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Chapter 3 Pressure and Fluid Statics

Solutions Manual for

Essentials of Fluid Mechanics:

Fundamentals and Applications

by Cimbala & Çengel

CHAPTER 3PRESSURE AND FLUID STATICS

PROPRIETARY AND CONFIDENTIAL

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*-+




Pressure, Manometer, and Barometer

3,.C Solution e are to iscuss the ifference !etween "a"e pressure an a!solute pressure.

Analysis The pressure relative to the at(ospheri$ pressure is calle the gage pressure, an the pressure relativeto an absolute va$uu( is calle absolute pressure.

Discussion Most pressure "a"es (lie your !icycle tire "a"e) rea relatie to atmospheric pressure, an therefore reathe "a"e pressure.

3,/C Solution e are to e/plain nose !leein" an shortness of !reath at hi"h eleation.

Analysis 0tmospheric air pressure which is the e/ternal pressure e/erte on the sin ecreases with increasin"eleation. Therefore, the pressure is lo*er at higher elevations' As a result) the i##eren$e bet*een the bloo pressurein the veins an the air pressure outsie in$reases. This pressure i(balan$e (a! $ause so(e thin,*alle veins su$has the ones in the nose to burst) $ausing bleeing. The shortness of !reath is cause !y the lower air ensity at hi"her eleations, an thus lower amount of o/y"en per unit olume.

Discussion 1eople who clim! hi"h mountains lie Mt. 2erest suffer other physical pro!lems ue to the low pressure.

3,3C Solution e are to e/amine a claim a!out a!solute pressure.

Analysis No) the absolute pressure in a li0ui o# $onstant ensit! oes not ouble *hen the epth is ouble . Itis the gage pressure that ou!les when the epth is ou!le.

Discussion This is analo"ous to temperature scales 3 when performin" analysis usin" somethin" lie the ieal "aslaw, you must use a!solute temperature (4), not relatie temperature (oC), or you will run into the same in of pro!lem.

3,1C Solution e are to compare the pressure on the surfaces of a cu!e.

Analysis 5ince pressure increases with epth, the pressure on the botto( #a$e o# the $ube is higher than that onthe top' The pressure varies linearl! along the sie #a$es. Howeer, if the len"ths of the sies of the tiny cu!e suspenein water !y a strin" are ery small, the ma"nitues of the pressures on all sies of the cu!e are nearly the same.

Discussion In the limit of an “infinitesimal cu!e”, we hae a flui particle, with pressure P efine at a “point”.

3,2C Solution e are to efine 1ascal6s law an "ie an e/ample.

Analysis Pascal’s law states that the pressure applie to a $on#ine #lui in$reases the pressure throughout b!the sa(e a(ount. This is a conse7uence of the pressure in a flui remainin" constant in the hori8ontal irection. 0ne/ample of 1ascal6s principle is the operation of the hyraulic car 9ac.

Discussion The a!oe iscussion applies to fluis at rest (hyrostatics). hen fluis are in motion, 1ascal6s principleoes not necessarily apply. Howeer, as we shall see in later chapters, the ifferential e7uations of incompressi!le fluiflow contain only pressure gradients, an thus an increase in pressure in the whole system oes not affect flui motion.


*-&



Chapter 3 Pressure and Fluid Statics3,C Solution e are to compare the olume an mass flow rates of two fans at ifferent eleations.

Analysis The ensity of air at sea leel is hi"her than the ensity of air on top of a hi"h mountain. Therefore, theolume flow rates of the two fans runnin" at ientical spees will !e the same, !ut the mass flow rate of the fan at sealeel will !e hi"her.

Discussion In reality, the fan !laes on the hi"h mountain woul e/perience less frictional ra", an hence the fanmotor woul not hae as much resistance 3 the rotational spee of the fan on the mountain woul !e sli"htly hi"her than

that at sea leel.

3,4 Solution The pressure in a acuum cham!er is measure !y a acuum "a"e. Thea!solute pressure in the cham!er is to !e etermine.

Analysis The a!solute pressure in the cham!er is etermine from

kPa68=−=−= &:;&acatma!s P P P

Discussion e must remem!er that “acuum pressure” is the ne"atie of "a"e pressure 3 hence the ne"atie si"n.

3,5E Solution The pressure in a tan is measure with a manometer !y measurin" the ifferential hei"ht of themanometer flui. The a!solute pressure in the tan is to !e etermine for two cases$ the manometer arm with the (a)hi"her an (b) lower flui leel !ein" attache to the tan.

Assumptions The flui in the manometer is incompressi!le.

Properties The specific "raity of the flui is "ien to !e 5G < +.&=. The ensity of water at *&°> is ?&.: l!m@ft*.

Analysis The ensity of the flui is o!taine !y multiplyin" its specific "raity !y the ensity of water,

&

* *5G (+.&=)(?&.: l!m@ft ) A ' l!m@ft H O . ρ ρ = × = =

The pressure ifference corresponin" to a ifferential hei"ht of & in !etween the two arms of the manometer is

in+::

ft+

ft@sl!m*&.+A:

l!f +ft))(&@+&ft@s)(*&.+A:l!m@ft(A

&

&

&

&*

⋅==∆ gh P ρ

Then the a!solute pressures in the tan for the two cases !ecome$

(a) The flui leel in the arm attache to the tan is hi"her (acuum)$

a!s atm ac +& A + &? ++ :: psia P P P . . .= − = − = ≅ 11.4 sia

(b) The flui leel in the arm attache to the tan is lower$

a!s "a"e atm +& A + &? +* ;? psia P P P . . .= + = + = ≅ 14.! sia

Discussion The final results are reporte to three si"nificant i"its. Bote that we can etermine whether the pressure in a tan is a!oe or !elow atmospheric pressure !y simply o!serin" the sie of the manometer arm with the hi"her flui leel.


*-*

Pabs

Patm

= 92 kPa

24 kPa

0ir

5G< +.&=

P atm < +&.A psia

& in

P atm



Chapter 3 Pressure and Fluid Statics3,6 Solution The pressure in a pressuri8e water tan is measure !y a multi-flui manometer. The "a"e pressure of air in the tan is to !e etermine.

Assumptions The air pressure in the tan is uniform (i.e., its ariation with eleation is ne"li"i!le ue to its lowensity), an thus we can etermine the pressure at the air-water interface.

Properties The ensities of mercury, water, an oil are "ien to !e +*,?'', +''', an =' "@m*, respectiely.

Analysis 5tartin" with the pressure at point + at the air-water interface, an moin" alon" the tu!e !y ain" (as we

"o own) or su!tractin" (as we "o up) the gh ρ terms until we reach point &, an settin" the result e7ual to P atm since thetu!e is open to the atmosphere "ies

atm P gh gh gh P =−++ *mercury&oil+water + ρ ρ ρ

5olin" for P +,

*mercury&oil+water atm+ gh gh gh P P ρ ρ ρ +−−=

or,

)( &oil+water *mercuryatm+ hhh g P P ρ ρ ρ −−=−

Botin" that P +,"a"e < P + - P atm an su!stitutin",

kPa"6.#=

⋅

−=

&&

*

**&,+

B@m+'''

1a+

m@s"+

B+m)C*.')("@m(='-

m)&.')("@m(+'''m):?.')("@m)D(+*,?''m@s(;.+ gage P

Discussion Bote that 9umpin" hori8ontally from one tu!e to the ne/t an reali8in" that pressure remains the same inthe same flui simplifies the analysis "reatly.

3,.7

Solution The !arometric reain" at a location is "ien in hei"ht of mercury column. The atmospheric pressure is to !e etermine.

Properties The ensity of mercury is "ien to !e +*,?'' "@m*.

Analysis The atmospheric pressure is etermine irectly from

( ) ( ) ( )* &

& &

+ B + 1a+*,?'' "@m ; + m@s ' A=' m

+ " m@s +''' B@m

+'' + 1a

atm P gh . .

.

ρ = = ÷ ÷×

= ≅ 1!! kPa

Discussion e roun off the final answer to three si"nificant i"its. +'' 1a is a fairly typical alue of atmospheric pressure on lan sli"htly a!oe sea leel.


*-:

ater

h+

0ir +

h*

h&



Chapter 3 Pressure and Fluid Statics3,.. Solution The "a"e pressure in a li7ui at a certain epth is "ien. The "a"e pressure in the same li7ui at a ifferentepth is to !e etermine.

Assumptions The ariation of the ensity of the li7ui with epth is ne"li"i!le.

Analysis The "a"e pressure at two ifferent epths of a li7ui can !e e/presse as ++ gh P ρ = an && gh P ρ = .

Tain" their ratio,

+

&

+

&

+

&

h

h

gh

gh

P

P

== ρ

ρ

5olin" for P & an su!stitutin" "ies

&Pa ../=== 1a)&(m*

m+&+

+

&& P

h

h P

Discussion Bote that the "a"e pressure in a "ien flui is proportional to epth.

3,./ Solution The a!solute pressure in water at a specifie epth is "ien. The local atmospheric pressure an thea!solute pressure at the same epth in a ifferent li7ui are to !e etermine.

Assumptions The li7ui an water are incompressi!le.

Properties The specific "raity of the flui is "ien to !e 5G < '.=. e tae the ensity of water to !e +''' "@m *.Then ensity of the li7ui is o!taine !y multiplyin" its specific "raity !y the ensity of water,

** "@m=')"@m'('.=)(+''5G&

==×= O H ρ ρ

Analysis (a) 4nowin" the a!solute pressure, the atmospheric pressure can !e etermine from

* &

&

+ 1a(+:= 1a) (+''' "@m )(;.+ m@s )(= m)

+''' B@m

atm P P gh ρ = −

= − = ÷

6'7 &Pa

(b) The a!solute pressure at a epth of = m in the other li7ui is

* &

&

+1a(;?.' 1a) (=' "@m )(;.+ m@s )(= m)

+''' B@m+*A A 1a

atm P P gh

.

ρ = +

= + ÷ = ≅ .35 &Pa

Discussion Bote that at a "ien epth, the pressure in the li"hter flui is lower, as e/pecte.

3,.3E Solution It is to !e shown that + "f@cm& < +:.&&* psi.

Analysis Botin" that + "f < ;.'??= B, + B < '.&&:+ l!f, an + in < &.=: cm, we hae

l!f &':?*.& B+

l!f '.&&:+) B;.'??=( B;.'??="f +

=

==an

si14.$$%==

== &

&

&&& l!f@in&&*.+:in+

cm&.=:)l!f@cm&':?*.&(l!f@cm&':?*.&"f@cm+

Discussion This relationship may !e use as a conersion factor.


*-=

h+

+ h&

&

P atm

h

P



Chapter 3 Pressure and Fluid Statics3,.1E Solution The wei"ht an the foot imprint area of a person are "ien. The pressures this man e/erts on the "rounwhen he stans on one an on !oth feet are to !e etermine.

Assumptions The wei"ht of the person is istri!ute uniformly on foot imprint area.

