Labatorial1:DensityofGranularSolids1Preparedby:FrancoLaBraca,ConcordiaUniversity

Pre-Reading:

• PhysicsforScientistsandEngineersbySerwayandJewett(10thed.),Section:o 1.4–ConversionofUnits

o 1.6–SignificantFigures

• Thefirsttwosectionsofserc.carleton.edu/mathyouneed/density/index.html

• Sections1and2ofhttp://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm

Equipment:Smallbeaker,thermometer,triplebeambalance,pycnometer,glassbeads,strainer,

acetone,calculator

LearningGoals:

• Understandinghowsolidsimmersedinaliquiddisplacetheliquid

• Understandinghowmeasurementerrorispropagatedthroughcalculations

Pycnometers:Apycnometerisasmallflaskwithaglassstopper.Acapillary(i.e.thin)opening,

whichrunsalongthelengthofthestopper,makesitpossibletofillthepycnometercompletely–

thatis,withoutleavingabubbleofairintheflask.(Note:Afteruse,apycnometercanbedried

byrinsingitwithsomeacetoneandthendumpingtheliquidinawastebasket.)

Activity1:SomeConceptualWarm-Ups(15-25min.)

Question1:Abeakerisfilledwith100 !"ofwater,whosedensityis1 !/!".Ifthewaterandthebeakertogetherhaveamassof0.250 !",whatisthemassofthebeaker?

Question2:Anirregularchunkofmetalhasadensityof7800 !"/!!

.A15 !chunkofthemetalisadded

intoagraduatedcylinderfilledwithenoughliquidtosubmergethechunk.Ifthefinalvolumeof

thecylinderreads40.2 !",whatwastheinitialvolume?(Recallthat1 !!! = 1 !".)

1Theexperimentperformedinthislabatorialhasbeenadaptedfromthe2018-2019labmanualforthecourse“PHYS

224:IntroductiontoExperimentalMechanics”atConcordiaUniversity.

Checkpoint1:Beforemovingontothenextpart,haveyourinstructorchecktheresultsyouobtainedsofar.

Activity2:ThreeBlindMice(25-35min.)

Problem:Ateamofscientistswishtomeasureoutaprecisequantityofaliquidthatthey

producedinthelab,aswellasquantifytheuncertaintiesintheirmeasurement.(Theamount

itselfisimportant,buttheywanttobeabletousethesamevolumeconsistentlyforfuture

experimentsandknowhowprecisetheirprocedureis.)However,theymiraculouslycrashinto

eachother,shatteringalloftheirreadingglassesintheprocess.Theycanstillreadthereadings

fromtheirdigitalthermometerandtriple-beambalance,butthelightmarkingsonagraduated

cylinderarenolongerlegibletothem.Stilldeterminedtotakethemeasurement,however,they

decidetousetheremainingtoolsattheirdisposal:apycnometer,adigitalthermometer,anda

triple-beambalance.

Question3:Duetovariouseffectsincludingthermalexpansion,thedensityofwaterchangeswith

temperature.Therefore,weneedtoknowthetemperatureofthewaterforthevolume

measurementstobeaccurate.(Thermalexpansioncanalsocausethepycnometeritselfto

deform,affectingitsvolume,butcanbeavoidedbyhandlingthepycnometerbyitsneckwith

oneortwolayersofpapertowel.)Fillthelargebeakerhalfwaywithwaterfromthesink.Leave

thethermometerinthebeakerforafewminutesandthentakeareadingofitstemperature.

Whatisthedensityofthewater?Circleyouranswerandconvertittounitsof!\!".(Whileyou

waitforthethermometer,feelfreetobeginworkingonthenextquestion.)

Table1–Densityofwater(in!"/!!,allwithuncertainty± 0.02)atvarioustemperatures.

