Statics Simplified

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1

    Pressure and Fluid Statics. Pressure

    Pressure is defined as a normal forceexerted by a fluid per unit area .Units of pressure are N/m ! which is called

    a pascal "Pa#.Since the unit Pa is too small for pressuresencountered in practice! kilopascal "1 $Pa% 1&3 Pa# and megapascal "1 MPa % 1& ' Pa# are commonl( used.)ther units include bar ! atm, kgf/cm 2 ,lbf/in 2 =psi .

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow

    *+solute! ,a,e! and -acuum pressures

    *ctual pressure at a ,i-e point is calledthe absolute pressure .Most pressure measurin, de-ices are

    cali+rated to read ero in the atmosphere!and therefore indicate gage pressure !P ,a,e %Pa+s Patm .

    Pressure +elow atmospheric pressure arecalled vacuum pressure ! P -ac %Patm Pa+s .

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3

    *+solute! ,a,e! and -acuum pressures

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 0

    ariation of Pressure with 2epth

    n the presence of a ,ra-itationalfield! pressure increases withdepth +ecause more fluid restson deeper la(ers.

    4o o+tain a relation for the-ariation of pressure with depth!consider rectan,ular element

    Force +alance in z direction ,i-es

    2i-idin, +( ∆ x and rearran,in,,i-es

    2 1

    0

    0

    z z F ma

    P x P x g x z ρ

    = =

    ∆ − ∆ − ∆ ∆ =∑

    2 1 s P P P g z z ρ γ ∆ = − = ∆ = ∆

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 5

    4he 6arometer

    *tmospheric pressure ismeasured +( a de-ice called abarometer 7 thus! atmosphericpressure is often referred to asthe barometric pressure .P C can +e ta$en to +e ero

    since there is onl( 8, -apora+o-e point C! and it is -er(low relati-e to P atm .Chan,e in atmosphericpressure due to ele-ation hasman( effects: Coo$in,! nose

    +leeds! en,ine performance!aircraft performance.C atm

    atm

    P gh P

    P gh

    ρ

    ρ

    + ==

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow '

    Pressure at a Point

    Pressure at an( point in a fluid is the samein all directions.Pressure has a ma,nitude! +ut not a

    specific direction! and thus it is a scalar9uantit(.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow

    ariation of Pressure with 2epth

    Pressure in a fluid at rest is independent of theshape of the container.Pressure is the same at all points on a hori ontalplane in a ,i-en fluid.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow ;

    Pascalatio * /* 1 is called idealmechanical advantage

    1 2 2 21 2

    1 2 1 1

    F F F A P P

    A A F A= → = → =

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow ?

    Scu+a 2i-in, and 8(drostatic Pressure

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1&

    Pressure on di-er at1&& ft "≈3&.5 m#@

    2an,er of emer,enc(ascent@

    ( ),2 3 2

    ,2 ,2

    1998 9.81 100

    3.28

    1298.5 2.95

    101.3252.95 1 3.95

    gage

    abs gage atm

    kg m m P gz ft

    m s ft

    atmkPa atm

    kPa P P P atm atm atm

    ρ = = ÷ ÷ ÷

    = = ÷ = + = + =

    Scu+a 2i-in, and 8(drostatic Pressure

    1 1 2 2

    1 2

    2 1

    3.954

    1

    PV PV V P atmV P atm

    == = ≈

    1&& ft

    1

    6o(le

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 11

    4he Manometer

    *n ele-ation chan,e of∆z in a fluid at restcorresponds to ∆P/ ρ g.

    * de-ice +ased on this iscalled a manometer.

    * manometer consists ofa U tu+e containin, oneor more fluids such asmercur(! water! alcohol!or oil.8ea-( fluids such asmercur( are used if lar,epressure differences areanticipated.1 2

    2 atm

    P P

    P P gh ρ

    == +

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1

    Mutlifluid Manometer

    For multi fluid s(stemsPressure chan,e across a fluidcolumn of hei,ht h is ∆P = ρ gh.Pressure increases downward! anddecreases upward.

    4wo points at the same ele-ation in acontinuous fluid are at the samepressure.Pressure can +e determined +(addin, and su+tractin, ρ gh terms.

    2 1 1 2 2 3 3 1 P gh gh gh P ρ ρ ρ + + + =

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 13

    Measurin, Pressure 2rops

    Manometers are wellsuited to measurepressure drops across-al-es! pipes! heateAchan,ers! etc.

    >elation for pressuredrop P !P 2 is o+tained +(startin, at point 1 andaddin, or su+tractin, ρ gh terms until we reach point

    .f fluid in pipe is a ,as!