Analysis The wei"ht of the man is "ien to !e &'' l!f. Botin" that pressure is force per unit area, the pressure this man e/erts on the "roun is

(a) En one foot$ si"."6==== l!f@in=?.=in*?

l!f &'' &

& A

W

P

(a) En !oth feet$ si$.&8==×

== l!f@inA.&in*?&

l!f &''

&

&

& A

W P

Discussion Bote that the pressure e/erte on the "roun (an on the feet) is reuce !y half when the person stans on !oth feet.

3,.2 Solution The mass of a woman is "ien. The minimum imprint area per shoe neee to ena!le her to wal on thesnow without sinin" is to !e etermine.

Assumptions . The wei"ht of the person is istri!ute uniformly on the imprint area of the shoes. / Ene foot carries theentire wei"ht of a person urin" walin", an the shoe is si8e for walin" conitions (rather than stanin"). 3 Thewei"ht of the shoes is ne"li"i!le.

Analysis The mass of the woman is "ien to !e A' ". >or a pressure of '.= 1a on thesnow, the imprint area of one shoe must !e

$m1.%&=

⋅===

&&

&

B@m+'''

1a+

m@s"+

B+

1a'.=

)m@s")(;.+(A'

P

mg

P

W A

Discussion This is a ery lar"e area for a shoe, an such shoes woul !e impractical to use. Therefore, some sinin" of the snow shoul !e allowe to hae shoes of reasona!le si8e.

3,. Solution The acuum pressure reain" of a tan is "ien. The a!solute pressure in the tan is to !e etermine.

Properties The ensity of mercury is "ien to !e ρ < +*,=;' "@m*.

Analysis The atmospheric (or !arometric) pressure can !e e/presse as

-1a+''.? B@m+'''

-1a+

m@s-"+

B+m))('.A==m@s)(;.'A-"@m(+*,=;'

&&

&*

=

⋅=

= h g Patm ρ

Then the a!solute pressure in the tan !ecomes

kPa &!.6=−=−= *'+''.?vacatmabs P P P


*-?

Pabs

Patm

= 755mmHg

30kPa




Discussion The "a"e pressure in the tan is the ne"atie of the acuum pressure, i.e., P "a"e < −*'.' 1a.


*-A



Chapter 3 Pressure and Fluid Statics3,.4E Solution 0 pressure "a"e connecte to a tan reas =' psi. The a!solute pressure in the tan is to !e etermine.

Properties The ensity of mercury is "ien to !e ρ < :.: l!m@ft*.

Analysis The atmospheric (or !arometric) pressure can !e e/presse as

&* &

& &

+l!f +ft(:.: l!m@ft )(*&.+A: ft@s )(&;.+@+& ft)

*&.+A: l!m ft@s +:: in+:.&; psia

atm P g h ρ =

= ÷ ÷×

=Then the a!solute pressure in the tan is

=' +:.&; ?: &; psiaabs gage atm P P P .= + = + = ≅ 64.% sia

Discussion This pressure is more than four times as much as stanar atmospheric pressure.

3,.5 Solution 0 pressure "a"e connecte to a tan reas ='' 1a. The a!solute

pressure in the tan is to !e etermine.

Analysis The a!solute pressure in the tan is etermine from

kPa"#4=+=+= ;:=''atm"a"ea!s P P P

Discussion This pressure is almost si/ times "reater than stanar atmospheric pressure.

3,.6 Solution 0 mountain hier recors the !arometric reain" !efore an after a hiin" trip. The ertical istanceclim!e is to !e etermine.

Assumptions The ariation of air ensity an the "raitational acceleration withaltitue is ne"li"i!le.

Properties The ensity of air is "ien to !e ρ < +.&' "@m*.

Analysis Tain" an air column !etween the top an the !ottom of the mountainan writin" a force !alance per unit !ase area, we o!tain

!ar '.A')('.;*' B@m+'','''

!ar +

m@s"+

B+))(m@s)(;.+"@m(+.&'

)(

@

&&

&*

top !ottomair

top !ottomair

−=

⋅

−=

−=

h

P P gh

P P AW

ρ

It yiels h < +&A: m ./47 ( (to * si"nificant i"its), which is also the istance clim!e.

Discussion 0 similar principle is use in some aircraft instruments to measure eleation.


*-

h = ?

70 mba!

930 mba!

Pabs

50 "s#a

Pabs

Patm

= 94 kPa

500 kPa



Chapter 3 Pressure and Fluid Statics3,/7 Solution 0 !arometer is use to measure the hei"ht of a !uilin" !y recorin" reain" at the !ottom an at the topof the !uilin". The hei"ht of the !uilin" is to !e etermine.

Assumptions The ariation of air ensity with altitue is ne"li"i!le.

Properties The ensity of air is "ien to !e ρ < +.+ "@m*. The ensity of mercury is +*,?'' "@m *.

Analysis 0tmospheric pressures at the top an at the !ottom of the !uilin" are

top top

* &

& &

!ottom !ottom

* &

& &

( ) + B + 1a(+*,?'' "@m )(;.'A m@s )('.A*' m)

+ " m@s +''' B@m;A.*? 1a

+ B + 1a(+*,?'' "@m )(;.'A m@s )('.A== m)

+ " m@s +''' B@m+''.A' 1a

Pρ g h

P ( g h ) ρ

= = ÷ ÷× ==

= ÷ ÷× =

Tain" an air column !etween the top an the !ottom of the !uilin", we write a force !alance per unit !ase area,

air !ottom top air !ottom top

* &

& &

an

+ B + 1a

(+.+ "@m )(;.'A m@s )( ) (+''.A' ;A.*?) 1a+ " m@s +''' B@m

W A P P ( gh ) P P

h

ρ = − = −

= − ÷ ÷×

which yiels h < &.? m ≅ /56 (, which is also the hei"ht of the !uilin".

Discussion There are more accurate ways to measure the hei"ht of a !uilin", !ut this metho is 7uite simple.

3,/.

Solution The preious pro!lem is reconsiere. The 225 solution is to !e printe out, incluin" proper units.

Analysis The 225 !"uations winow is printe !elow, followe !y the #olution winow.

P_bottom=755"[mmHg]"P_top=730"[mmHg]"g=9.807 "[m/s^2]" "local acceleratio o! grait# at sea leel"r$o=%.%8"[&g/m^3]" '()*+P_abs=,P_bottom-P_top1(4*,mmHg6&Pa "[&Pa]" "'elta P reaig !rom t$ebarometers6 coerte !rom mmHg to &Pa."'()*+P_$ =r$og$/%000 "[&Pa]" "(. %-%:. 'elta P e to t$e air ;i colm$eig$t6 $6 bet<ee t$e top a bottom o! t$e bilig.""stea o! iiig b# %000 Pa/&Pa <e col $ae mltiplie r$og$ b# t$e ((> !ctio61(4*,Pa6&Pa"'()*+P_abs='()*+P_$

SOLUTION

ariables i ?ai'()*+P_abs=3.333 [&Pa] '()*+P_$=3.333 [&Pa]g=9.807 [m/s^2] $=288 [m]P_bottom=755 [mmHg] P_top=730 [mmHg]r$o=%.%8 [&g/m^3]

Discussion To o!tain the solution in 225, simply clic on the icon that loos lie a calculator, or Calculate-5ole.


*-;

730 mmHg

755 mmHg

h




3,// Solution 0 ier is moin" at a specifie epth from the water surface. The pressure e/erte on the surface of theier !y the water is to !e etermine.

Assumptions The ariation of the ensity of water with epth is ne"li"i!le.

Properties The specific "raity of sea water is "ien to !e 5G < +.'*. e tae the ensity of water to !e +''' "@m *.

Analysis The ensity of the sea water is o!taine !y multiplyin" its specific "raity !y the ensity of water which is taen to !e +''' "@m*$

&

* *5G (+.'*)(+''' "@m ) +'*' "@m H O ρ ρ = × = =

The pressure e/erte on a ier at *' m !elow the free surface of the sea isthe a!solute pressure at that location$

* &

&

+1a(+'+1a) (+'*' "@m )(;.'A m@s )(*' m)

+''' B@m

atm P P gh ρ = +

= + ÷ = 4!4 kPa

Discussion This is a!out : times the normal sea leel alue of atmospheric pressure.

3,/3E Solution 0 su!marine is cruisin" at a specifie epth from the water surface. The pressure e/erte on the surface of the su!marine !y water is to !eetermine.

Assumptions The ariation of the ensity of water with epth is ne"li"i!le.

Properties The specific "raity of sea water is "ien to !e 5G < +.'*. The

ensity of water at *&°> is ?&.: l!m@ft*.

Analysis The ensity of the seawater is o!taine !y multiplyin" its specific"raity !y the ensity of water,

** l!m@ft?:.&A)l!m@ft:(+.'*)(?&.5G& ==×= O H ρ ρ

The pressure e/erte on the surface of the su!marine cruisin" *'' ft !elow the free surface of the sea is the a!solute pressure at that location$

&* &

& &

+ l!f + ft(+:.A psia) (?:.&A l!m@ft )(*&.+A: ft@s )(*'' ft)

*&.+A: l!m ft@s +:: in+: ? psia

atm P P gh

.

ρ = +

= + ÷ ÷× = ≅ 14# sia

where we hae roune the final answer to three si"nificant i"its.

Discussion This is more than +' times the alue of atmospheric pressure at sea leel.


*-+'

P atm

5eah

P

P atm

5eah

P



Chapter 3 Pressure and Fluid Statics3,/1 Solution 0 "as containe in a ertical piston-cyliner eice is pressuri8e !y a sprin" an !y the wei"ht of the

piston. The pressure of the "as is to !e etermine.

Analysis Frawin" the free !oy ia"ram of the piston an !alancin" the ertical forces yiels

PA P A W F atm spring= + +Thus,

sprin"

atm

&

: & &

(: ")(;.'A m@s ) ?' B + 1a(;= 1a) +&* : 1a

*= +' m +''' B@m

mg $ P P

A

.−

+= +

+= + = ≅ ÷×

1$% kPa

Discussion This setup represents a crue !ut functional way to control the pressure in a tan.

3,/2

Solution The preious pro!lem is reconsiere. The effect of the sprin" force in the ran"e of ' to ='' B on the pressure insie the cyliner is to !e inesti"ate. The pressure a"ainst the sprin" force is to !e plotte, an results are to !e iscusse.

Analysis The 225 !"uations winow is printe !elow, followe !y the ta!ulate an plotte results.

g=9.807"[m/s^2]"P_atm= 95"[&Pa]"m_pisto=@"[&g]"AB_sprig=:0"[]"C+=351(4*,cm^26m^2"[m^2]"D_pisto=m_pistog"[]"B_atm=P_atm+1(4*,&Pa6/m^2"[]""Brom t$e !ree bo# iagram o! t$e pisto6 t$e balacig ertical !orces #ielE"B_gas= B_atmFB_sprigFD_pisto"[]"P_gas=B_gas/+1(4*,/m^26&Pa"[&Pa]"

%esults$

Fspring [N] Pgas [kPa]

0 %0:.255.5: %22.%%%%.% %38%::.7 %53.8222.2 %:9.7277.8 %85.:333.3 20%[email protected] 2%7.3@@@.@ 233.2

500 2@9.%

Discussion The relationship is linear, as e/pecte.


