北京邮电大学英文版物理实验

ReproducedfromtheUnitGuidesforSEP101andSEP115

physicslabreport

DeterminationoftheGravitationConstantg

byMeansofaSimplePendulum

Aim

Thisexperimentwasperformedtodeterminethegravitationalaccelerationofobjectsclosetothesurfaceoftheearth,byobservingthemotionofasimplependulum.

Introduction

Asimplependulum,figure1,displacedthroughasmallangle,willoscillatebackandforthaboutitsequilibriumpositionwithperiodT.Tisthetimethependulumtakestomakeonecompleteback-and-forthmotion.ThebobishungfromarigidsupportonastringoflengthL.

Figure1:Thesimplependulum.

Foroscillationswheretheangleissmall,theperiodTisrelatedtothelengthLofthestringandthegravitationconstantgby

Squaringbothsidesofthisequationyields

Ifonemeasurestheperiodofapendulumasafunctionofthelengthofthestring,thenaplotofT2asafunctionofLwillyieldastraightlinewithagradientG;and

ExperimentalMethod

Asimplependulumwasproducedfromalengthofstringandafishingsinker.Thesinkerwasdisplacedthroughananglelessthan10degreesandreleased.Forfivedifferentlengthsofstringbetween23and100cm,theperiodofoscillationwasmeasured.Ineachmeasurement,thependulumwasallowedtooscillate50times.Thetotaltimefor50oscillationswasmeasuredwithastopwatch,andtheperiodwascalculatedbydividingthetotaltimefor50oscillationsby50.Thestopwatchmeasurestimeto0.01seconds.However,itisestimatedthatthetotalreactiontimeoftheexperimenterwas0.2seconds.Thustheuncertaintyofanyoriginalmeasurementoftimewastakentobe0.2seconds.Withametrerule,thelengthofthestringwasmeasuredtothenearestmillimetre.Thelengthofthestringwasmeasuredfromthesupporttothecentreofthebob.

ResultsandCalculations

Table1showstheresultsoftheexperiment,andtheplotofT2versusLisgiveninfigure2.

String

Length(m)

Timefor50

oscillations(s)

TimeTforone

oscillation(s)

Period

squared(s2)

0.975

97.3

1.95

3.80

0.812

88.7

1.77

3.13

0.597

75.6

1.51

2.28

0.411

61.9

1.24

1.54

0.235

45.9

0.918

0.843

Table1:Pendulumdata.

Theuncertaintyineachmeasurementoflengthis0.001m.Theuncertaintyineachmeasurementoftimefor50oscillations,T1is0.2

s.Thustheuncertaintyinanymeasurementoftheperiodis

Theperiodissquaredpriortoplotting.Therelativeuncertaintyintheperiodsquaredistwicetherelativeuncertaintyintheperiod:

SolvingforT,

ThustheuncertaintyinT2isproportionaltoT.ThelargestdatapointisforT2

=

3.80

s2.Theuncertaintyinthisdatumis.ThemaximumuncertaintyinT2isthus0.4

%.TheerrorbarsassociatedwithT2aretoosmalltoplotonagraph.Similarly,theerrorbarsassociatedwithLarealsotoosmalltoplotonthegraph.

Figure2:PlotofT2asafunctionofL.

Astraightlinefitsthedatawell.Thegradientofthelineofbestfitcanbecalculatedfrom

and

Toworkouttheuncertaintyinthegradient,analternativelineofbestfitwasselected,anditsgradientisgivenby

TheuncertaintyinGisthedifferenceofthese2gradients:

ThepercentuncertaintyinG,andthusingis

Thustheexperimentallydeterminedvalueofthegravitationconstantisg

=

9.99

m/s2

2%.

Discussion

Theacceptedvalue

ofgis9.81m/s2.Theaccuracyoftheresultsis

Theexperimentallydeterminedvalueofgagreeswiththeacceptedvaluetowithintheexperimentaluncertainty.Thusthisexperimentwasasuccessfulandaccuratedeterminationofg,evenwiththesimpleapparatus.

Thebobusedinthisexperimentisintheshapeofatriangularwedge.Thecentreofmasswasestimated(guessed)forthebob,andthelengthofthestringwasconsistentlymeasuredtothatpoint.Theaccuracyofthelengthofthestringdidnotmatterinthisexperimentsolongasthelengthwasalwaysmeasuredinthesameway.ThegravitationconstantwasdeterminedfromthechangeinT2asLchanged.Thischangeisstatic,regardlessofwheretheendofthestringwastakentobe.Anyerrorsinestimatingwherethestringendedwillmerelyshifttheplotupordown.Itwillnotaffectthegradient.AninterestingfurtherexperimentwouldbetocollectmoredatapointsforsmallL,andseeiftheplotteddatapassthroughtheorigin.

Anotherinterestinginvestigationwouldbetoperformtheexperimentforlargeanglesofdisplacement.Thetheoryassumesthatthisangleissmall.Furtherexperimentscouldinvestigatehowthedeterminationofginthistechniqueisaffectedbyanincreasingangleofdisplacement.

Conclusion

Bymeansofasimplependulum,thevalueofthegravitationconstantwasdeterminedtobeg

=

9.99

m/s2

2%.Thisagreedwiththeacceptedvalue,9.81m/s2,towithintheexperimentalunc