    ρ 2 "" ρ and P !P 2 = ρ gh

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 10

    Pro+lem

    4he manometer fluid in Fi,. is mercur(. Estimate the -olumeflow in the tu+e if the flowin, fluid is "a# ,asoline and "+#nitro,en! at &BC and 1 atm.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 15

    Fluid Statics

    Fluid Statics deals with pro+lems associated withfluids at rest.n fluid statics! there is no relati-e motion +etween

    ad acent fluid la(ers.

    4herefore! there is no shear stress in the fluidtr(in, to deform it.4he onl( stress in fluid statics is normal stress

    Normal stress is due to pressureariation of pressure is due onl( to the wei,ht of the

    fluid D fluid statics is onl( rele-ant in presence of,ra-it( fields. *pplications: Floatin, or su+mer,ed +odies!water dams and ,ates! li9uid stora,e tan$s! etc.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1'

    8oo-er 2am

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1

    8(drostatic Forces on Plane Surfaces

    )n a plane surface! theh(drostatic forces form as(stem of parallel forcesFor man( applications!ma,nitude and location ofapplication! which iscalled center ofpressure ! must +edetermined.

    *tmospheric pressureP atm can +e ne,lectedwhen it acts on +oth sidesof the surface.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1;

    >esultant Force

    4he ma,nitude of # $ actin, on a plane surface of a

    completel( su+mer,ed plate in a homo,enous fluidis e9ual to the product of the pressure P C at thecentroid of the surface and the area % of thesurface

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1?

    Formula deduction

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow &

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 1

    Center of Pressure

    =ine of action of resultant force# $ =P C % does not pass throu,hthe centroid of the surface. n,eneral! it lies underneathwhere the pressure is hi,her.

    ertical location of Center of

    Pressure is determined +(e9uation the moment of theresultant force to the momentof the distri+uted pressureforce.

    AA!C is ta+ulated for simple,eometries.

    , xx C p C

    c

    I y y

    y A= +

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow

    Pro+lemate *6 is hin,ed at point 6. 4he ,ate is 1. m lon, and &.; m into the paper. Compute

    the mass of the ,ate that would not let the oil to escape.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3

    8(drostatic Forces on Cur-ed Surfaces

    # $ on a cur-ed surface is more in-ol-ed since itre9uires inte,ration of the pressure forces thatchan,e direction alon, the surface.Easiest approach: determine hori ontal and-ertical components # & and # ' separatel(.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 0

    8(drostatic Forces on Cur-ed Surfaces

    8ori ontal force component on cur-ed surface:# & =# x . =ine of action on -ertical plane ,i-es y coordinate of center of pressure on cur-edsurface.

    ertical force component on cur-ed surface:# ' =# y () ! where ) is the wei,ht of the li9uid inthe enclosed +loc$ )= ρ g' . x coordinate of thecenter of pressure is a com+ination of line ofaction on hori ontal plane "centroid of area# and

    line of action throu,h -olume "centroid of -olume#.Ma,nitude of force # $ =*# & 2 (# ' 2 + /2 *n,le of force is α = tan ! *# ' /# & +

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 5

    Pro+lem

    * water trou,h tr α f, canal!comedero- of semi circular crosssection of radius &.5 m consists of two s(mmetric parts hin,edto each other at the +ottom. 4he two parts are held to,ether +(a ca+le and turn+uc$les *tensores+ placed e-er( 3 m alon, thelen,th of the trou,h. Calculate the tension in each ca+le when

    the trou,h is filled to the rim.1m

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow '

    6uo(anc( and Sta+ilit(

    6uo(anc( is due to the fluid displaced +( a+od(. # . = ρ f g' .

    Archimedes principal : 4he +uo(ant

    force actin, on a +od( immersed in a fluidis e9ual to the wei,ht of the fluid displaced+( the +od(! and it acts upward throu,h

    the centroid of the displaced -olume.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow

    4he olden Crown of 8iero ! in, ofS(racuse

    *rchimedes! ; 1 6.C.8iero! 3&' 15 6.C.8iero learned of a rumor wherethe ,oldsmith replaced some ofthe ,old in his crown with sil-er.8iero as$ed *rchimedes todetermine whether the crown was

    pure ,old. *rchimedes had to de-elop anondestructi-e testin, method

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow ;

    4he olden Crown of 8iero ! in, ofS(racuse

    4he wei,ht of the crown andnu,,et are the same in air: ) c = ρ c ' c = ) n = ρ n ' n.

    f the crown is pure ,old! ρ c % ρ n which means that the -olumes

    must +e the same! ' c =' n.n water! the +uo(anc( force is= ρ &2/ '.f the scale +ecomes un+alanced!

    this implies that the c G n ! which

    in turn means that the ρ c G ρ noldsmith was shown to +e a

    fraudH

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow ?