*-++

$ sprin"

P atm

P

W < mg

! 1!! $!! %!! 4!! "!!1!!

1$!

14!

16!

18!

$!!

$$!

$4!

$6!

'sring

()*

P g a s

( k P a *



Chapter 3 Pressure and Fluid Statics3,/ 8 Also solved using !!# on enclosed &'&9

Solution #oth a pressure "a"e an a manometer are attache to a tan of "as to measure its pressure. >or a specifiereain" of "a"e pressure, the ifference !etween the flui leels of the two arms of the manometer is to !e etermine for mercury an water.

Properties The ensities of water an mercury are "ien to !e ρ water < +''' "@m* an !e ρ H" < +*,?'' "@m*.

Analysis The "a"e pressure is relate to the ertical istance h !etween the two flui leels !y

"a"e

"a"e

P

P g h h g ρ ρ = → =(a) >or mercury,

m!.6!=

⋅

==

B+

s"@m+'''

1a+

B@m+

)m@s)(;.'A"@m(+*?''

1a' &&

&*

"a"e

g

P h

Hg ρ

(b) >or water,

m8.16=

⋅

==

B+

s"@m+'''

1a+

B@m+

)m@s)(;.'A"@m(+'''

1a' &&

&*

"a"e

& g

P h

O H ρ

Discussion The manometer with water is more precise since the column hei"ht is !i""er (!etter resolution). Howeer,a column of water more than meters hi"h woul !e impractical, so mercury is the !etter choice of manometer flui here.


*-+&

0I

h

0 kPa




3,/4

Solution The preious pro!lem is reconsiere. The effect of the manometer flui ensity in the ran"e of '' to+*,''' "@m* on the ifferential flui hei"ht of the manometer is to !e inesti"ate. Fifferential flui hei"ht is to !e

plotte as a function of the ensity, an the results are to !e iscusse.


Bctio ;i_esit#,BliG! ;iG=?ercr# t$e ;i_esit#=%3:00 else ;i_esit#=%000

e

Apt !rom t$e iagram <io<. ! t$e iagram <io< is $ie6 t$e all o! t$e ipt mstcome !rom t$eeatios <io<. +lso ote t$at brac&ets ca also eote commets - bt t$ese commetso ot appear i t$e !ormatte eatios <io<.C

ABliG=?ercr#P_atm = %0%.325 "&pa"'()*+P=80 "&Pa ote $o< '()*+P is ispla#e o t$e Bormatte (atiosDio<."C

g=9.807 "m/s26 local acceleratio o! grait# at sea leel"r$o=Bli_esit#,BliG "et t$e ;i esit#6 eit$er Hg or H216 !rom t$e !ctio""*o plot ;i $eig$t agaist esit# place AC aro t$e aboe eatio. *$e set p t$eparametric table a sole." '()*+P = 4H1g$/%000"stea o! iiig b# %000 Pa/&Pa <e col $ae mltiplie b# t$e ((> !ctio61(4*,Pa6&Pa"$_mm=$coert,m6mm "*$e ;i $eig$t i mm is !o sig t$e bilt-i 1(4*!ctio."P_abs= P_atm F '()*+P

"*o ma&e t$e grap$6 $ie t$e iagram <io< a remoe t$e ACbrac&ets !rom BliG a

!rom P_atm. >elect e< Parametric *able !rom t$e *ables me. $oose P_abs6 '()*+P a$ to be i t$e table. $oose +lter ales !rom t$e *ables me. >et ales o! $ to rage!rom 0 to % i steps o! 0.2. $oose >ole *able ,or press B3 !rom t$e alclate me.$oose e< Plot Dio< !rom t$e Plot me. $oose to plot P_abs s $ a t$e c$oose1erla# Plot !rom t$e Plot me a plot '()*+P o t$e same scale."

%esults$

hmm

[mm]

[kg/m3]

%0%97 800378@ 2%5:2323 35%%

%:7: @8:7%3%% :222%07: 75789%3.% 8933792.8 %0289700.5 %%:@@:27.5 %3000


*-+*0$00 0$20 0$40 0$%0 0$0 &$000

20

40

%0

0

&00

&20

&40

&%0

&0

200

220

240

Manometer 'luid +eigt, m

P r e s s u r e ,

k P a

Ta'k F()#* +ag, a'* Abs-()t, P!,ss)!,s .s /a'-m,t,! F()#* H,#gt

Abs-()t, P!,ss)!,

+ag, P!,ss)!,

/a'-m,t,! F()#*1 /,!)!




Discussion Many comments are proie in the 27uation winow a!oe to help you learn some of the features of 225.

3,/5 Solution The air pressure in a tan is measure !y an oil manometer. >or a "ien oil-leel ifference !etween the two columns, the a!solute pressure in thetan is to !e etermine.

Properties The ensity of oil is "ien to !e ρ < =' "@m*.

Analysis The a!solute pressure in the tan is etermine from

* &

&

+ 1a(; 1a) (=' "@m )(;.+ m@s )('.:=m)

+''' B@m+'+ A= 1a

atm P P gh

.

ρ = +

= + ÷ = ≅ 1!$ kPa

Discussion If a heaier li7ui, such as water, were use for the manometer flui, the column hei"ht woul !e smaller,an thus the reain" woul !e less precise (lower resolution).


*-+:

0I

P atm < ; 1a

'.:= m

! $!!! 4!!! 6!!! 8!!! 1!!!! 1$!!! 14!!!!

$$!!

44!!

66!!

88!!

11!!!

ρ (kg-m%*

m m

( m m *

Manometer 'luid +eigt /s Manometer 'luid 0ensity



Chapter 3 Pressure and Fluid Statics3,/6 Solution The air pressure in a uct is measure !y a mercury manometer. >or a "ien mercury-leel ifference

!etween the two columns, the a!solute pressure in the uct is to !e etermine.

Properties The ensity of mercury is "ien to !e ρ < +*,?'' "@m*.

Analysis (a) The pressure in the uct is a!oe atmospheric pressure since theflui column on the uct sie is at a lower leel.

(b) The a!solute pressure in the uct is etermine from

* &

& &

+ B + 1a(+'' 1a) (+*,?'' "@m )(;.+ m@s )('.'+= m)

+ " m@s +''' B@m+'&.'' 1a

atm P P gh ρ = +

= + ÷ ÷× = ≅ 1!$ kPa

Discussion hen measurin" pressures in a flui flow, the dierence !etween two pressures is usually esire. In thiscase, the ifference is !etween the measurement point an atmospheric pressure.

3,37 Solution The air pressure in a uct is measure !y a mercury manometer. >or a "ien mercury-leel ifference

!etween the two columns, the a!solute pressure in the uct is to !e etermine.

Properties The ensity of mercury is "ien to !e ρ < +*,?'' "@m*.

Analysis (a) The pressure in the uct is a!oe atmospheric pressure since the flui column on the uct sie is at alower leel.

(b) The a!solute pressure in the uct is etermine from

* &

& &

+ B + 1a

(+'' 1a) (+*,?'' "@m )(;.+ m@s )('.'*' m) + " m@s +''' B@m+':.'' 1a

atm P P gh ρ = +

= + ÷ ÷× = ≅ 1!4 kPa

Discussion The final result is "ien to three si"nificant i"its.


*-+=

0ir

P

+= mm



Chapter 3 Pressure and Fluid Statics3,3.

Solution The systolic an iastolic pressures of a healthy person are "ien in mm of H". These pressures are to !ee/presse in 1a, psi, an meters of water column.

Assumptions #oth mercury an water are incompressi!le su!stances.

Properties e tae the ensities of water an mercury to !e +''' "@m* an +*,?'' "@m*, respectiely.

Analysis sin" the relation gh P ρ = for "a"e pressure, the hi"h an low pressures are e/presse as

1=

⋅==

=

⋅==

&&

&*lowlow

&&

&*hi"hhi"h

B@m+'''

1a+

m@s"+

B+m))('.'m@s)(;.+"@m(+*,?''

B@m'''+

1a+

m@s"+

B+m))('.+&m@s)(;.+"@m(+*,?''

gh P

gh P

ρ

ρ

Botin" that + psi < ?.;= 1a,

si$.%$=

=

-1a?.;=

psi+-1a) '.(+?hi"h P an

si1.""=

=

-1a?.;=

psi+1a)- (+'.Alow P

>or a "ien pressure, the relation gh P ρ = is e/presse for mercury an water as

water water gh P ρ = an mercurymercury gh P ρ = . 5ettin" these two relations e7ual to each other ansolin" for water hei"ht "ies

mercury

water

mercury

water mercurymercurywater water hh gh gh P ρ

ρ ρ ρ =→==

Therefore,

m 1.!#

m 1.6%

===

===

m)'.'("@m+'''

"@m?'',+*

m)+&.'("@m+'''

"@m?'',+*

*

*

lowmercury,

water

mercury

lowwater,

*

*

hi"hmercury,

water

mercury

hi"hwater,

hh

hh

ρ

ρ

ρ

ρ

Discussion Bote that measurin" !loo pressure with a water monometer woul inole water column hei"hts hi"her

than the person6s hei"ht, an thus it is impractical. This pro!lem shows why mercury is a suita!le flui for !loo pressuremeasurement eices.

3,3/ Solution 0 ertical tu!e open to the atmosphere is connecte to the ein in the arm of a person. The hei"ht that the

!loo rises in the tu!e is to !e etermine.

Assumptions . The ensity of !loo is constant. / The "a"e pressure of !loo is +&' mmH".

Properties The ensity of !loo is "ien to !e ρ < +'=' "@m*.

Analysis >or a "ien "a"e pressure, the relation gh P ρ = can !e e/presse for

mercury an !loo as !loo !loo gh P ρ = an mercurymercury gh P ρ = . 5ettin" these

two relations e7ual to each other we "et

mercurymercury !loo !loo gh gh P ρ ρ ==

5olin" for !loo hei"ht an su!stitutin" "ies

m1.""=== m)+&.'("@m+'='

"@m?'',+**

*

mercury

!loo

mercury

!loo hh ρ

ρ

Discussion Bote that the !loo can rise a!out one an a half meters in a tu!e connecte to the ein. This e/plains whyI tu!es must !e place hi"h to force a flui into the ein of a patient.


*-+?

h#loo

h



Chapter 3 Pressure and Fluid Statics3,33 Solution 0 man is stanin" in water ertically while !ein" completely su!mer"e. The ifference !etween the

pressure actin" on his hea an the pressure actin" on his toes is to !e etermine.

Assumptions ater is an incompressi!le su!stance, an thus the ensity oes not chan"e with epth.

Properties e tae the ensity of water to !e ρ <+''' "@m*.

