Kenyon alumni working in food and drink industries are sharing their go-to happy hour pairings. Interested in making videos for your class projects or lectures? Just need a basic editor to be able…. Do you use online videos in your lessons or presentations? Would you like to direct your viewers to specific…. ThemomentsoftheseforcesaboutB areequalwhenthegateopens: M B 0 H 0. Thissolutionisindependentofboththewater densityandthegatewidthbintothepaper.

Determine therequiredforcePforequilibrium. Thus it is 3. Findthehydrostaticforceonsurface AB and its moment about C. Could this forcetipthedamover? Wouldfluidseepage underthedamchangeyourargument? As showninthefigure,thelineofactionofFis2. Iftherewereseepage underthedam,themainsupportforceatthebottomofthedamwouldshifttotheleftof pointCandmightindeedcausethedamtotipover.

If the dam material has specific gravity SG, with no seepage, find the critical angle c for which the dam will just tip overtotheright. Evaluatethisexpression forSG2. Solution:Bygeometry, L Hcos and theverticalheightofthedamisHsin.

SG Anyanglegreaterthan c willcausetipovertotheright. Fortheparticular caseof concrete,SG2. WhathorizontalforcePisrequiredat pointBforequilibrium? Theforceonthegateis F h CG A 0. Thegateweightof Nisassumedatthecentroidoftheplate,withmomentarm0. Ifnot, Fig.

Themissingwaterweightequalsthe vertical componentofthe force. Thehinge 15cm h hinge is15cmfromthecenterline,asshown. Whatspherediameterisjustrightto closethegate? Whatisthewaterdepth h whichwillfirst causethegatetoopen? Solution:Theminimumheightneededto openthegatecanbeassessedbycalculating thehydrostaticforceoneachsideofthegate andequatingmomentsaboutthehinge.

Fair 10,Pa 0. Sincetheairpressureisuniform,Fairactsatthecentroidofthegate,or15cmbelowthe hinge. Summomentsaboutthehingeandsetequaltozerotofindtheminimumheight: M hinge 0 Thehingeat Ais2ftabovethefreshwaterlevel. Findh whenthegateopens.

Solution:Therearetwodifferenthydro static forces and two different lines of action. Onthewaterside, Fw h CG A Thenthedislodging:pressurejustundercapBwillbe F Compute a thehydrostaticforceofthe wateronthepanel; b itscenterofpressure;and c themomentofthisforceabout pointB. Compute theforcejustsufficient tokeep Fig. Solution: a The writer is unable to deduce by inspection which panel force is larger. CD is longer than AB, but its centroid is not as deep.

If you have a great insight, let me know. Calculate the two forces:. It turns out that panel AB has the larger force, but it is only 4 percent larger. It opens auto Fig. Neglecting atmospheric pressure, determinethelowestlevelhforwhichthe gatewillopen. Isyourresultindependent oftheliquiddensity? Solution:The gate will open when the hydrostaticforceFonthegateisaboveB, thatis,when. Analyzethisgeometrytofind L F therangeofanglesforwhichthe W. Solution: Thecriticalangleiswhenthehydrostaticforce F causesaclockwisemoment equaltothecounterclockwisemomentofthedamweightW.

Whenthemomentisnegative small,thedamisstable,itwillnottipover. Atanyposition, A asinFig. Determinethe horizontalandverticalcomponentsofhydrostaticforceagainstthedamandthepointCP wheretheresultantstrikesthedam. The resultant 2 2 hydrostaticforceisF [ FindtheforceFjust Fig. ThenwemaysummomentsabouthingeB to find the force F, using the freebody of the gate as sketched at the topright of thispage:. M B clockwise 0 F 8. A C Neglectatmosphericpressure. Find a thevertical.

Solution: b Thehorizontalforceiscalculatedfromtheverticalprojectionofthepanel frompointAdowntothebottom. Thisisarectangle,75cmbycm,andits centroidis Thisisintwoparts 1 the weightoftherectangularportionabovethelineAC;and 2 thelittlecurvypieceabove theparabolaandbelowlineAC.

Find the Fig. The width b into the paper is3m. MC 0 2m P N 0. Compute the netforce onthe2inradius hemisphericalendcapatthebottomofthe bottle. Solution:First,fromthemanometer,com putethegagepressureatsectionAAinthe champagne6inchesabovethebottom: 2 4 p AA 0. Thentheforceonthebottomendcapisverticalonly duetosymmetry andequalsthe forceatsectionAAplustheweightofthechampagnebelowAA:. For the position Fig. Doesthe forcepassthroughpointO? Theverticalforceisupwardandequaltothe goodhandbookwillgiveyouthegeometric weightofthemissingwaterinthesegment properties of a circular segment, and you ABCshownshadedbelow.