    6uo(anc( and Sta+ilit(

    6uo(anc( force # . is e9ualonl( to the displaced-olume ρ f g' displaced .4hree scenarios possi+le

    1. ρ body 0 ρ fluid : Floatin, +od(2. ρ body = ρ fluid : Neutrall( +uo(ant

    3. ρ body " ρ fluid 1 Sin$in, +od(

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3&

    Pro+lem

    4he densit( of a li9uid is to +edetermined +( an old 1 cm diameterc(lindrical h(drometer whose di-isionmar$s are completel( wiped out. 4heh(drometer is first dropped in water! andthe water le-el is mar$ed. 4heh(drometer is then dropped into theother li9uid! and it is o+ser-ed that themar$ for water has risen &.5 cm a+o-ethe li9uid air interface. f the hei,ht of thewater mar$ is 1& cm! determine the

    densit( of the li9uid.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 31

    Sta+ilit( of mmersed 6odies

    >otational sta+ilit( of immersed +odies depends uponrelati-e location of center of gravity and center of

    buoyancy . +elow : sta+le a+o-e : unsta+le or sta+le "it depends on the case# coincides with : neutrall( sta+le.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3

    Sta+ilit( of Floatin, 6odies

    f +od( is +ottom hea-(" lower than #! it isalwa(s sta+le.Floatin, +odies can +esta+le when is hi,herthan due to shift inlocation of center+uo(anc( and creation ofrestorin, moment.Measure of sta+ilit( is themetacentric hei,ht 3 . f

    3 I&! ship is sta+le.

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 33

    Pressure Force on a Fluid Element

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 30

    4here are special cases where a +od( of fluid can under,o ri,id+od( motion: linear acceleration! and rotation of a c(lindricalcontainer.

    n these cases! no shear is de-eloped.NewtonJs nd law of motion can +e used to deri-e an equation ofmotion for a fluid that acts as a ri,id +od(

    n Cartesian coordinates"up directed #:

    >i,id 6od( Motion

    ( ), , x y x P P P

    a a g a x y z

    ρ ρ ρ ∂ ∂ ∂= − = − = − +∂ ∂ ∂

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 35

    =inear *cceleration

    Particular case : container is mo-in, on a strai,ht path

    4otal differential of P

    Pressure difference +etween points

    Find the rise +( selectin, points on free surface P %P 1

    0, 0

    , 0,

    x y z

    x

    a a a

    P P P a g

    x y z ρ ρ

    ≠ = =∂ ∂ ∂= = = −∂ ∂ ∂

    xdP a dx gdz ρ ρ = − −

    ( ) ( )2 1 2 1 2 1 x P P a x x g z z ρ ρ − = − − − −

    ( )2 1 2 1 x s s sa

    z z z x x g ∆ = − = − −

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3'

    Pro+lem

    * fish tan$ 1' in +( in +( 10 in deep iscarried in a car which ma( eAperienceacceleration as hi,h as ' m/s . "a#

    *ssumin, ri,id +od( motion! estimate themaAimum water depth to a-oid spillin,."+#Khich is the +est wa( to ali,n the tan$@

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3

    4he tan$ of water is fulland open to theatmosphere of 15 psi atpoint *. For whathori ontal acceleration!in ft/s ! will the pressureat point 6 +e "a#atmospheric7 and "+#

    ero a+solute

    "ne,lectin, ca-itation#@

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3;

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    Chapter 3: Pressure and Fluid StaticsME33 : Fluid Flow 3?

    >otation in a C(lindrical Container

    Container is rotatin, a+out the aAis

    4otal differential of P

    )n an iso+ar! dP % &

    E9uation of the free surface " %h&L∆h! r%>#

    2

    2

    , 0

    , 0,

    r z a r a a

    P P P r g

    r z

    θ ω

    ρ ω ρ θ

    = − = =∂ ∂ ∂= = = −∂ ∂ ∂

    2dP r dr gdz ρ ω ρ = −

    2 22

    12isobar

    isobar

    dz r z r C dr g g

    ω ω = → = +

    ( )2

    2 20 24 s

    z h R r g

    ω = − −

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    Pro+lem

    * 1' cm diameter open c(linder cmhi,h is full of water. Find the central ri,id+od( rotation rate for which "a# one third of

    the water will spill out7 and "+# the +ottoncenter of the can will +e eAposed.