Analysis The pressures at the hea an toes of the person can !e e/presse as

heaatmhea gh P P ρ

+= an toeatmtoe gh P P ρ

+=where h is the ertical istance of the location in water from the free surface. The pressureifference !etween the toes an the hea is etermine !y su!tractin" the first relationa!oe from the secon,

)( heatoeheatoeheatoe hh g gh gh P P −=−=− ρ ρ ρ

5u!stitutin",

1&.&=

⋅=−

&&

&*heatoe

B@m+'''

1a+

m@s"+

B+')-m)(+.'m@s)(;.+"@m(+''' P P

Discussion This pro!lem can also !e sole !y notin" that the atmospheric pressure (+ atm < +'+.*&= 1a) is

e7uialent to +'.*-m of water hei"ht, an finin" the pressure that correspons to a water hei"ht of +. m.

3,31 Solution ater is poure into the -tu!e from one arm an oil from the other arm. The water column hei"ht in onearm an the ratio of the hei"hts of the two fluis in the other arm are "ien. The hei"ht of each flui in that arm is to !eetermine.

Assumptions #oth water an oil are incompressi!le su!stances.

Properties The ensity of oil is "ien to !e ρ oil < A;' "@m*. e tae the ensity of water to !e ρ w <+''' "@m*.

Analysis The hei"ht of water column in the left arm of the manometer is "ien to !e hw+ < '.A' m. e let the hei"ht

of water an oil in the ri"ht arm to !e hw& an ha, respectiely. Then, ha < ?hw&. Botin" that !oth arms are open to theatmosphere, the pressure at the !ottom of the -tu!e can !e e/presse as

w+watm !ottom gh P P ρ += an aaw&watm !ottom gh gh P P ρ ρ ++=

5ettin" them e7ual to each other an simplifyin",

aaw&w+aaw&ww+waaw&ww+w )@( hhhhhh gh gh gh w ρ ρ ρ ρ ρ ρ ρ ρ +=→+=→+=

Botin" that ha < ?hw& an we tae ρ a < ρ oil, the water an oil column hei"hts in the

secon arm are etermine to !e

m!.1$$=→+= &&& ?(A;'@+''')m'.A www hhh

m!.&%$=→+= aa hh (A;'@+''')m+&&.'m'.A

Discussion Bote that the flui hei"ht in the arm that contains oil is hi"her. This ise/pecte since oil is li"hter than water.


*-+A

ha

hw&

hw+

oilater

hhea

htoe



Chapter 3 Pressure and Fluid Statics3,32 Solution The hyraulic lift in a car repair shop is to lift cars. The flui "a"e pressure that must !e maintaine in thereseroir is to !e etermine.

Assumptions The wei"ht of the piston of the lift is ne"li"i!le.

Analysis 1ressure is force per unit area, an thus the "a"e pressure re7uire is simply theratio of the wei"ht of the car to the area of the lift,

kPa$&8==

⋅===

&

&&

&

&"a"e B@m&Am@s"+'''

B+

:@m)*'.'(

)m@s")(;.+&'''(

:@ π π &

mg

A

W

P

Discussion Bote that the pressure leel in the reseroir can !e reuce !y usin" a piston with a lar"er area.

3,3 Solution >resh an seawater flowin" in parallel hori8ontal pipelines are connecte to each other !y a ou!le -tu!emanometer. The pressure ifference !etween the two pipelines is to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / The effect

of air column on pressure is ne"li"i!le. Properties The ensities of seawater an mercury are "ien to

!e ρ sea < +'*= "@m* an ρ H" < +*,?'' "@m*. e tae the ensity

of water to !e ρ w <+''' "@m*.

Analysis 5tartin" with the pressure in the fresh water pipe(point +) an moin" alon" the tu!e !y ain" (as we "o own) or

su!tractin" (as we "o up) the gh ρ terms until we reach the sea

water pipe (point &), an settin" the result e7ual to P & "ies

&seaseaair air H"H"w+ P gh gh gh gh P w =+−−+ ρ ρ ρ ρ

earran"in" an ne"lectin" the effect of air column on pressure,

)( seaseawH"H"seaseaH"H"w&+ hhh g gh gh gh P P ww ρ ρ ρ ρ ρ ρ −−=−+−=−

5u!stitutin",

kPa%.%#==

⋅−−

=−

&

&

**

*&&+

B@m*;.*

m@s"+'''

B+m)C:.')("@m(+'*=m)?.')("@m(+'''

m)+.')("@m)D(+*?''m@s(;.+ P P

Therefore, the pressure in the fresh water pipe is *.*; 1a hi"her than the pressure in the sea water pipe.

Discussion 0 '.A'-m hi"h air column with a ensity of +.& "@m* correspons to a pressure ifference of '.'' 1a.

Therefore, its effect on the pressure ifference !etween the two pipes is ne"li"i!le.


*-+

W < mg

P atm

P

>reshwater

hsea

hair

5eawater

Mercury

0ir

hH"

hw



Chapter 3 Pressure and Fluid Statics3,34 Solution >resh an seawater flowin" in parallel hori8ontal pipelines are connecte to each other !y a ou!le -tu!emanometer. The pressure ifference !etween the two pipelines is to !e etermine.

Assumptions 0ll the li7uis are incompressi!le.

Properties The ensities of seawater an mercury are "ien to !e ρ sea < +'*= "@m* an ρ H" < +*,?'' "@m*. e tae

the ensity of water to !e ρ w <+''' "@m*. The specific "raity of oil is "ien to !e '.A&, an thus its ensity is A&' "@m*.

Analysis 5tartin" with the pressure in the fresh water pipe (point +) an moin" alon" the tu!e !y ain" (as we "o

own) or su!tractin" (as we "o up) the gh ρ terms until we reach the sea water pipe (point &), an settin" the resulte7ual to P & "ies

&seaseaoiloilH"H"w+ P gh gh gh gh P w =+−−+ ρ ρ ρ ρ

earran"in",

)( seaseawoiloilH"H"

seaseaoiloilH"H"w&+

hhhh g

gh gh gh gh P P

w

w

ρ ρ ρ ρ

ρ ρ ρ ρ

−−+=

−++−=−

5u!stitutin",

kPa8.%4==

⋅−

−+=−

&

&

*

***&&+

B@m*:.

m@s"+'''

B+m)C:.')("@m(+'*=

m)?.')("@m(+'''m)A.')("@m(A&'m)+.')("@m)D(+*?''m@s(;.+ P P

Therefore, the pressure in the fresh water pipe is .*: 1a hi"her than the pressure in the sea water pipe.

Discussion The result is "reater than that of the preious pro!lem since the oil is heaier than the air.


*-+;

>reshwater

hw

hsea

hoil

5eawater

Mercury

Eil

hH"



Chapter 3 Pressure and Fluid Statics3,35E Solution The pressure in a natural "as pipeline is measure !y a ou!le -tu!e manometer with one of the armsopen to the atmosphere. The a!solute pressure in the pipeline is to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / The effect of air column on pressure is ne"li"i!le. 3 The pressurethrou"hout the natural "as (incluin" the tu!e) is uniform since its ensity is low.

Properties e tae the ensity of water to !e ρ w < ?&.: l!m@ft*. The specific "raity of mercury is "ien to !e +*.?,

an thus its ensity is ρ H" < +*.?×?&.: < :.? l!m@ft*.

Analysis 5tartin" with the pressure at point + in the natural "as pipeline, an moin" alon" the tu!e !y ain" (aswe "o own) or su!tractin" (as we "o up) the gh ρ terms until we reach the free surface of oil where the oil tu!e is

e/pose to the atmosphere, an settin" the result e7ual to P atm "ies

atm P gh gh P =−− water water H"H"+ ρ ρ

5olin" for P +,

+water H"H"atm+ gh gh P P ρ ρ ++=

5u!stitutin",

sia18.1=

++= **&

*&.

ft)C)(&A@+&l!m@ft(?&.:ft))(?@+&l!m@ft)D(:.?ft@s&.*&( psia+:.& P

Discussion Bote that 9umpin" hori8ontally from one tu!e to the ne/t an reali8in" that pressure remains the same inthe same flui simplifies the analysis "reatly. 0lso, it can !e shown that the +=-in hi"h air column with a ensity of '.'A=l!m@ft* correspons to a pressure ifference of '.'''?= psi. Therefore, its effect on the pressure ifference !etween the two

pipes is ne"li"i!le.


*-&'

hw

Batural"as

hH"

+'in

Mercury

ater

0ir



Chapter 3 Pressure and Fluid Statics3,36E Solution The pressure in a natural "as pipeline is measure !y a ou!le -tu!e manometer with one of the armsopen to the atmosphere. The a!solute pressure in the pipeline is to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / The pressure throu"hout the natural "as (incluin" the tu!e) isuniform since its ensity is low.

Properties e tae the ensity of water to !e ρ w < ?&.: l!m@ft*. The specific "raity of mercury is "ien to !e +*.?,

an thus its ensity is ρ H" < +*.?×?&.: < :.? l!m@ft*. The specific "raity of oil is "ien to !e '.?;, an thus its ensity is

ρ oil < '.?;×

?&.: < :*.+ l!m@ft*.

Analysis 5tartin" with the pressure at point + in the natural "as pipeline, an moin" alon" the tu!e !y ain" (as

we "o own) or su!tractin" (as we "o up) the gh ρ terms until we reach the free surface of oil where the oil tu!e is

e/pose to the atmosphere, an settin" the result e7ual to P atm "ies

atm P gh gh gh P =−+− water water oiloilH"H"+ ρ ρ ρ

5olin" for P +,

oiloil+water H"H"atm+ gh gh gh P P ρ ρ ρ −++=

5u!stitutin",

sia1&.&=

⋅−

++=

&

&

&

*

**&

+

in+::

ft+

ft@sl!m*&.&

l!f +ft)C)(+=@+&l!m@ft(:*.+

ft))(&A@+&l!m@ft(?&.:ft))(?@+&l!m@ft)D(:.?ft@s&.*&( psia:.&+ P



*-&+

hw

Batural"as

hH"

Mercury

ater

Eil

hoil



Chapter 3 Pressure and Fluid Statics3,17 Solution The "a"e pressure of air in a pressuri8e water tan is measure simultaneously !y !oth a pressure "a"ean a manometer. The ifferential hei"ht h of the mercury column is to !e etermine.

Assumptions The air pressure in the tan is uniform (i.e., its ariation with eleation is ne"li"i!le ue to its lowensity), an thus the pressure at the air-water interface is the same as the inicate "a"e pressure.

Properties e tae the ensity of water to !e ρ w <+''' "@m*. The specific "raities of oil an mercury are "ien to !e

'.A& an +*.?, respectiely.

Analysis 5tartin" with the pressure of air in the tan (point +), an moin" alon" the tu!e !y ain" (as we "oown) or su!tractin" (as we "o up) the gh ρ terms until we reach the free surface of oil where the oil tu!e is e/pose to

the atmosphere, an settin" the result e7ual to P atm "ies

atmw P gh gh gh P =−−+ oiloilH"H"w+ ρ ρ ρ

earran"in" ,

w gh gh gh P P wH"H"oiloilatm+ ρ ρ ρ −+=−

or,

whhh g

P −+= H"H"s,oiloils,

w

"a"e,+ ρ ρ

ρ

5u!stitutin",

m*.'+*.?m)('.A=A&.'m1a.+

m@s "+'''

)m@s(;.+)"@m(+'''

1a?=H"&

&

&* −×+×=

⋅⋅

h

5olin" for hH" "ies hH" < !.4 m. Therefore, the ifferential hei"ht of the mercury column must !e :A cm.

