introductory physics - home | 13.4 pressure in a tube with two liquids a u-shaped tube is closed at...

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PHYS101

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PHYS101

IntroductoryPhysics

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DouglasAdamsHitchhiker’sGuidetotheGalaxy

You’realreadyknowphysics!Youjustdon’tnecessarilyknowtheterminologyandlanguageweuse!!!

PhysicsofNASCARPhysicsofAngerBirds

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Inclass!!

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Thislecturewillhelpyouunderstand:FluidsandDensityPressureMeasuringandUsingPressureBuoyancy

PHYS101

Section13.1FluidsandDensity

©2015PearsonEducation,Inc.

FluidsandDensityAfluid isasubstancethatflows.

Liquidsandgasesarefluids.

Gasesarecompressible;thevolumeofagasiseasilyincreasedordecreased.

Liquidsarenearlyincompressible;themoleculesarepackedclosely,yettheycanmovearound.


DensityThemassdensity istheratio ofmasstovolume:

TheSIunitsofmassdensityarekg/m3.

Gasolinehasamassdensityof680kg/m3,meaningthereare680kgofgasolineforeach1cubicmeteroftheliquid.


Density


Example13.1WeighingtheairinalivingroomWhatisthemassofairinalivingroomwithdimensions4.0m´ 6.0m´ 2.5m?PREPARE Table13.1givesairdensityatatemperatureof20°C,whichisaboutroomtemperature.SOLVE Theroom’svolumeis

V=(4.0m)´ (6.0m)´ (2.5m)=60m3

Themassoftheairism=ρV=(1.20kg/m3)(60m3)=72kg

ASSESS Thisisperhapsmoremass—aboutthatofanadultperson—thanyoumighthaveexpectedfromasubstancethathardlyseemstobethere.Forcomparison,aswimmingpoolthissizewouldcontain60,000kgofwater.


Section13.2Pressure


PressureLiquidsexertforcesonthewallsoftheircontainers.

Thepressureistheratiooftheforcetotheareaonwhichtheforceisexerted:

Thefluid’spressurepushesonall partsofthefluiditself,forcingthefluidoutofacontainerwithholes.


PressureWecanmeasurethepressureinaliquidwithasimpledevice.Wefindthatpressureiseverywhereinthefluid;differentpartsofafluidarepushingagainsteachother.


PressureinLiquidsTheforceofgravity(theweightoftheliquid)isresponsibleforthepressureintheliquid.

Thehorizontalforcescanceleachotherout.

Theverticalforcesbalance:

pA =p0A +mg


PressureinLiquidsTheliquidisacylinderofcross-sectionareaA andheightd.Themassism =ρAd. Thepressureatdepthd is

Becauseweassumedthatthefluidisatrest,thispressureisthehydrostaticpressure.


PressureinLiquidsAconnectedliquidinhydrostaticequilibriumrisestothesameheightinallopenregionsofthecontainer.

Inhydrostaticequilibrium,thepressureisthesameatallpointsonahorizontallinethroughaconnectedliquidofasinglekind.


QuickCheck 13.1

Anicebergfloatsinashallowsea.Whatcanyousayaboutthepressuresatpoints1and2?

A. p1 >p2B. p1 =p2C.p1 <p2


QuickCheck 13.1

Anicebergfloatsinashallowsea.Whatcanyousayaboutthepressuresatpoints1and2?

A. p1 >p2B. p1 =p2C.p1 <p2


Hydrostatic pressure is the same at all points on a horizontal line through a connected fluid.

QuickCheck 13.2

Whatcanyousayaboutthepressuresatpoints1and2?

A. p1 >p2B. p1 <p2C.p3 =p1


QuickCheck 13.2

Whatcanyousayaboutthepressuresatpoints1and2?

A. p1 >p2B. p1 <p2C.p3 =p1


Hydrostaticpressureisthesameatallpointsonahorizontallinethroughaconnectedfluid.

