhave foundapendulumthatproduces thesameperiodwith twodifferenthangingpoints. Determine theperiodat thiscrossingpoint,eitherbyreading it off the graph directly or by using the fitting equations to seek the commonsolution.Sincethisfittingprocedureaccountsfortrendsinseveraldatapoints,itproducesamore accurate value thanmerely looking for the slider position in your raw data thatproduces nearly identical periods. The final step in the calculation is straightforward:Returningto

€

g = 4π 2 d1 + d2T *( )2

,

let’s set D=d1+d2. There is uncertainty in values for both D and T*. The value of, anduncertainty in, D has been given by the manufacturer: 0.2481 +/- 0.00005 m. In thisexperiment,theuncertaintyinT*isabithardtodeterminesinceitisextractedfromyourplot. You can use your own best judgment on this, but it will probably be somewherearound0.0001s,theprecisionlimitoftheelectronictimer.Oncetheuncertaintyhasbeenestablished,thenyoucanperformtheremaininganalysisoftheuncertaintyasdescribedinPart1.ThemuchsmalleruncertaintyingcalculatedusingtheKaterpendulumwillallowyoutodisplaymoresignificantdigitsinyourreportedresult.

13

A word on significant digits Recallthatthenumberofsignificantdigitsisthenumberofdigitsshownwhenthenumberiswritteninscientificnotation.Thatis,0.035=3.5x10-2hastwosignificantdigits.Notethatinthiscontextthat0.35≠0.350sincethelatterexpressioncontainsmoresignificantdigitsandrepresentsaclaimtoknowthevaluetothenearestthousandth,notmerelythenearesthundredth.Ingeneral,thenumberofsignificantdigitsyoushowwilldependonthelevelofuncertaintyinyourdata.Therearetwoprinciplestoremember:1.Weusuallyshowoneormaybetwosignificantdigitsintheuncertaintycalculation,butinthiscoursetherewillneverbeanypointinshowingmorethantwodigits. Thatis, ifyouareestimatingthenumberofpenniesinalargejar,itmaybereasonabletosayyourcountisgoodtothenearest100pennies,orperhapsthatyourcountisgoodto±110pennies,butit is unreasonable to write ±113 pennies because if you know your uncertainty to thenearestpenny,thenyouprobablycouldhavemadeamoreaccuratecountinthefirstplace!2.Make the level of precision in your best value consistentwith the uncertainty. If youcalculateaforcewhoseuncertaintyis0.3N,thenyoushouldshowtheforcetothenearesttenthofaNewton,sincethatiswheretheuncertaintylies.Thefollowingaresomesimpleexamples:Correct Incorrect23±4 23.1±423.2±1.2 23±1.23.50±0.03 3.5±0.033.55±0.12 3.552±0.123These rules imply that you will need to calculate your uncertainty first before youdeterminehowmany significant digits in your best value to use in your tables. If usingExceltosetupyourtables,rememberthat ithasatendencytotruncatetrailingzeros. Ifthesezeroesaresignificantdigits,youneedtoforceExceltoretainthembyusingaFormat:Numberprocess.

14

WhatyouwillturninforthisexperimentAllyouneedtosubmitforthislabisapartialResultssection:Youshouldhavetableswithyourrawdata,aproperlylabeledplotshowingtheintersectionbetweenthetwodifferentcurves, and your calculations of g from Part 1 and Part 2, including the uncertaintycalculation, as outlined above. I need to see the details of the calculations (i.e., whatnumbersyoupluggedinwhere),not justtheanswers. Irefertothisasa“partialResultssection”sinceinaregularreport,theResultssectionwillcontaintextaswellasdataandcalculations.You’lldothatforyournextexperiment.Addtotheendofyourcalculationaparagraphdiscussingthehighlightsoftheuncertaintyanalysis. In particular, discuss the role of the instrument and of the operator of theinstrumentinproducingbothsystematicandrandomerror,thewayinwhichdataanalysiscanintroducesystematicerror(evenifyoudonotmakemistakeswiththeequations!)andtowhatextenttheseeffectsareaccountedforinyourvaluesofg.Basedonyournumericalresults(notonyourprocedureandpriorexpectations)whichvalueofgshouldyouuseinyourfuturecalculations,1A,1B,or2?Finally,don’tforgettoindicateyourlabpartner.Infuturereports,thisinformationwillgoontheTitle/Abstractpage.