Discussion Fou!le instrumentation lie this allows one to erify the measurement of one of the instruments !y themeasurement of another instrument.


*-&&

0ir

ater

hoil

%5 kPa

hH"h

w



Chapter 3 Pressure and Fluid Statics3,1. Solution The "a"e pressure of air in a pressuri8e water tan is measure simultaneously !y !oth a pressure "a"ean a manometer. The ifferential hei"ht h of the mercury column is to !e etermine.

Assumptions The air pressure in the tan is uniform (i.e., its ariation with eleation is ne"li"i!le ue to its lowensity), an thus the pressure at the air-water interface is the same as the inicate "a"e pressure.

Properties e tae the ensity of water to !e ρ w <+''' "@m*. The specific "raities of oil an mercury are "ien to !e

'.A& an +*.?, respectiely.

Analysis 5tartin" with the pressure of air in the tan (point +), an moin" alon" the tu!e !y ain" (as we "oown) or su!tractin" (as we "o up) the gh ρ terms until we reach the free surface of oil where the oil tu!e is e/pose to

the atmosphere, an settin" the result e7ual to P atm "ies

atmw P gh gh gh P =−−+ oiloilH"H"w+ ρ ρ ρ

earran"in" ,

w gh gh gh P P wH"H"oiloilatm+ ρ ρ ρ −+=−

or,

whh#h# g

P −+= H"H"oiloil

w

"a"e,+

ρ

5u!stitutin",

m*.'+*.?m)('.A=A&.'m1a.+

m@s" +'''C

)m@s(;.+)"@m(+'''

1a:=H"&

&

&* −×+×=

⋅

⋅h

5olin" for hH" "ies hH" < !.%$ m. Therefore, the i##erential height o# the (er$ur! $olu(n (ust be 3/ $(.

Discussion Fou!le instrumentation lie this allows one to erify the measurement of one of the instruments !y themeasurement of another instrument.

3,1/

Solution The top part of a water tan is iie into two compartments, an a flui with an unnown ensity is poure into one sie. The leels of the water an the li7ui are measure. The ensity of the flui is to !e etermine.

Assumptions . #oth water an the ae li7ui are incompressi!le su!stances. / The ae li7ui oes not mi/ withwater.


Analysis #oth fluis are open to the atmosphere. Botin" that the pressureof !oth water an the ae flui is the same at the contact surface, the

pressure at this surface can !e e/presse as

wwatmf f atmcontact gh P gh P P ρ ρ +=+=

5implifyin", we hae f f w w gh gh ρ ρ = . 5olin" for ρ f "ies

( )* *:= cm+''' "@m =?& = "@m

' cm

w

( w

(

h.

h ρ ρ = = = ≅ %"6% kg-m

Discussion Bote that the ae flui is li"hter than water as e/pecte (a heaier flui woul sin in water).


*-&*

hf

hw

>lui

ater

0ir

hoil

45 kPa

hH"h

w

ater



Chapter 3 Pressure and Fluid Statics3,13 Solution 0 loa on a hyraulic lift is to !e raise !y pourin" oil from a thin tu!e. The hei"ht of oil in the tu!ere7uire in orer to raise that wei"ht is to !e etermine.

Assumptions . The cyliners of the lift are ertical. / There are no leas. 3 0tmospheric pressure act on !oth sies, anthus it can !e isre"are.

Properties The ensity of oil is "ien to !e ρ <A' "@m*.

Analysis Botin" that pressure is force per unit area, the "a"e pressure in the flui uner the loa is simply the ratio

of the wei"ht to the area of the lift,

1a:.*:B@m*:.:m@s"+'''

B+

:@m)&'.+(

)m@s")(;.+=''(

:@

&

&&

&

&"a"e ==

⋅===

π π &

mg

A

W P

The re7uire oil hei"ht that will cause :.*: 1a of pressure rise is

m!."6&=

⋅==→=

&

&

&*

&"a"e

"a"eB@m+

m@s"'''+

)m@s)(;.+"@m(A'

B@m*:.:

g

P h gh P

ρ ρ

Therefore, a ='' " loa can !e raise !y this hyraulic lift !y simply raisin" the oil leel in the tu!e !y =?.A cm.

Discussion Bote that lar"e wei"hts can !e raise !y little effort in hyraulic lift !y main" use of 1ascal6s principle.


*-&:

LOAD

277 &g h

+.& m + cm



Chapter 3 Pressure and Fluid Statics3,11E Solution Two oil tans are connecte to each other throu"h a mercury manometer. >or a "ien ifferential hei"ht,the pressure ifference !etween the two tans is to !e etermine.

Assumptions . #oth the oil an mercury are incompressi!le fluis. /The oils in !oth tans hae the same ensity.

Properties The ensities of oil an mercury are "ien to !e ρ oil <

:= l!m@ft* an ρ H" < : l!m@ft*.

Analysis 5tartin" with the pressure at the !ottom of tan +(where pressure is P +) an moin" alon" the tu!e !y ain" (as we "o

own) or su!tractin" (as we "o up) the gh ρ terms until we reach the

!ottom of tan & (where pressure is P &) "ies

&+oil&H"&+oil+ )( P gh ghhh g P =−−++ ρ ρ ρ

where h+ < +' in an h* < *& in. earran"in" an simplifyin" ,

&oilH"&oil&H"&+ )( gh gh gh P P ρ ρ ρ ρ −=−=−

5u!stitutin",

⋅=−=∆

&

&

&*

&+ in+::

ft+

ft@sl!m*&.&

l!f +

ft))(*&@+&ft@s&.*&()l!m@ft:=-(: P P P

Therefore, the pressure in the left oil tan is +:.; psia hi"her than the pressure in the ri"ht oil tan.

Discussion Bote that lar"e pressure ifferences can !e measure coneniently !y mercury manometers. If a water manometer were use in this case, the ifferential hei"ht woul !e oer *' ft.

3,12 Solution The stanar atmospheric pressure is e/presse in terms of mercury, water, an "lycerin columns.

Assumptions The ensities of fluis are constant.

Properties The specific "raities are "ien to !e 5G < +*.? for mercury, 5G < +.' for water, an 5G < +.&? for

"lycerin. The stanar ensity of water is +''' "@m*, an the stanar atmospheric pressure is +'+,*&= 1a.

Analysis The atmospheric pressure is e/presse in terms of a flui column hei"ht as

gh# gh P watm ρ ρ == → g #

P h

w

atm

ρ =

5u!stitutin",

(a) Mercury$

& &

atm

* & &

+'+ *&= B@m + " m@s

5G +*.?(+''' "@m )(;.+ m@s ) + B@mw

P +h

g ρ

×= = = ÷

!."# m

(b) ater$

& &

atm

* & &

+'+ *&= B@m + " m@s

5G +(+''' "@m )(;.+ m@s ) + B@mw

P +h

g ρ

×= = = ÷

1!.% m

(c) Glycerin$

& &

atm

* & &+'+ *&= B@m + " m@s

5G +.&?(+''' "@m )(;.+ m@s ) + B@mw

P +h g ρ

×= = = ÷

8.$! m

Discussion sin" water or "lycerin to measure atmospheric pressure re7uires ery lon" ertical tu!es (oer +' m for water), which is not practical. This e/plains why mercury is use instea of water or a li"ht flui.


*-&=

*& in

+' in

Mercury



Chapter 3 Pressure and Fluid Statics3,1 Solution 0 "lass fille with water an coere with a thin paper is inerte. The pressure at the !ottom of the "lassis to !e etermine.

Assumptions . ater is an incompressi!le su!stance. / The wei"ht of the paper is ne"li"i!le. 3 The atmospheric pressure is +'' 1a.


Analysis The paper is in e7uili!rium, an thus the net force actin" on the

paper must !e 8ero. 0 ertical force !alance on the paper inoles the pressureforces on !oth sies, an yiels

"lassatm"lass+ A P A P = → atm+ P P =

That is, the pressures on !oth sies of the paper must !e the same.The pressure at the !ottom of the "lass is etermine from the hyrostatic

pressure relation to !e

"lass !ottomatm gh P P ρ += → "lassatm !ottom gh P P ρ −=

5u!stitutin",

#=

⋅−=

&&

&* !ottom

B@m+'''

1a+

m@s"+

B+m))('.+m@s)(;.+"@m(+'''1a)+''( P

Discussion Bote that there is a acuum of + 1a at the !ottom of the "lass, an thus there is an upwar pressure forceactin" on the water !oy, which !alance !y the wei"ht of water. 0s a result, the net ownwar force on water is 8ero, anthus water oes not flow own.


*-&?

P atm

P +

P !ottom



Chapter 3 Pressure and Fluid Statics3,14 Solution Two cham!ers with the same flui at their !ase are separate !y a piston. The "a"e pressure in each air cham!er is to !e etermine.

Assumptions . ater is an incompressi!le su!stance. / Theariation of pressure with eleation in each air cham!er isne"li"i!le !ecause of the low ensity of air.

Properties e tae the ensity of water to !e ρ <+'''

"@m*.

Analysis The piston is in e7uili!rium, an thus the netforce actin" on the piston must !e 8ero. 0 ertical force

!alance on the piston inoles the pressure force e/erte !ywater on the piston face, the atmospheric pressure force, anthe piston wei"ht, an yiels

piston pistonatm piston W A P A P , += →

piston

piston

atm A

W P P , +=

The pressure at the !ottom of each air cham!er is eterminefrom the hyrostatic pressure relation to !e

,! g A

W P ,! g P P P , ! ρ ρ ++=+==

piston

piston

atm0air → ,! g A

W P ρ +=

piston

piston

"a"e0,air

,& g A

W P ,& g P P P , & ρ ρ −+=−==

piston

piston

atm#air → ,& g A

W P ρ −=

piston

piston

"a"e#,air

5u!stitutin",

* & &

air 0, "a"e & &

&= B + B(+''' "@m )(;.+ m@s )('.&= m) &'? B@m

' * m) : + " m@s P

( . π

= + = = ÷×

$.81 kPa

* & &air #, "a"e & &

&= B + B(+''' "@m )(;.+ m@s )('.&= m) &';; B@m' * m) : + " m@s

P ( . π

= − = − = − ÷× $.1! kPa

Discussion Bote that there is a acuum of a!out & 1a in tan # which pulls the water up.


*-&A

air air

water

1iston

=' cm

&= cm*' cm

A -

!

&

*' cm

;' cm

,



Chapter 3 Pressure and Fluid Statics3,15 Solution 0 ou!le-flui manometer attache to an air pipe is consiere. The specific "raity of one flui is nown,an the specific "raity of the other flui is to !e etermine.

Assumptions . Fensities of li7uis are constant. / The air pressure in the tan is uniform (i.e., its ariation witheleation is ne"li"i!le ue to its low ensity), an thus the pressure at the air-water interface is the same as the inicate"a"e pressure.

Properties The specific "raity of one flui is "ien to !e +*.==. e tae the stanar ensity of water to !e +'''"@m*.