Example13.3PressureinaclosedtubeWaterfillsthetubeshowninFIGURE13.7.Whatisthepressureatthetopoftheclosedtube?

PREPARE Thisisaliquidinhydrostaticequilibrium.Theclosedtubeisnotanopenregionofthecontainer,sothewatercannotrisetoanequalheight.Nevertheless,thepressureisstillthesameatallpointsonahorizontalline.Inparticular,thepressureatthetopoftheclosedtubeequalsthepressureintheopentubeattheheightofthedashedline.Assumep0 =1atm.


Example13.3Pressureinaclosedtube(cont.)SOLVE Apoint40cmabovethebottomoftheopentubeisatadepthof60cm.Thepressureatthisdepthis

p=p0 +ρgd

=(1.01´ 105 Pa)+(1000kg/m3)(9.80m/s2)(0.60m)

=1.07´ 105 Pa=1.06atm

ASSESS Thewatercolumnthatcreatesthispressureisnotverytall,soitmakessensethatthepressureisonlyalittlehigherthanatmosphericpressure.


QuickCheck 13.3

Whatcanyousayaboutthepressuresatpoints1,2,and3?

A. p1 =p2 =p3B. p1 =p2 >p3C. p3 >p1 =p2D. p3 >p1 >p2E. p2 =p3 >p1


QuickCheck 13.3

Whatcanyousayaboutthepressuresatpoints1,2,and3?

A. p1 =p2 =p3B. p1 =p2 >p3C. p3 >p1 =p2D. p3 >p1 >p2E. p2 =p3 >p1


Hydrostatic pressure is the same at all points on a horizontal line through a connected fluid, and pressure increases with depth.

PressureinLiquids


AtmosphericPressureGasiscompressible,sotheairintheatmospherebecomeslessdensewithincreasingaltitude.

99%oftheairinouratmosphereisbelow30km.

Atmosphericpressurevarieswithaltitudeandwithchangesintheweather.


Section13.3MeasuringandUsingPressure


MeasuringandUsingPressure


Text:p.404

ManometersandBarometersAmanometermeasuresthegaspressure.

Thetubeisfilledwithliquid(oftenmercury).Sincepressuresonahorizontallineareequal,p1 isthegaspressure,p2isthehydrostaticpressureatdepthd=h.

Equatingthetwopressuresgives

pgas =1atm +ρgh


ManometersandBarometersAbarometermeasurestheatmosphericpressurepatmos.

Aglasstubeisplacedinabeakerofthesameliquid.Some,butnotallliquidleavesthetube.

p2 isthepressureduetotheweightoftheliquidinthetubeandp1 =patmos.

Equatingthetwopressuresgives

patmos =ρgh


Example13.4PressureinatubewithtwoliquidsAU-shapedtubeisclosedatoneend;theotherendisopentotheatmosphere.Waterfillsthesideofthetubethatincludestheclosedend,whileoil,floatingonthewater,fillsthesideofthetubeopentotheatmosphere.Thetwoliquidsdonotmix.Theheightoftheoilabovethepointwherethetwoliquidstouchis75cm,whiletheheightoftheclosedendofthetubeabovethispointis25cm.Whatisthegaugepressureattheclosedend?


Example13.4Pressureinatubewithtwoliquids(cont.)PREPARE FollowingthestepsinTacticsBox13.1,westartbydrawingthepictureshowninFIGURE13.12.Weknowthatthepressureattheopensurfaceoftheoilisp0 =1atm.Pressuresp1 andp2 arethesamebecausetheyareonahorizontallinethatconnectstwopointsinthesamefluid.

(ThepressureatpointAisnotequaltop3,eventhoughpointAandtheclosedendareonthesamehorizontalline,becausethetwopointsareindifferentfluids.)


Example13.4Pressureinatubewithtwoliquids(cont.)Wecanapplythehydrostaticpressureequationtwice:oncetofindthepressurep1 byitsknowndepthbelowtheopenendatpressurep0,andagaintofindthepressurep3 attheclosedendonceweknowp2 adistancedbelowit.We’llneedthedensitiesofwaterandoil,whicharefoundinTable13.1tobeρw =1000kg/m3 andρo =900kg/m3.