Determine the vertical hydrostatic force on circulararc sectionABanditslineofaction. Solution:Assume unit depth into the paper. Theverticalforceistheweightof benzene plus the force due to the air pressure:. Determinethe Missing horizontalandverticalhydrostatic cm water. Missing water 1. What is the force in each bolt requiredtoholdthedomedown? Iftheendcapsareneglected,computethe forceineachbolt. Solution:Consider a 25cm width of uppercylinder,asseenbelow. Thewater pressureintheboltplaneis.

Derivean analytic expression for the hydrodynamic forceFontheshellanditslineofaction. Thereisnoneedtofindthe complicated centersofpressureforthesethreecomponents, forweknowthattheresultantonasphericalsurfacemustpassthroughthecenter.

B B Fig. Solution: It looks deceiving, since the bulging panel on the right has more water nearby,butthesetwoforcesarethesame,exceptfortheirdirection. Theleftsidefigure isthesameasExample2. Therightsidefigurehasthe sameverticalforce,butitisdown. Bothverticalforcesequaltheweightofwaterinside, ordisplacedby,thehalfcylinderAB.

Theirhorizontalforcesequaltheforceonthe projectedplaneAB. Whatisthespecificgravityofthisbodywhen a h0;and b hR?

If the shownhere. Theverticalcomponentequalstheweight ofwaterabovethegate,whichisthesum oftherectangularpieceaboveBC,andthe curvytriangularpieceofwaterjustabove arc BCsee figure at right. The curvy. Computethe a horizontaland 1m B b verticalwaterforcesonthecurvedpanelAB.

Computethe Fig. Water 20ft Determine a thehorizontaland b thevertical. Solution:SincetheproblemasksforBGunits, Fig. F V pair Asurface water water lbf lbf 1 4 2 6 ft 2 Computethehydrostaticforce Fig. Then the vertical force on the cone equals the weight of fictitiouswateraboveABC:. FV Volabove 2 1 2 2 8. Estimatethe A gagepressureoftheairif a the 90cm Water. Theweddingcake shapeoftheboxhasnothingtodowiththeproblem. Neglect atmosphericpressure. Find a thehydrostaticforceonthebottom;and b theforceona sidepanel.

Fside h CG Aside 0. Whatisthe specificgravityoffluidX? What is its weight in newtons? W displaced 0. Was itgold? Solution:Thebuoyancyisthedifferencebetweenairweightandunderwaterweight: B Wair Wwater Solution:Thealtitudecanbedeterminedbycalculatingtheairdensitytoprovidethe properbuoyancyandthenusingTableA.

Solution:Refer back to Prob. Theonlydifferenceisthattheforce applied togateABbytheweightisless duetobuoyancy: SG 1 2. According to Table A3, this fluid is probably carbon tetrachloride. For stem diameter D and total weightW,if h 0represents SG 1.

Solution:Let submerged volume be o 2 whenSG1. Ifso,howmanycubiccentimeterswill beexposed? Thereforethesphereis floatingexactlyhalfinandhalfoutofthewater. The volumeexposedis.

Ambient air is at 25C and 1atm. Whatdiameterspherical balloonwilljustsupporttheweight? Solution:Thebuoyancyisduetothedifferencebetweenhotandcoldairdensity: p kg kg cold 1. Checktomakesuretheballoonmaterialisnotexcessivelyheavy: W balloon 0.

Determine a the stringtension; and b the specific gravity ofthewood. Is italso possibletodeterminetheinclinationangle? SummingmomentsaboutCgives MC 0 W 2. Thus W 0. Letthe Fig. Howmanypoundsofsteel SG 7. Wsteel or: 1. Whatisthespecificgravityoftherod material? What is peculiar about the rest angle30?

If the rock exerts no moments on the spar, computetheangleofinclination. We see that the buoyancy exceeds the weight by 19 kN, or more than 2 tons. The sphere would float nicely. W 9. Slightly less than doubling the wall thickness would create neutral buoyancy for the sphere.

Estimate a thetensioninthe mooring line, and b the height in the standardatmospheretowhichtheballoon willriseifthemooringlineiscut. Theyestimatetheweightofthe boxandtreasure inair atlbf.

Theirplanistoattachtheboxtoasturdyballoon, inflatedwithairto3atmpressure. Theboxis2ftwide, 5ftlong,and18inhigh. Theboxvolumeis 2ft 5ft 1. Thusthe balloonmustdevelopanetupwardforceof1. Theairweightin theballoonisnegligible,butwecancomputeitanyway. Fnet 64 0. Determine a ; b wood. A Summoments abouttheleftcorner,point uniformheavyspheretiedtotheleftcorner C: causes the beam to float exactly on its diagonal.