!(°!) 18 19 20 21 22 23 24 25 26. 0 998.60 998.40 998.20 998.00 997.80 997.60 997.40 997.20 997.00. 1 998.58 998.38 998.18 997.98 997.78 997.58 997.38 997.18 996.98. 2 998.56 998.36 998.16 997.96 997.76 997.56 997.36 997.16 996.96. 3 998.54 998.34 998.14 997.94 997.74 997.54 997.34 997.14 996.94. 4 998.52 998.32 998.12 997.92 997.72 997.52 997.32 997.12 996.92. 5 998.50 998.30 998.10 997.90 997.70 997.50 997.30 997.10 996.90. 6 998.48 998.28 998.08 997.88 997.68 997.48 997.28 997.08 996.88. 7 998.46 998.26 998.06 997.86 997.66 997.46 997.26 997.06 996.86. 8 998.44 998.24 998.04 997.84 997.64 997.44 997.24 997.04 996.84. 9 998.42 998.22 998.02 997.82 997.62 997.42 997.22 997.02 996.82

ReviewofMeasurements,Uncertainties,andErrorPropagation:

Thesmallestdivisionofscalemarkingsonaninstrumentiscalledtheabsoluteerror.Whenwe

wanttoreportameasurementsuchasthewidthofapaper,werepresentitas:! ± ∆!,

whereWisthemeasurementvalueandΔWisthemeasurementuncertaintyorabsoluteerror.

Thismeansthatthetruewidthwillprobablyliesomewhereintheinterval(! − ∆!,! +∆!).Ifyoutaketwodifferentwidthmeasurementswiththeirownabsoluteerrorsandtheir

intervalsoverlap,thenwesaythatthetwomeasurementsagreewithinuncertainty.

IfyouhavemeasuredvaluesforthequantitiesX,Y,andZwithuncertaintiesΔX,ΔY,andΔZ,whenaddingorsubtractingtwomeasurementsthathaveanerror,addtheabsoluteerrorstogiveyoutheabsoluteerrorinthesumordifference.

If! = ! + !then!" = !" + !"

If! = ! − !then!" = !" + !"(1)

Therelativeerrorofameasurementisdefinedas:

!"#$%&'" !""#" = !"#$%&'( !""#"!"#$%&"' !"#$%

Whenmultiplyingordividingtwomeasurementsthathaveanerror,addtherelativeerrorstogiveyoutherelativeerrorintheproductorquotient.

If! = ! · !then∆!! = ∆!! + ∆!

!

If! = !!then

∆!! = ∆!

! + ∆!! (2)

Therefore,theabsoluteerrorinthiscasewillbegivenby

∆! = ! ∆!! + ∆!

! (3)

Question4:Returningbacktotheexperimentandkeepinginmindthatwealwaysneedtoallowairbubbles

torisetothetopofapycnometerafteritisfilledwithliquid,letusdevisetheprocedurethe

blindedscientistswouldhavetofollowinordertomeasurethevolumeofthepycnometer,and

useittomeasureyourpycnometer’svolume.

a. Measurethemassoftheemptypycnometerwithitsstopperandthepycnometerfilled

withwater(closedwiththestopper).

!!=__________________±0.1 !!!"=__________________± 0.1 !

b. Calculatethemassofthewaterinthepycnometer,usingEquation1tocalculateits

uncertainty.

!!=__________________±__________!

c. Calculatethevolumeofthepycnometer,usingEquation3tocalculateitsuncertainty.

(AsktheTAforhowtoquotethisuncertaintyproperly.)

!!=__________________±__________!"

d. Calculatethepercentagedifferencebetweenyourmeasuredvaluefor!!andthecalibratedvalueof50 !"at20 ℃.

e. Howdoesusingpercentagedifferencedifferfromusinguncertaintyinquantifying

measurementerrorandinterpretingtheprecisionofresults?

Checkpoint2:Beforemovingontothenextpart,haveyourinstructorchecktheresultsyouobtainedsofar.

Activity3:ExploringWorldsUnknown(60-70min.)

Problem:InanongoingspacesearchbyNASA,apotentiallyhabitableplanetisdiscoveredbyan

extra-solarprobe.Oneofthecrucialconditionsforthehabitabilityofaplanetisthesuitable

compositionofitssoil(i.e.containingcarbon-basedcompoundssimilartotheEarth’ssoil).To

verifythis,theprobelaunchesalandertothesurfaceofthisrockyplanetinordertotakesome

measurements.Luckily,ithappenstofindsomenearbydrysoilthatitcaneasilycollect.

