Lec4-Hydrostatics-Forces on Planes (3)

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MECH 222

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Jan 15, 2025

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MECH 222F – HYDROSTATICS –FORCES ON PLANESRef.: Wikipedia

LECTURE OVERVIEW•HYDROSTATIC FORCES ACTING ON FLAT SURFACES IN CONSTANT DENSITY FLUID•HYDROSTATIC FORCES ACTING ON SURFACES IN STRATIFIED FLUID

WHY CARE ABOUT HYDROSTATIC FORCES?Ref.: wikipediaRef.: National Geographic•Liquids like water are dense and hydrostatic forces can be largeQ: Are hydrostatic forces due to gases large (hint: think about a blimp)?A:Yes,itscaleisbig

REMINDER: PRESSURE VARIATION WITH DEPTHQ: How does the pressure vary with depth in a constant density fluid?A:Q: How does the pressure vary with depth if the density is not constant?A:Q: In what direction does the pressure on a surface act?A:P=P-89242=-49alwaysnormal/perpendiculartosurfaceP=Po-syz-9---

FORCES ON A SUBMERGED PLATECONSIDER A PLATE SUBMERGED IN A FLUID OF CONSTANT ࠵༌࠵?. THE SURFACE IS VERTICAL AND RECTANGULAR WITH HEIGHT ࠵఻࠵?AND BREADTH ࠵వ࠵?. LET’S TAKE ࠵ౝ࠵?࠵౎࠵?࠵౎࠵?࠵౎࠵?= 0࠵ౝ࠵?࠵౧࠵?=−࠵༌࠵?࠵༌࠵?࠵౧࠵?࠵఻࠵?࠵ౝ࠵?=࠵༌࠵?࠵༌࠵?࠵఻࠵?࠵౧࠵?࠵౥࠵?࠵హ࠵?࠵౥࠵?=࠵ౝ࠵?࠵శ࠵?࠵శ࠵?࠵ఴ࠵?What is the net force acting on the plate? ࠵ౝ࠵?࠵శ࠵?࠵శ࠵?is the pressure at the centre of gravity (centroid) of the plateA=HBHe3Fu=1**incrementarea99t=Pca--------=-9231I2=pyIB=P9I.A↓Lookslikeplzz-I->->->>--->>-uregzPatm-Patr=O--Patm=0↓dz--=F=q-=-----#z=F=pA⊥hear-PG=g.

FORCES ON A SUBMERGED PLATEAt what location does this net force act? Σ࠵ీ࠵?࠵హ࠵?࠵హ࠵?==࠵༌࠵?࠵༌࠵?࠵఻࠵?33࠵వ࠵?∴ ࠵౧࠵?࠵శ࠵?࠵శ࠵?=−࠵༌࠵?࠵༌࠵?࠵఻࠵?3࠵వ࠵?3࠵༌࠵?࠵༌࠵?࠵఻࠵?2࠵వ࠵?2=−2࠵఻࠵?3=−࠵఻࠵?2−࠵఻࠵?6=࠵౧࠵?࠵శ࠵?࠵శ࠵?−࠵వ࠵?࠵఻࠵?312࠵఻࠵?2×࠵఻࠵?࠵వ࠵?=࠵౧࠵?࠵శ࠵?࠵శ࠵?−࠵఼࠵?࠵౥࠵?࠵౥࠵?ℎ࠵శ࠵?࠵శ࠵?࠵ఴ࠵?࠵ౝ࠵?࠵౧࠵?=−࠵༌࠵?࠵༌࠵?࠵౧࠵?࠵఻࠵?࠵ౝ࠵?=࠵༌࠵?࠵༌࠵?࠵఻࠵?࠵౧࠵?•To find out, let’s take the moment about the free surface:•By definition, this must be ࠵హ࠵? �(−࠵౧࠵?࠵శ࠵?࠵శ࠵?), where ࠵౧࠵?࠵శ࠵?࠵శ࠵?is the “centre of pressure”where ࠵఼࠵?࠵౥࠵?࠵౥࠵?is the area moment of inertia of the plate about a horizontal axis through the centroid and ℎ࠵శ࠵?࠵శ࠵?is the depth (positive) of the centroid of the platedNEs=dF,Z--dztdi=P2)Bd=-92BdziF=ByEMB⑰S:-1-92B2dz--/-momentOOnforceᶑ*-zFleft----G<*7CPmonentBQ~&Force.↑[MEs=Ex.)-zcp)-FxS--ZcGZcla-~-