Analysis 5tartin" with the pressure of air in the tan, an moin" alon" the tu!e !y ain" (as we "o own) or

su!tractin" (as we "o up) the gh ρ terms until we reach the free surface where the oil tu!e is e/pose to the atmosphere,

an settin" the result e7ual to P atm "ie

atm&&++air P gh gh P =−+ ρ ρ → air atm & w & + +5G 5G w P P gh gh ρ ρ − = −

earran"in" an solin" for 5G&,

( ) &

air atm+& + * & &

& w &

A? +'' 1a'.&& m +''' " m@s5G 5G +*.==

'.:' m (+''' "@m )(;.+ m@s )('.:' m) + 1a m

P P h

h gh ρ

− − ×= + = + = ÷ ÷×

1.%4

Discussion Bote that the ri"ht flui column is hi"her than the left, an this woul imply a!oe atmospheric pressurein the pipe for a sin"le-flui manometer.


*-&

0ir P < A? 1a

&& cm

:' cm

>lui +5G

+

>lui &5G

&



Chapter 3 Pressure and Fluid Statics3,16 Solution The pressure ifference !etween two pipes is measure !y a ou!le-flui manometer. >or "ien fluihei"hts an specific "raities, the pressure ifference !etween the pipes is to !e calculate.

Assumptions 0ll the li7uis are incompressi!le.

Properties The specific "raities are "ien to !e +*.= for mercury, +.&? for "lycerin, an '. for oil. e tae the

stanar ensity of water to !e ρ w <+''' "@m*.

Analysis 5tartin" with the pressure in the water pipe (point 0) an moin" alon" the tu!e !y ain" (as we "o

own) or su!tractin" (as we "o up) the gh ρ terms until we reach the oil pipe (point #), an settin" the result e7ual to P - "ie

-w A P gh gh gh gh P =+−++ oil pil"ly"lyH"H"w ρ ρ ρ ρ

earran"in" an usin" the efinition of specific "raity,

( )H" "ly oil

H" "ly oil

5G 5G 5G 5G

5G 5G 5G 5G

- A w w w Hg w gly w oil w

w w w Hg gly oil

P P gh gh gh gh

g h h h h

ρ ρ ρ ρ

ρ

− = + − +

= + − +

5u!stitutin",

kPa$&.&==

+−+=−&

*&

B@mA.&A

"+'''

+m)C+.'(.'m):=.'(&?.+m)&.'(=.+*m)?.'(+)D"@m)(+'''m@s(;.+

A - P P

Therefore, the pressure in the oil pipe is &A.A 1a hi"her than the pressure in the water pipe.

Discussion sin" a manometer !etween two pipes is not recommene unless the pressures in the two pipes arerelatiely constant. Etherwise, an oer-rise of pressure in one pipe can push the manometer flui into the other pipe,creatin" a short circuit.


*-&;

Mercury,5G<+*.=?

Glycerin,5G<+.&?

Eil5G<'.

ater,5G<+.'

A

:

?' cm

&' cm

+= cm

+' cm



Chapter 3 Pressure and Fluid Statics3,27 Solution The flui leels in a multi-flui -tu!e manometer chan"e as a result of a pressure rop in the trappe air space. >or a "ien pressure rop an !rine leel chan"e, the area ratio is to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / 1ressure in the !rine pipe remains constant. 3 The ariation of pressure in the trappe air space is ne"li"i!le.

Properties The specific "raities are "ien to !e +*.=? for mercury an +.+ for !rine. e tae the stanar ensity of

water to !e ρ w <+''' "@m*.

Analysis It is clear from the pro!lem statement an the fi"ure that the !rine pressure is much hi"her than the air pressure, an when the air pressure rops !y '.A 1a, the pressure ifference !etween the !rine an the air space alsoincreases !y the same amount. 5tartin" with the air pressure (point 0) an moin" alon" the tu!e !y ain" (as we "o

own) or su!tractin" (as we "o up) the gh ρ terms until we reach the !rine pipe (point #), an settin" the result e7ual to

P - !efore an after the pressure chan"e of air "ie

-eore$ -w A P gh gh gh P =−++ !r,+ !r +H",H"w+ ρ ρ ρ

Ater $ -w A P gh gh gh P =−++ !r,& !r &H",H"w& ρ ρ ρ

5u!tractin",

' !r !r H"H"+& =∆−∆+− h g h g P P A A ρ ρ → + &

H" H" !r !r 5G 5G ' A A

w

P P h h

g ρ

−= ∆ − ∆ = (+)

where H"h∆ an !r h∆ are the chan"es in the ifferential mercury an !rine column hei"hts, respectiely, ue to the

rop in air pressure. #oth of these are positie 7uantities since as the mercury-!rine interface rops, the ifferential fluihei"hts for !oth mercury an !rine increase. Botin" also that the olume of mercury is constant, we hae

ri"htH",&leftH",+ h Ah A ∆=∆ an

&&+& s"@mA'' B@mA''1aA.' ⋅−=−=−=− A A P P

m''=.' !r =∆h)@0+(@0 +& !r +& !r !r leftH",ri"htH",H" Ah Ahhhhh +∆=∆+∆=∆+∆=∆

5u!stitutin",

m'.''=C.+.+-)@'.''=(++*.=?D)m@s )(;.+"@m+'''(

s"@mA''+&&*

&

×+×=⋅

A A

It "ies A&@ A+ < !.1%4

Discussion In aition to the e7uations of hyrostatics, we also utili8e conseration of mass in this pro!lem.


*-*'

ater

Mercury5G<+*.=?

5G<+.+

##rine pipe

00ir

0rea, A&

0rea, A+

∆hb < = mm



Chapter 3 Pressure and Fluid Statics3,2. Solution Two water tans are connecte to each other throu"h a mercury manometer with incline tu!es. >or a

"ien pressure ifference !etween the two tans, the parameters a an θ are to !e etermine.

Assumptions #oth water an mercury are incompressi!le li7uis.

Properties The specific "raity of mercury is "ien to !e +*.?. e tae the stanar ensity of water to !e ρ w <+'''

"@m*.

Analysis 5tartin" with the pressure in the tan 0 an moin" alon" the tu!e !y ain" (as we "o own) or

su!tractin" (as we "o up) the gh ρ terms until we reach tan #, an settin" the result e7ual to P - "ie

- A P gaa g ga P =−++ wH"w & ρ ρ ρ → A - P P ga −=H"& ρ

earran"in" an su!stitutin" the nown alues,

m&."!m'A='.' B+

m@s"+'''

)m@s(;.+)"@m''&(+*.?)(+'

B@m&'

&

&

&*

&

==

⋅=

−=

g

P P a

Hg

A -

ρ

>rom "eometric consierations,

a&sin.&? =θ (cm)

Therefore,

=?'.'.&?

='.A&

.&?

&sin =

×==

aθ → θ < %4.!°

Discussion Bote that ertical istances are use in manometer analysis. Hori8ontal istances are of no conse7uence.


*-*+

Mercury

ater 0

ater #a

a

θ

&a&?. cm



Chapter 3 Pressure and Fluid Statics3,2/ Solution 0 multi-flui container is connecte to a -tu!e. >or the "ien specific "raities an flui column hei"hts,the "a"e pressure at 0 an the hei"ht of a mercury column that woul create the same pressure at 0 are to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / The multi-flui container is open to the atmosphere.

Properties The specific "raities are "ien to !e +.&? for "lycerin an '.;' for oil. e tae the stanar ensity of

water to !e ρ w <+''' "@m*, an the specific "raity of mercury to !e +*.?.

Analysis 5tartin" with the atmospheric pressure on the top surface

of the container an moin" alon" the tu!e !y ain" (as we "o own) or su!tractin" (as we "o up) the gh ρ terms until we reach point A, an

settin" the result e7ual to P A "ie

Awatm P gh gh gh P =−++ "ly"lywoiloil ρ ρ ρ

earran"in" an usin" the efinition of specific "raity,

oil "ly5G 5G 5G A atm oil w w w w gly w P P gh gh gh ρ ρ ρ − = + −or

( )"a"e oil oil "ly "ly5G 5G 5G A+ w w w P g h h h ρ = + −

5u!stitutin",

kPa!.4&1==

⋅

−+=

&

&*&

,

B@m:A+.'

m@s"+''' B+m)CA'.'(&?.+m)*.'(+m)A'.'(;'.')D"@m)(+'''m@s(;.+ gage A P

The e7uialent mercury column hei"ht is

m!.%"%m''*=*.' B+

m@s"+'''

)m@s(;.+)"@m'(+*.?)(+''

B@m'.:A+&

&*

&, ==

⋅==

g

P h

Hg

gage A

Hg ρ

Discussion Bote that the hi"h ensity of mercury maes it a ery suita!le flui for measurin" hi"h pressures inmanometers.


*-*&

0

;' cm

A' cm

*' cm

+= cm

Glycerin5G<+.&?

Eil5G<'.;'

ater

&' cm




'luid Statis2 +ydrostati 'ores on Plane and Cur/ed Surfaes

3,23C Solution e are to efine resultant force an center of pressure.

Analysis The resultant hydrostatic orce actin" on a su!mer"e surface is the resultant o# the pressure #or$esa$ting on the sur#a$e. The point o# appli$ation o# this resultant #or$e is calle the center o pressure.

Discussion The center of pressure is "enerally not at the center of the !oy, ue to hyrostatic pressure ariation.

3,21C Solution e are to e/amine a claim a!out hyrostatic force.

Analysis Yes, !ecause the ma"nitue of the resultant force actin" on a plane surface of a completely su!mer"e !oy in a homo"eneous flui is e7ual to the prouct of the pressure P , at the centroi of the surface an the area A of the

surface. The pressure at the centroi of the surface is , , gh P P ρ += ' where , h is the ertical istance of the centroi

from the free surface of the li7ui.

Discussion e hae assume that we also now the pressure at the li7ui surface.

3,22C Solution e are to consier the effect of plate rotation on the hyrostatic force on the plate surface.

Analysis There will !e no $hange on the hyrostatic force actin" on the top surface of this su!mer"e hori8ontalflat plate as a result of this rotation since the ma"nitue of the resultant force actin" on a plane surface of a completelysu!mer"e !oy in a homo"eneous flui is e7ual to the prouct of the pressure P , at the centroi of the surface an thearea A of the surface.

Discussion If the rotation were not aroun the centroi, there would !e a chan"e in the force.

3,2C

Solution e are to e/plain why ams are !i""er at the !ottom than at the top.

Analysis Fams are !uilt much thicer at the !ottom !ecause the pressure #or$e in$reases *ith epth) an thebotto( part o# a(s are sub;e$te to largest #or$es.

Discussion Fam construction re7uires an enormous amount of concrete, so taperin" the am in this way saes a lot of concrete, an therefore a lot of money.

3,24C Solution e are to e/plain how to etermine the hori8ontal component of hyrostatic force on a cure surface.

Analysis The hori8ontal component of the hyrostatic force actin" on a cure surface is e7ual (in !oth ma"nituean the line of action) to the h!rostati$ #or$e a$ting on the verti$al pro;e$tion o# the $urve sur#a$e.

