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[SHIVOK SP211] October 22, 2015 CH 14 Fluids I. WhatisaFluid? A. Afluid,incontrasttoasolid,isasubstancethatcan_________________. B. Fluidsconformtotheboundariesof________________________________in whichweputthem.Theydosobecauseafluidcannotsustainaforce thatistangentialtoitssurface.Thatis,afluidisasubstancethatflows becauseitcannotwithstandashearingstress. C. Itcan,however,exertaforceinthedirection__________________________ toitssurface. II. DensityandPressure A. Tofindthedensityofafluidatanypoint,weisolateasmall volumeelementVaroundthatpointandmeasurethemassmofthe fluidcontainedwithinthatelement.Ifthefluidhasuniformdensity, then 1. Densityisascalarproperty;itsSIunitisthekilogrampercubicmeter. B. IfthenormalforceexertedoveraflatareaAisuniformoverthat area,thenthepressureisdefinedas: 1. TheSIunitofpressureisthenewtonpersquaremeter,whichisgivena specialname,thePascal(Pa). 2. 1atmosphere(atm)= Page1 [SHIVOK SP211] October 22, 2015 3. Exampleproblem: a) Apartiallyevacuatedairtightcontainerhasatightfittinglidof surfacearea77cm2andnegligiblemass.Iftheforcerequiredtoremove thelidis480Nandtheatmosphericpressureis1.0x105Pa,whatisthe internalairpressure? (1) Solution: III. FluidsatRest A. Thepressureatapointinafluidinstaticequilibriumdependson thedepthofthatpointbutnotonanyhorizontaldimensionofthe fluidoritscontainer. 1. Above:Atankofwaterinwhichasampleofwateriscontainedinan imaginarycylinderofhorizontalbaseareaA. 2. Below:Afree‐bodydiagramofthewatersample. Page2 [SHIVOK SP211] October 22, 2015 3. Thebalanceofthe3forcesiswrittenas: 4. Ifp1andp2arethepressuresonthetopandthebottomsurfacesofthe sample, 5. Sincethemassmofthewaterinthecylinderis,m=V,wherethe cylinder’svolumeVistheproductofitsfaceareaAanditsheight(y1‐y2),then 6. Therefore, 7. Ify1isatthesurfaceandy2isatadepthhbelowthesurface,then (wherepoisthepressureatthesurface,andpthepressureatdepthh). B. Exampleproblems: 1. Crewmembersattempttoescapefromadamagedsubmarine100.0m belowthesurface.Whatforcemustbeappliedtoapop‐outhatch,whichis 1.20mby0.60m,topushitoutatthatdepth?Assumethatthedensityofthe oceanwateris1024kg/m3andtheinternalairpressureisat1.00atm. Page3 [SHIVOK SP211] October 22, 2015 2. Atadepthof10.9km,theChallengerDeepintheMarianasTrenchof thePacificOceanisthedeepestsiteinanyocean.Yet,in1960,DonaldWalsh andJacquesPiccardreachedtheChallengerDeepinthebathyscaphTrieste. Assumingthatseawaterhasauniformdensityof1024kg/m3,approximate thehydrostaticpressure(inatmospheres)thattheTriestehadtowithstand. IV. Pascal’sPrinciple A. Achangeinthepressureappliedtoanenclosedincompressible fluidistransmittedundiminishedtoeveryportionofthefluidandto thewallsofitscontainer. Page4 [SHIVOK SP211] October 22, 2015 B. Pascal’sPrincipleandtheHydraulicLever 1. TheforceF isappliedontheleftandthedownwardforceF fromthe i o loadontherightproduceachangepinthepressureoftheliquidthatis givenby 2. Ifwemovetheinputpistondownwardadistanced ,theoutputpiston i movesupwardadistanced ,suchthatthesamevolumeVofthe o incompressibleliquidisdisplacedatbothpistons. 3. Thentheoutputworkis: Page5 [SHIVOK SP211] October 22, 2015 C. Sampleproblem: 1. Apistonofcross‐sectionalareaaisusedinahydraulicpresstoexerta smallforceofmagnitudefontheenclosedliquid.Aconnectingpipeleadstoa largerpistonofcross‐sectionalareaA(Fig.below).(a)WhatforcemagnitudeF willthelargerpistonsustainwithoutmoving?(b)Ifthepistondiametersare 3.80cmand53.0cm,whatforcemagnitudeonthesmallpistonwillbalancea 20.0kNforceonthelargepiston? a) Solution: (a) (b) Page6 [SHIVOK SP211] October 22, 2015 V. ArchimedesPrinciple A. Whenabodyisfullyorpartiallysubmergedinafluid,abuoyant forcefromthesurroundingfluidactsonthebody.Theforceis directedupwardandhasamagnitudeequaltotheweightofthefluid thathasbeendisplacedbythebody. 1. Thenetupwardforceontheobjectisthebuoyantforce,F . 