However,thesoil,beingverybrittle,crumblesuponbeinglifted,andsoonlythegranularform

ofthesolidcouldbecollected.Usingstandardizedprocedures,itbeginsmeasuringthedensity

ofthesoiltoverifywhethernotthenewplanetmeetshabitabilityconditions.Youwillnowput

yourselfintheshoesofthosedesigningtheprocedures,testingthemoutfirstinyourlab.

Question5:

a. Supposewepour20 !"ofwaterat20 ℃intoagraduatedcylinder.Wetheninserta

1 !!!cubeofleadwithdensity11.34 !/!!!

intothecylinder.Whatwillbethefinal

volumemeasuredinthegraduatedcylinder?

b. Nowsupposewefillthiscylindertomaximumcapacityandtheninsertthesamecube,

causingwatertooverflow.Comparethiscylinderwithanothercylinderfilledwithonly

water.Whatisthedifferenceinmassbetweenthetwocylinders?

Question6:Wenowwanttomeasurethedensityofthegranularsolid(which,forus,ismadeofglass).

a. Placeasmallbeaker(dryandempty)onthetriplebeambalance.Recorditsmass.

!!=_____________________±_______________!

b. Carefullyaddgranularsolidtothisbeakeruntilthereis50 !ofthesolid.(Ifthesolidsarenotdried,placetheminthestrainerandslowlypoursomeacetoneonthem.You

canshakethestraineraroundslightlywhiletheyevaporatetospeedupthedrying

process.)

c. Pouroutabouthalfofthewaterfromthepycnometerintothesink.Thenaddthe

granularsolidtothepycnometer,topitoffwithwater,andinsertthestopper.Whatis

thetotalmassofthepycnometerwithitscontent(i.e.waterandsolid)?

!!"#=_____________________±_______________!

d. Thedifferencebetweenthesolid’smass!!andthedisplacedwater’smass!!"#$canbeexpressedintermsofourmeasurementsas!! −!!"#$ = !!"# −!!".Thenwhatisthemassofthedisplacedwater?

!!"#$=_____________________±_______________!

e. Whatisthevolumeofthegranularsolidinthepycnometer?(Hint:Thewatergets

displacedbythesolid.)

!!=______________________±_______________!"

f. Whatisthedensityofthesolid?

!!=______________________±_______________!"/!!

g. Howdoesyouranswercomparetothetabulatedvalueof2400 !"/!!?Doesitagree

withinuncertainty?Whydoyouthinkthatis?

h. Givenwhatwehavelearnedabouthowerrorspropagatethroughcalculations,howcan

weimprovethisexperimenttoreducetheuncertaintiesinourresults?

Checkpoint3:Pleasecleanyourareaandthenhaveyourinstructorcheckyourworkbeforeleavingthelab.Component Explanations Points MarkWorksheet • Ifyoufinishallcheckpoints,youwillget4points. 4

Group • Allstudentsmustbeengagedinthelabactivity.

• Allstudentsmustwork,discuss,andsharetheir

informationinthelab.

• InteractionwithgroupmembersandTAismandatory.

• Allstudentsmustobtainanswerstothequestions

thatarethesameastheothergroupmembers.

3

Individual • Allappropriatedatamustbecollected.

• Datamustbewellorganizedandneatlydisplayed,

includinggraphs.

• Theresultsofcalculationsmustbepresentedwith

appropriateunits.

• Relatedphysicsconceptsmustbestatedcorrectly.

3

Pleasenotethat:• Notproperlycleaningtheworktableornotputtingawayequipmentthatwastakenout

willresultina1-pointdeductionfromthe“group”componentofallmembers’grades.

• AsofLabatorial2onward,notbringingyourlabatorialmanual(inwhichcase,a

separatelyprintedworksheetwillbeprovided)orpre-readingsummarytothelabwill

resultina1-pointdeductionfromthe“individual”componentofyourgrade.

• Progressingasagroupiscriticaltothesuccessofthelabatorial,andsobeingmorethan

15minuteslatewillresultina1-pointdeductionfromthe“individual”componentof

yourgrade.Beingmorethan20minuteslatemeansyoucannotperformthelabatorial

andyouwillreceivea0.