Discussion e coul also inte"rate pressure alon" the surface, !ut the metho iscusse here is much simpler anyiels the same answer.


*-**



Chapter 3 Pressure and Fluid Statics3,25C Solution e are to e/plain how to etermine the ertical component of hyrostatic force on a cure surface.

Analysis The ertical component of the hyrostatic force actin" on a cure surface is e7ual to the h!rostati$#or$e a$ting on the hori"ontal pro;e$tion o# the $urve sur#a$e, plus (minus, if actin" in the opposite irection) the*eight o# the #lui blo$& .

Discussion e coul also inte"rate pressure alon" the surface, !ut the metho iscusse here is much simpler anyiels the same answer.

3,26C Solution e are to e/plain how to etermine the line of action on a circular surface.

Analysis The resultant hyrostatic force actin" on a circular surface always passes throu"h the $enter o# the $ir$lesince the pressure forces are normal to the surface, an all lines normal to the surface of a circle pass throu"h the center of the circle. Thus the pressure forces form a concurrent force system at the center, which can !e reuce to a sin"lee7uialent force at that point. If the ma"nitues of the hori8ontal an ertical components of the resultant hyrostatic

force are nown, the tan"ent of the an"le the resultant hyrostatic force maes with the hori8ontal is H ' $ $ @tan =α .

Discussion This fact maes analysis of circular-shape surfaces simple. There is no corresponin" simplification for shapes other than circular, unfortunately.

3,7 Solution 0 car is su!mer"e in water. The hyrostatic force on the oor an its line of action are to !e eterminefor the cases of the car containin" atmospheric air an the car is fille with water.

Assumptions . The !ottom surface of the lae is hori8ontal. / The oor can !e appro/imate as a ertical rectan"ular plate. 3 The pressure in the car remains at atmospheric alue since there is no water leain" in, an thus no compressionof the air insie. Therefore, we can i"nore the atmospheric pressure in calculations since it acts on !oth sies of the oor.

Properties e tae the ensity of lae water to !e +''' "@m* throu"hout.

Analysis (a) hen the car is well-seale an thus the pressure insie the car is the atmospheric pressure, theaera"e pressure on the outer surface of the oor is the pressure at the centroi (mipoint) of the surface, an isetermine to !e

( )

( ) ( ) ( )

a"

* & &

&

&

+ B+''' "@m ; + m@s + + & m * B@m

+''' " m@s

, , P P gh g s b

. . .

ρ ρ = = = +

= + = ÷×

Then the resultant hyrostatic force on the oor !ecomes

k) 8%.!=×== m)+.+m ;.')(B@m.*( & A P $ ave %

The pressure center is irectly uner the mipoint of the plate, an its istancefrom the surface of the lae is etermine to !e

m8."6=+

++=+

++=)&@+.+(+&

+.+

&

+.+

)&@(+&&

&&

b s

bb s y P

(b) hen the car is fille with water, the net force normal to the surface of the oor is "ero since the pressure on !othsies of the oor will !e the same.

Discussion Bote that it is impossi!le for a person to open the oor of the car when it is fille with atmospheric air. #utit taes little effort to open the oor when car is fille with water, !ecause then the pressure on each sie of the oor is thesame.


*-*:

s < m

Foor, +.+ m × '.; m



Chapter 3 Pressure and Fluid Statics3,.E Solution The hei"ht of a water reseroir is controlle !y a cylinrical "ate hin"e to the reseroir. The hyrostaticforce on the cyliner an the wei"ht of the cyliner per ft len"th are to !e etermine.

Assumptions . The hin"e is frictionless. / 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !ei"nore in calculations for conenience.

Properties e tae the ensity of water to !e ?&.: l!m@ft * throu"hout.

Analysis (a) e consier the free !oy ia"ram of the li7ui !loc enclose !y the circular surface of the cyliner

an its ertical an hori8ontal pro9ections. The hyrostatic forces actin" on the ertical an hori8ontal plane surfaces aswell as the wei"ht of the li7ui !loc per ft len"th of the cyliner are$

Hori8ontal force on ertical surface$

l!f +A:A

ft@sl!m*&.&

l!f +ft)+ftft)(&&@&+*)(ft@s&.*&)(l!m@ft:.?&(

)&@(

&

&*

=

⋅

×+=

+==== A % s g A gh A P $ $ , ave H ρ ρ

ertical force on hori8ontal surface (upward )$

( ) ( ) ( ) ( )

a" !ottom

* &

&+ l!f ?& : l!m@ft *& & ft@s += ft & ft + ft

*&.& l!m ft@s

+A& l!f

y , $ P A gh A gh A

. .

ρ ρ = = =

= × ÷× =

ei"ht of flui !loc per ft len"th (downward)/

l!f =:

ft@sl!m*&.&

l!f +ft)@:)(+-(+ft)&)(ft@s&.*&)(l!m@ft:.?&(

ft)+)(:@+(ft)+)(:@(

&

&&*

&&&

=

⋅=

−=−===

π

π ρ π ρ ρ g% % % g g mg W V

Therefore, the net upwar ertical force is

l!f ++=:+A& =−=−= W $ $ y'

Then the ma"nitue an irection of the hyrostatic force actin" on the cylinrical surface !ecome

& & & &+A:A ++ &=&+ l!f % H ' $ $ $ = + = + = ≅ # $"$! lb

°=→=== +.:? ':+.+l!f +A:A

l!f ++tan θ θ

H

'

$

$

Therefore, the ma"nitue of the hyrostatic force actin" on the cyliner is &=&+ l!f per ft len"th of the cyliner, an its

line of action passes throu"h the center of the cyliner main" an an"le :?.+ ° upwars from the hori8ontal.

(b) hen the water leel is +=-ft hi"h, the "ate opens an the reaction force at the !ottom of the cyliner !ecomes 8ero.

Then the forces other than those at the hin"e actin" on the cyliner are its wei"ht, actin" throu"h the center, an thehyrostatic force e/erte !y water. Tain" a moment a!out the point A where the hin"e is an e7uatin" it to 8ero "ies

sin ' sin (&=&+ l!f)sin:? + ++A l!f % cyl cyl % $ % W % W $ .θ θ − = → = = ° = ≅ 18$! lbf (per ft)

Discussion The wei"ht of the cyliner per ft len"th is etermine to !e +&' l!f, which correspons to a mass of +&'l!m, an to a ensity of +:= l!m@ft* for the material of the cyliner.


*-*=

$ H

$ ' W

%<& ft

s < +* ft

b< %

<& ft



Chapter 3 Pressure and Fluid Statics3,/ Solution 0n a!oe the "roun swimmin" pool is fille with water. The hyrostatic force on each wall an theistance of the line of action from the "roun are to !e etermine, an the effect of ou!lin" the wall hei"ht on thehyrostatic force is to !e assesse.

Assumptions 0tmospheric pressure acts on !oth sies of the wall of the pool, an thus it can !e i"nore in calculationsfor conenience.

Properties e tae the ensity of water to !e +''' "@m * throu"hout.

Analysis The aera"e pressure on a surface is the pressure at the centroi(mipoint) of the surface, an is etermine to !e

( ) ( ) ( )

a"

* &

&

&

&

+ B+''' "@m ;.+ m@s + = & m

+ " m@s

A*=A = B@m

, , P P gh g( h )

.

.

ρ ρ = = =

= ÷× =

Then the resultant hyrostatic force on each wall !ecomes

( ) ( )&

a" A*=A = B@m : m + = m :: +:= B % $ P A . . += = × = ≅ 44.1 k)

The line of action of the force passes throu"h the pressure center, which is & h@* from the free surface an h@* from the

!ottom of the pool. Therefore, the istance of the line of action from the "roun is

m !."!===*

=.+

*

h y P (from the !ottom)

If the hei"ht of the walls of the pool is ou!le, the hyrostatic force 3uadrules since

&@))(&@(&

gwhwhh g A gh $ , % ρ ρ ρ =×==

an thus the hyrostatic force is proportional to the s7uare of the wall hei"ht, h&.

Discussion This is one reason why a!oe-"roun swimmin" pools are not ery eep, whereas in-"roun swimmin" pools can !e 7uite eep.


*-*?

$ %

h < +.= m&h@*

h@*



Chapter 3 Pressure and Fluid Statics3,3E Solution 0 am is fille to capacity. The total hyrostatic force on the am, an the pressures at the top an the

!ottom are to !e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of the am, an thus it can !e i"nore in calculations for conenience.


Analysis The aera"e pressure on a surface is the pressure at the centroi

(mipoint) of the surface, an is etermine to !e

( )

( ) ( ) ( )

a"

* &

&

&

&

+ l!f ?& : l!m@ft *& & ft@s &'' & ft

*&.& l!m ft@s

?&:' l!f@ft

, P gh g h

. .

ρ ρ = =

= ÷× =

Then the resultant hyrostatic force actin" on the am !ecomes

( ) ( )&

ae ?&:' l!f@ft &'' ft +&'' ft % $ P A= = × = × #1."! 1! lbf

esultant force per unit area is pressure, an its alue at the top an the !ottom of the am !ecomes

$lbf-ft!== toptop gh P ρ

( ) ( ) ( )* & &

!ottom !ottom &+ l!f ?& : l!m@ft *& & ft@s &'' ft +& :' l!f@ft

*&.& l!m ft@s P gh . . + ρ = = = ≅ ÷×

$1$,"!! lbf-ft

Discussion The alues a!oe are "ae pressures, of course. The "a"e pressure at the !ottom of the am is a!out ?.? psi", or +'+.: psia, which is almost seen times "reater than stanar atmospheric pressure.

3,1 Solution 0 room in the lower leel of a cruise ship is consiere. The hyrostatic force actin" on the winow anthe pressure center are to !e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of the winow, an thus it can !e i"nore in calculations for conenience.

Properties The specific "raity of sea water is "ien to !e +.'&=, an thus its ensity is +'&= "@m *.


( ) ( ) ( )* & &

&

+ B+'&= "@m ; + m@s = m =' &A? B@m

+ " m@savg , , P P gh . + ρ

= = = = ÷×


& & &

a" a"D : (=' &A? B@m )D ' * m) : *==: B % $ P A P & + ( . π π = = = = ≅ %""! )

The line of action of the force passes throu"h the pressure center, whose erticalistance from the free surface is etermine from

: & &

&

: '.+= m= = ''++ m: : = m

+,

P , , ,

, , ,

0 % % ( ) y y y y . y A y ( ) y %

π

π = + = + = + = + = ≅ ".!! m

Discussion >or small surfaces eep in a li7ui, the pressure center nearly coincies with the centroi of the surface.Here, in fact, to three si"nificant i"its in the final answer, the center of pressure an centroi are coincient.


*-*A

$ %

h<&'' ft&h@*

h@*

$ %

= m

&<'.* m



Chapter 3 Pressure and Fluid Statics3,2 Solution The cross-section of a am is a 7uarter-circle. The hyrostatic force on the am an its line of action are to

!e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of the am, an thus it can !e i"nore in calculations for conenience.