2. Thebuoyantforceonabodyinafluidhasthemagnitude b wheremfisthemassofthefluidthatisdisplacedbythebody. Page7 [SHIVOK SP211] October 22, 2015 B. FloatingandApparentWeight 1. Whenabodyfloatsinafluid,themagnitudeF ofthebuoyantforceon b thebodyisequaltothemagnitudeF ofthegravitationalforceonthebody. g 2. Thatmeans,whenabodyfloatsinafluid,themagnitudeF ofthe g gravitationalforceonthebodyisequaltotheweightm gofthefluidthathas f beendisplacedbythebody,wherem isthemassofthefluiddisplaced. f 3. Thatis,afloatingbodydisplacesitsownweightoffluid. 4. Theapparentweightofanobjectinafluidislessthantheactualweight oftheobjectinvacuum,andisequaltothedifferencebetweentheactual weightofabodyandthebuoyantforceonthebody. Page8 [SHIVOK SP211] October 22, 2015 C. Sampleproblem: 1. Whatistheapparentweightofasphereofradius r 0.020m and kg thatisfullysubmergedinwater m3 kg (density f 1000 3 )? m density 1500 a) Solution: Page9 [SHIVOK SP211] October 22, 2015 VI. IdealFluidsinMotion A. Steadyflow:Insteady(orlaminar)flow,thevelocityofthemoving fluidatanyfixedpointdoesnotchangewithtime. B. Incompressibleflow:Weassume,asforfluidsatrest,thatourideal fluidisincompressible;thatis,itsdensityhasaconstant,uniform value. C. Nonviscousflow:Theviscosityofafluidisameasureofhow resistivethefluidistoflow;viscosityisthefluidanalogoffriction betweensolids.Anobjectmovingthroughanonviscousfluidwould experiencenoviscousdragforce—thatis,noresistiveforcedueto viscosity;itcouldmoveatconstantspeedthroughthefluid. D. Irrotationalflow:Inirrotationalflowatestbodysuspendedinthe fluidwillnotrotateaboutanaxisthroughitsowncenterofmass. VII. TheEquationofContinuity Page 10 [SHIVOK SP211] October 22, 2015 A. Theamountofvolume(DeltaV)offluidflowingpastthedotted lineinthetimeintervaldeltatis: B. Applyingtheaboveequationforboththeleftandrightdotted linesweget: C. Whichyields 1. Whatitsaysinlaymen’stermsisthatthevolumetricflowrateis constant;thusthespeedincreasesifwerestrictthearea(i.e.youputyour thumbovertheendofthegardenhoseandthewatercameoutatahigher speed): 2. Ifwemultiplybothsidesoftheequationbydensityofthefluidweget massflowrate: Page 11 [SHIVOK SP211] October 22, 2015 D. Example:WaterStream 1. Whydoesthewaterstreamnarrowasitfallsfromthefaucet? 2. Solution: Page 12 [SHIVOK SP211] October 22, 2015 VIII. Bernoulli’sEquation A. Ifthespeedofafluidelementincreasesastheelementtravels alongahorizontalstreamline,thepressureofthefluidmustdecrease, andconversely. a) Fig.14‐19FluidflowsatasteadyratethroughalengthLofa tube,fromtheinputendatthelefttotheoutputendattheright.From timetin(a)totimet+tin(b),theamountoffluidshowninpurple enterstheinputendandtheequalamountshowningreenemergesfrom theoutputend. 2. Thatmeansthat Or Page 13 [SHIVOK SP211] October 22, 2015 B. Proof 1. Thechangeinkineticenergyofthesystemistheworkdoneonthe system. 2. Ifthedensityofthefluidis, 3. Theworkdonebygravitationalforcesis: 4. Thenetworkdonebythefluidis: 5. Therefore, 6. Finally, Page 14 [SHIVOK SP211] October 22, 2015 C. Sampleproblems: 1. AcylindricaltankwithalargediameterisfilledwithwatertoadepthD =0.30m.Aholeofcross‐sectionalareaA=6.5cm2inthebottomofthetank allowswatertodrainout.(a)Whatistherateatwhichwaterflowsout,in cubicmeterspersecond?(b)Atwhatdistancebelowthebottomofthetankis thecross‐sectionalareaofthestreamequaltoone‐halftheareaofthehole? where h1 is the height of the water in the tank, p1 is the pressure there, and v1 is the speed of the water there; h2 is the altitude of the hole, p2 is the pressure there, and v2 is the speed of the water there. is the density of water. The pressure at the top of the tank and at the hole is atmospheric, so p1 = p2. Since the tank is large we may neglect the water speed at the top; it is much smaller than the speed at the hole. The Bernoulli equation then becomes (b) We use the equation of continuity: A2v2 = A3v3, where A3 12 A2 and v3 is the water speed where the area of the stream is half its area at the hole. Thus The water is in free fall and we wish to know how far it has fallen when its speed is doubled to 4.84 m/s. Since the pressure is the same throughout the fall, Page 15 [SHIVOK SP211] October 22, 2015 2. Figurebelowshowsastreamofwaterflowingthroughaholeatdepth h 10cm inatankholdingwatertoheight H 40cm .(a)Atwhatdistancex doesthestreamstrikethefloor?(b)Describetheprocessyouwoulduseto findthevalueofhwhichmaximizesthedistancex. a) Solution Page 16