activity 1: some conceptual warm-ups (15-25 min.) · 2019. 5. 15. · pycnometers: a pycnometer is...
TRANSCRIPT
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.