Example13.4Pressureinatubewithtwoliquids(cont.)SOLVE Thepressureatpoint1,75cmbelowtheopenend,is

p1 =p0 +ρogh

=1atm +(900kg/m3)(9.8m/s2)(0.75m)

=1atm +6620Pa

(Wewillkeepp0 =1atm separateinthisresultbecausewe’lleventuallyneedtosubtractexactly1atm tocalculatethegaugepressure.)


Example13.4Pressureinatubewithtwoliquids(cont.)Wecanalsousethehydrostaticpressureequationtofind

p2 =p3 +ρwgd

=p3 +(1000kg/m3)(9.8m/s2)(0.25m)

=p3 +2450Pa


Example13.4Pressureinatubewithtwoliquids(cont.)Butweknowthatp2 =p1,so

p3 =p2 - 2450Pa=p1 - 2450Pa

=1atm +6620Pa- 2450Pa

=1atm +4200Pa

Thegaugepressureatpoint3,theclosedendofthetube,isp3 - 1atm or4200Pa.


Example13.4Pressureinatubewithtwoliquids(cont.)

ASSESS Theoil’sopensurfaceis50cmhigherthanthewater’sclosedsurface.Theirdensitiesarenottoodifferent,soweexpectapressuredifferenceofroughly ρg(0.50m)=5000Pa.Thisisnottoofarfromouranswer,givingusconfidencethatit’scorrect.


PressureUnits


BloodPressureBloodpressureismeasuredbypressurizingacuffaroundapatient’sarm.Thecuffsqueezesthearteryshut.Whenthecuffpressuredropsbelowthesystolic(max)bloodpressure,thearterypushesbloodthroughinpulses,whichcanbeheardthroughastethoscope.Whenthecuffpressuredropsbelowthediastolicpressure,bloodflowssmoothly.


BloodPressureWhenadoctorornursegivesyouyourbloodpressure,thefirstnumberisthesystolicbloodpressureandthesecondnumberisthediastolicpressure.


ConceptualExample13.5InFigure13.14,thepatient’sarmisheldataboutthesameheightasherheart.Why?


ConceptualExample13.5REASON Thehydrostaticpressureofafluidvarieswithheight.Althoughflowingbloodisnotinhydrostaticequilibrium,itisstilltruethatbloodpressureincreaseswiththedistancebelowtheheartanddecreasesaboveit.

Becausetheupperarmwhenheldbesidethebodyisatthesameheightastheheart,thepressurehereisthesameasthepressureattheheart.Ifthepatientheldherarmstraightup,thepressurecuffwouldbeadistanced≈25cmaboveherheartandthepressurewouldbelessthanthepressureattheheartbyΔp =ρblood gd ≈20mmHg.


ConceptualExample13.5ASSESS 20mmHgisasubstantialfractionoftheaveragebloodpressure.Measuringpressureaboveorbelowheartlevelcouldleadtoamisdiagnosisofthepatient’scondition.


Section13.4Buoyancy


BuoyancyBuoyancy istheupwardforceofaliquid.

Thepressureinaliquidincreaseswithdepth,sothepressureinaliquid-filledcylinderisgreateratthebottomthanatthetop.

Thepressureexertsanetupwardforce onasubmergedcylinderof

Fnet =Fup – Fdown


BuoyancyIfanisolatedparcelofafluidisinstaticequilibrium,thentheparcel’sweightforcepullingitdownmustbebalancedbyanupwardforce:thebuoyantforce

Thebuoyantforcematchesthefluidweight:FB=w.

Ifwereplacetheparcelofliquidwithanobjectofthesameshapeandsize,thebuoyantforceonthenewobjectisexactlythesame asbefore.