FORCES ON PLANAR SUBMERGED SURFACESCONSIDER A SURFACE SUBMERGED IN A FLUID OF CONSTANT ߩ. THE SURFACE IS VERTICAL AND RECTANGULAR WITH HEIGHT ܪAND BREADTH ܤ. WE SET ݌௔௧௠= 0:݌ݖ=െߩ݃ݖܨ௫=න଴ு݌ݕ ܤ݀ݕ=ߩ݃ݕଶܤ/2଴ு=ߩ݃ܪ2×ܪܤܪ݌=ߩ݃ܪݖݔܨ௫=݌஼ீܣWhere is the location through which the resultant force acts?Measure everything relative to free surface and define ݕpointing down (ݕ=െݖ):ȭܯிௌ=න଴ு݌ݕ ݕܤ݀ݕ=න଴ுߩ݃ݕ ݕܤ݀ݕ׵ ݕ஼௉=ʹܪ3=ܪ2+ܪ6=ݕ஼ீ+ܤܪଷ12ܪ2×ܪܤ=ݕ஼ீ+ܫ௬௬ݕ஼ீܣ݌ݕ=ߩ݃ݕ=ߩ݃ܪଷ3ܤ=ߩ݃ܪ2×ܪܤ×ʹܪ3=ܨ௫ȉ ݕ஼௉-Ycp=Yc+#pees-freesurface~Y#S<p(y)=197vY⊥11/11/IIi#ply)=PgyyYcenteofSmomentarmFre,ExPRESSURE-EyaF--pressuetheareaofplate-plateFx=Sp.dAPressureatcentreofplate-*8LaAreaofplate

FORCES ON PLANAR SUBMERGED SURFACESIF THE SURFACE IS PLANAR, HAS VARIABLE BREADTH, ܤ(ݕ), AND IS INCLINED AT AN ANGLE OF ߠRELATIVE TO THE HORIZONTAL THE MATH GETS MORE COMPLICATED BUT THE RESULTS ARE ALMOST THE SAME:݌ݖ=݌௔௧௠െ ߩ݃ݖܨୄ=න௬೅ೀು௬ಳೀ೅೅ೀಾ݌ݕ ܤ(ݕ)݀ݕݖݔ ݋ݑݐ ݋݂ ݏܿݎ݁݁݊ܨୄ=݌஼ீܣݕ஼௉=െߩ݃ݏ݅݊ߠܫ௬௬݌஼ீܣ݌ݕ=݌௔௧௠+ߩ݃ݕ ȉ ݏ݅݊ߠݕݔ஼௉=െߩ݃ݏ݅݊ߠܫ௫௬݌஼ீܣRELATIVE TO THE CENTROID:VIEW NORMAL TO PLANE:(ݔ஼ீ,ݕ஼ீ)ܤ(ݕ)(ݔ஼௉,ݕ஼௉)ߠ#pre-classz==ysinf-!CoGEv-ICop·/co