BuoyancyWhenanobjectisimmersedinafluid,itdisplaces thefluidthatwouldotherwisefillthatregionofspace.Thefluidiscalledthedisplacedfluid:

Archimedes’principleinequationformis

FB =ρfVfg


QuickCheck13.4

Aheavyleadblockandalightaluminumblockofequalsizesarebothsubmergedinwater.Uponwhichisthebuoyantforcegreater?

◦ Ontheleadblock◦ Onthealuminumblock◦ Theybothexperiencethesamebuoyantforce.


QuickCheck13.4

Aheavyleadblockandalightaluminumblockofequalsizesarebothsubmergedinwater.Uponwhichisthebuoyantforcegreater?

◦ Ontheleadblock◦ Onthealuminumblock◦ Theybothexperiencethesamebuoyantforce.


Same size Þ both displace the same volume and weight of water.

QuickCheck13.5

Twoblocksareofidenticalsize.Oneismadeofleadandsitsonthebottomofapond;theotherisofwoodandfloatsontop.Uponwhichisthebuoyantforcegreater?

◦ Ontheleadblock◦ Onthewoodblock◦ Theybothexperiencethesamebuoyantforce


QuickCheck13.5

Twoblocksareofidenticalsize.Oneismadeofleadandsitsonthebottomofapond;theotherisofwoodandfloatsontop.Uponwhichisthebuoyantforcegreater?

◦ Ontheleadblock◦ Onthewoodblock◦ Theybothexperiencethesamebuoyantforce


The fully submerged lead block displaces more much water than the wood block.

QuickCheck13.6

Abargefilledwithorefloatsinacanallock.Iftheoreistossedoverboardintothelock,thewaterlevelinthelockwill

◦ Rise.◦ Fall.◦ Remainconstant.


Example13.6Isthecrowngold?

LegendhasitthatArchimedeswasaskedbyKingHiero ofSyracusetodeterminewhetheracrownwasofpuregoldorhadbeenadulteratedwithalessermetalbyanunscrupulousgoldsmith.Itwasthisproblemthatledhimtotheprinciplethatbearshisname.Inamodernversionofhismethod,acrownweighing8.30Nissuspendedunderwaterfromastring.Thetensioninthestringismeasuredtobe7.81N.Isthecrownpuregold?


Example13.6Isthecrowngold?(cont.)

PREPARE Todiscoverwhetherthecrownispuregold,weneedtodetermineitsdensityρo andcompareittotheknowndensityofgold.FIGURE13.17showstheforcesactingonthecrown.Inadditiontothefamiliartensionandweightforces,thewaterexertsanupwardbuoyantforceonthecrown.ThesizeofthebuoyantforceisgivenbyArchimedes’principle.



SOLVE Becausethecrownisinstaticequilibrium,itsaccelerationandthenetforceonitarezero.Newton’ssecondlawthenreadsSFy =FB +T- wo =0fromwhichthebuoyantforceis

FB =wo - T=8.30N- 7.81N=0.49N



AccordingtoArchimedes’principle,FB =ρfVf g,whereVf isthevolumeofthefluiddisplaced.Here,wherethecrowniscompletelysubmerged,thevolumeofthefluiddisplacedisequaltothevolumeVo ofthecrown.Nowthecrown’sweightiswo =mog =ρoVog,soitsvolumeis


Example13.6Isthecrowngold?(cont.)InsertingthisvolumeintoArchimedes’principlegives

or,solvingforρo,



Thecrown’sdensityisconsiderablylowerthanthatofpuregold,whichis19,300kg/m3.Thecrownisnotpuregold.

ASSESS Foranobjectmadeofadensematerialsuchasgold,thebuoyantforceissmallcomparedtoitsweight.


FloatorSink?Whetheranobjectreleasedunderwaterwillheadtothesurfaceortothebottomdependsonwhethertheupwardbuoyantforceontheobjectislargerorsmallerthanthedownwardweightforce.