FORMULAS FOR INCLINED FLAT PLATES࠵౧࠵?࠵౥࠵? ࠵౜࠵?࠵౜࠵?࠵౜࠵? ࠵౜࠵?࠵౜࠵? ࠵ౠ࠵?࠵ౠ࠵?࠵ౠ࠵?࠵ౠ࠵?࠵ౠ࠵?࠵ౠ࠵?࠵హ࠵?⊥=࠵ౝ࠵?࠵శ࠵?࠵శ࠵?࠵ఴ࠵?࠵౦࠵?࠵శ࠵?࠵శ࠵?=−࠵༌࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵఼࠵?࠵౥࠵?࠵౥࠵?࠵ౝ࠵?࠵శ࠵?࠵శ࠵?࠵ఴ࠵?࠵౦࠵?࠵౥࠵?࠵శ࠵?࠵శ࠵?=−࠵༌࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵఼࠵?࠵౥࠵?࠵౥࠵?࠵ౝ࠵?࠵శ࠵?࠵శ࠵?࠵ఴ࠵?RELATIVE TO THE CENTROID:VIEW NORMAL TO PLATE FROM ABOVE:(࠵౥࠵?࠵శ࠵?࠵శ࠵?,࠵౦࠵?࠵శ࠵?࠵శ࠵?)(࠵౥࠵?࠵శ࠵?࠵శ࠵?,࠵౦࠵?࠵శ࠵?࠵శ࠵?)࠵༌࠵?Q: For the plate shown above, is the ࠵౥࠵?࠵శ࠵?࠵శ࠵?to the left of right of CG?A:Ixy=SSMdA↑totheleft(alwaysbelow(G)momentofinertaabt-X-axiszinin t1/1lProveninmomentofficeabthorizontalareatextbook:-axis(x-axis)aboutmoreareatotheleft/GICGQ:Ingened,isYoppositiveornegative?measuredfromLOGalwayspositive(alwaysbelow(G)

(35z=-ysin35W=7Xcp=-gsinf.Xp=0Ath=3XoutofscreenQ:WhatisPco?Yo:(2)+)-1.5)=-3.5mZ-77P26=epgzia==g(Ycbin357Fᶖ>350=998(981))-3.5 sin35=350&-*&:HowtofindopandFBT?=19.7LPaYup=-0.21m2m#=Fpessue=PcaA~=>Frop(3)-Fpressure·(1.5-021=0--SEIMBoT=03=19700x(7x3)=414Will-Fop=414000(15-0.21=177kNParAFFB01+178=414Yup,BysinGExIxx=&whY=-5m[FX=0:Fir+Fop:Fpress=>y(p=998(9.81) sin35.2(7)(3:Fot=236WNV19700(73)==0Zim

VariableDensityExample.z-Assumegate=f-weight#Patm=o=ForestS350↓~↓For7X-dy~FBot3mdA=7 dy.Y=-5mBOTQ:Wheredowestart?&HowtofindFropandFioT?Momentarmofinfinitesimal9(z)=998-20z;(z+0)#EMBOT=0:ply)dA=7 dyelementatlocationy~~~~P=g=-998+20(Frop3)-Y=Y-5615y+32.3 y).(Edy)(4)=0Y=YBOT:p(z)=(-998z+10229#HARD!!TFORMULA=>p(y)=(-998ysin35+10y'sin35)gFrop+3=2)-5015y+32·3yrly-isi=-5615y+32.342:Frop=180./M(slightlyhigherear)

RECAP࠵హ࠵?⊥=࠵ౝ࠵?࠵శ࠵?࠵శ࠵?࠵ఴ࠵?࠵౦࠵?࠵శ࠵?࠵శ࠵?=−࠵༌࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵఼࠵?࠵౥࠵?࠵౥࠵?࠵ౝ࠵?࠵శ࠵?࠵శ࠵?࠵ఴ࠵?࠵౥࠵?࠵శ࠵?࠵శ࠵?=−࠵༌࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵ౠ࠵?࠵༌࠵?࠵఼࠵?࠵౥࠵?࠵౥࠵?࠵ౝ࠵?࠵శ࠵?࠵శ࠵?࠵ఴ࠵?RELATIVE TO THE CENTROID:FOR CONSTANT DENSITY FLUID:FOR VARIABLE DENSITY FLUID: NO CHOICE BUT TO ESTIMATE OR INTEGRATE!