Someobjectsarenotuniform.Wethereforedefinetheaveragedensity tobeρavg =mo/Vo.Theweightofacompoundobjectcanbewrittenaswo =ρavgVog.


FloatorSink?Anobjectwillfloatorsinkdependingonwhetherthefluiddensityislargerorsmallerthantheobject’saveragedensity.

Ifthedensitiesareequal,theobjectisinstaticequilibriumandhangsmotionless.Thisiscalledneutralbuoyancy.


FloatorSink?


Text:p.410

QuickCheck13.7

Whichfloatingblockismostdense?

◦ Blocka◦ Blockb◦ Blockc◦ Blocksaandbaretied.◦ Blocksbandcaretied.


Example13.8MeasuringthedensityofanunknownliquidYouneedtodeterminethedensityofanunknownliquid.Younoticethatablockfloatsinthisliquidwith4.6cmofthesideoftheblocksubmerged.Whentheblockisplacedinwater,italsofloatsbutwith5.8cmsubmerged.Whatisthedensityoftheunknownliquid?


Example13.8Measuringthedensityofanunknownliquid(cont.)PREPARE Assumethattheblockisanobjectofuniformcomposition.FIGURE13.19showstheblockaswellasthecross-sectionareaAandsubmergedlengthshu intheunknownliquidandhw inwater.


Example13.8Measuringthedensityofanunknownliquid(cont.)SOLVE Theblockisfloating,soEquation13.10applies.TheblockdisplacesvolumeVu =Ahu oftheunknownliquid.Thus

Similarly,theblockdisplacesvolumeVw =Ahw ofthewater,leadingto


Example13.8Measuringthedensityofanunknownliquid(cont.)Becausetherearetwofluids,we’veusedsubscriptswforwaterandufortheunknowninplaceofthefluidsubscriptf.TheproductρoVo appearsinbothequations.InthefirstρoVo =ρu Ahu,andinthesecondρoVo =ρw Ahw.Equatingtheright-handsidesgives

ρuAhu =ρw Ahw


Example13.8Measuringthedensityofanunknownliquid(cont.)TheareaAcancels,andthedensityoftheunknownliquidis

ASSESS ComparisonwithTable13.1showsthattheunknownliquidislikelytobeglycerin.


BoatsandBalloonsThehullofaboatisreallyahollowshell,sothevolumeofwaterdisplacedbytheshellismuchlargerthanthevolumeofthehullitself.

Theboatsinksuntiltheweightofthedisplacedwaterexactlymatchestheboat’sweight.Itistheninstaticequilibriumandfloats.


BoatsandBalloonsThedensityofairislowsothebuoyantforceisgenerallynegligible.

Balloonscannotbefilledwithregularairbecauseitwouldweighthesameamountasthedisplacedairandthereforehavenonetupwardforce.

Foraballoontofloat,itmustbefilledwithagasthathasalower densitythanthatofair.


QuickCheck 13.8

Blocksa,b,andcareallthesamesize.Whichexperiencesthelargestbuoyantforce?

◦ Blocka◦ Blockb◦ Blockc◦ Allhavethesamebuoyantforce.

◦ Blocksaandchavethesamebuoyantforce,butthebuoyantforceonblockbisdifferent.


QuickCheck13.9Blocksa,b,andcareallthesamesize.Whichisthecorrectorderofthescalereadings?

◦ a=b=c◦ c>a=b◦ c>a>b◦ b>c>a◦ a=c>b


ExampleProblemTheenvelopeofatypicalhotairballoonhasavolumeof2500m3.AssumethatsuchaballoonisflyinginFortCollins,Colorado,wherethedensityofairisapproximately1.0kg/m3.

◦ Whatmassofairdoestheballoondisplace?◦ Ifheatedtothemaximumtemperature,theairinsidetheballoonhasadensityofabout80%thatofthesurroundingair.Whatisthemassofairintheballoon?

◦ Howmuchmasscantheballoonlift?


introductory physics - home | 13.4 pressure in a tube with two liquids a u-shaped tube is closed at...

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