Analysis e consier the free !oy ia"ram of the li7ui !loc enclose !y the circular surface of the am an its

ertical an hori8ontal pro9ections. The hyrostatic forces actin" on the ertical an hori8ontal plane surfaces as well asthe wei"ht of the li7ui !loc are$


( ) ( ) ( ) ( )

a"

* &

&

A

&

+ B+''' "@m ; + m@s +' & m +' m +'' m

+ " m@s

: ;'= +' B

H , $ $ P A gh A g( % )A

.

.

ρ ρ = = = =

= × ÷× = ×

ertical force on hori8ontal surface is 8ero since it coincies with the freesurface of water. The wei"ht of flui !loc per m len"th is

B+'A'=.A

m@s"+

B+@:Cm)(+'m)+'')D(m@s+.;)("@m+'''(

C:@D

A

&

&&*

&

×=

⋅=

×===

π

π ρ ρ %w g g W $ ' V

Then the ma"nitue an irection of the hyrostatic force actin" on the surface of the am !ecome

( ) ( )& && & A A A

A

A

: ;'= +' B A A'= +' B ; +*: +' B

A A'= +' Btan + =A+

: ;'= +' B

% H '

'

H

$ $ $ . . .

$ ..

$ .θ θ

= + = × + × = × ≅ ×

×= = = → =

×

&#.1% 1! )

"."

Therefore, the line of action of the hyrostatic force passes throu"h the center of the curature of the am, main" =A.= °ownwars from the hori8ontal.

Discussion If the shape were not circular, it woul !e more ifficult to etermine the line of action.


*-*

% < +' m

$ H

$ y < '

W



Chapter 3 Pressure and Fluid Statics3, Solution 0 rectan"ular plate hin"e a!out a hori8ontal a/is alon" its upper e"e !locs a fresh water channel. The

plate is restraine from openin" !y a fi/e ri"e at a point -. The force e/erte to the plate !y the ri"e is to !eetermine.

Assumptions 0tmospheric pressure acts on !oth sies of the plate, an thus it can !e i"nore in calculations for conenience.



( ) ( ) ( )

a"

* & &

&

&

+ B+''' "@m ; + m@s : & m +; ?& B@m

+''' " m@s

, , P P gh g( h )

. .

ρ ρ = = =

= = ÷×


( ) ( )&

a" +; ?& B@m : m = m *;& B % $ P A .= = × =

The line of action of the force passes throu"h the pressure center, which is & h@*from the free surface,

m&.??A*

m):(&*

& =×== h y P

Tain" the moment a!out point A an settin" it e7ual to 8ero "ies

A- $ y s $ 1 P % A ri"e)( ' =+→=∑5olin" for $ ri"e an su!stitutin", the reaction force is etermine to !e

k)$88=+

=+

= B)*;&(m=

m)??A.&+(ri"e %

P $

A-

y s $

Discussion The ifference !etween $ % an $ ri"e is the force actin" on the hin"e at point A.


*-*;

$ %

$ ri"e

s < + m

h < : m

A

-




3,4

Solution The preious pro!lem is reconsiere. The effect of water epth on the force e/erte on the plate !y theri"e as the water epth aries from ' to = m in increments of '.= m is to !e inesti"ate.


g=9.8% "m/s2"

r$o=%000 "&g/m3"s=%"m"

<=5 "m"+=<$P_ae=r$og$/2000 "&Pa"B_4=P_ae+ "&"#_p=2$/3B_rige=,sF#_pB_4/,sF$

Fepth, m

P ae,1a

$ %B

y p

m $ ri"e

B

'.''.=

+.'+.=&.'&.=*.'*.=:.':.==.'

'&.:=*

:.;'=A.*=;.+

+&.&?+:.A&+A.+A+;.?&&&.'A&:.=*

'.'?.+

&:.===.&;.+

+=*.*&&'.A*''.:*;&.::;?.??+*.+

'.'''.**

'.?A+.''+.**+.?A&.''&.**&.?A*.''*.**

'=

&'::A?++A+??&&*&*?+::*

Discussion The force on the ri"e oes not increase linearly, as we may hae suspecte.


*-:'

0 & 2 3 4 5

0

50

&00

&50

200

250

300

350

400

450

, m

' r i d g e ,

k )



Chapter 3 Pressure and Fluid Statics3,5E Solution The flow of water from a reseroir is controlle !y an -shape "ate hin"e at a point A. The re7uirewei"ht W for the "ate to open at a specifie water hei"ht is to !e etermine.

Assumptions . 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !e i"nore in calculations for conenience. / The wei"ht of the "ate is ne"li"i!le.




( )

( ) ( ) ( )

a"

* &

&

&

&

+ l!f ?& : l!m@ft *& & ft@s +& & ft

*&.& l!m ft@s

*A: : l!f@ft

, P gh g h

. .

.

ρ ρ = =

= ÷× =


( ) ( )&

a" *A: : l!f@ft +& ft = ft &&,:?: l!f % $ P A .= = × =

The line of action of the force passes throu"h the pressure center, which is & h@*

from the free surface,

ft*

ft)+&(&

*

&=

×==

h y P


A-W y s $ 1 P % A =+→=∑ )( '

5olin" for W an su!stitutin", the re7uire wei"ht is etermine to !e

lbf %!,#!!=+

=+

= l!f):?:,&&(ft

ft)*( %

P $ A-

y sW

Discussion Bote that the re7uire wei"ht is inersely proportional to the istance of the wei"ht from the hin"e.


*-:+

B

$ %

s < * ft

h<+& ft

A

ft



Chapter 3 Pressure and Fluid Statics3,6E Solution The flow of water from a reseroir is controlle !y an -shape "ate hin"e at a point A. The re7uirewei"ht W for the "ate to open at a specifie water hei"ht is to !e etermine.

Assumptions . 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !e i"nore in calculations for conenience. / The wei"ht of the "ate is ne"li"i!le.




( )

( ) ( ) ( )

a"

* &

&

&

&

+ l!f ?& : l!m@ft *& & ft@s & ft

*&.& l!m ft@s

&:; ? l!f@ft

, P gh g h

. .

.

ρ ρ = =

= ÷× =


( ) ( )&

a" &:; ? l!f@ft ft = ft ;;: l!f % $ P A .= = × =

The line of action of the force passes throu"h the pressure center, which is & h@*

from the free surface,

ft***.=*

ft)(&

*

&=

×==

h y P


A-W y s $ 1 P % A =+→=∑ )( '

5olin" for W an su!stitutin", the re7uire wei"ht is etermine to !e

( )( )

A = *** ft;;: l!f += *;' l!f

ft

P %

. s yW $ +

A-

++= = = ≅ 1",4!! lbf

Discussion Bote that the re7uire wei"ht is inersely proportional to the istance of the wei"ht from the hin"e.


*-:&

$ %

s < A ft

h< ft

A B

ft



Chapter 3 Pressure and Fluid Statics3,47 Solution Two parts of a water trou"h of semi-circular cross-section are hel to"ether !y ca!les place alon" thelen"th of the trou"h. The tension 2 in each ca!le when the trou"h is full is to !e etermine.

Assumptions . 0tmospheric pressure acts on !oth sies of the trou"h wall, an thus it can !e i"nore in calculations for conenience. / The wei"ht of the trou"h is ne"li"i!le.


Analysis To e/pose the ca!le tension, we consier half of the trou"h whose cross-section is 7uarter-circle. The

hyrostatic forces actin" on the ertical an hori8ontal plane surfaces as well as the wei"ht of the li7ui !loc are$


( ) ( ) ( )

a"

* &

&

&

+ B+''' "@m ; + m@s ' = & m ('.= m * m)

+ " m@s

*?A; B

H , $ $ P A gh A g( % )A

. .

ρ ρ = = = =

= × ÷× =

The ertical force on the hori8ontal surface is 8ero, since it coincies with thefree surface of water. The wei"ht of flui !loc per *-m len"th is

B=AA;

m@s"+

B+@:Cm)('.=m)*)D(m@s+.;)("@m+'''(

C:@D

&

&&*

&

=

⋅=

×===

π

π ρ ρ %w g g W $ ' V

Then the ma"nitue an irection of the hyrostatic force actin" on the surface of the *-m lon" section of the trou"h !ecome

°=→===

=+=+=

=A.= =A+.+ B*?A;

B=AA;tan

B?=+ B)=AA;( B)*?A;( &&&&

θ θ H

'

' H %

$

$

$ $ $

Therefore, the line of action passes throu"h the center of the curature of the trou"h, main" =A.= ° ownwars from thehori8ontal. Tain" the moment a!out point A where the two parts are hin"e an settin" it e7ual to 8ero "ies

' sin ;' =A = A % 1 $ % ( . ) 2%= → − ° =∑5olin" for 2 an su!stitutin", the tension in the ca!le is etermine to !e

( ) ( ) ( )sin ;' =A = ?=+ B sin ;' =A = *?+ B %2 $ . .= − ° = − ° = ≅ %68! )

Discussion This pro!lem can also !e sole without finin" $ % !y finin" the lines of action of the hori8ontalhyrostatic force an the wei"ht.


*-:*

% < '.= m

$ H

T

W A

$ %

θ



Chapter 3 Pressure and Fluid Statics3,4. Solution Two parts of a water trou"h of trian"ular cross-section are hel to"ether !y ca!les place alon" the len"thof the trou"h. The tension 2 in each ca!le when the trou"h is fille to the rim is to !e etermine.



Analysis To e/pose the ca!le tension, we consier half of the trou"h whose cross-section is trian"ular. The water

hei"ht h at the misection of the trou"h an with of the free surface are

m=*'.'m)cos:=A=.'(cos

m=*'.'m)sin:=A=.'(sin

=°===°==

θ

θ

3b

3h

The hyrostatic forces actin" on the ertical an hori8ontal plane surfaces aswell as the wei"ht of the li7ui !loc are etermine as follows$


( )

( ) ( ) ( )

a"

* &

&

&

+ B+''' "@m ; + m@s ' =*' & m ('.=*' m ? m)

+ " m@s

&?A B

H , $ $ P A gh A g h A

. .

ρ ρ = = = =

= × ÷×

=

The ertical force on the hori8ontal surface is 8ero since it coincies with the free surfaceof water. The wei"ht of flui !loc per ?-m len"th is

B&?A

m@s"+

B+m)@&Cm)('.=*'m)('.=*'?)D(m@s+.;)("@m+'''(

C&@D

&

&*

=

⋅=

×=== bhw g g W $ ' ρ ρ V

The istance of the centroi of a trian"le from a sie is +@* of the hei"ht of the trian"le for that sie. Tain" the momenta!out point A where the two parts are hin"e an settin" it e7ual to 8ero "ies

'* *

A H

b h 1 W $ 2h= → + =∑

5olin" for 2 an su!stitutin", an notin" that h 4 b, the tension in the ca!le is etermine to !e

( )&?A &?A B==++ B

* *

H $ W 2

++= = = ≅ ""1! )

Discussion The analysis is simplifie !ecause of the symmetry of the trou"h.


*-::

$ H

W

'.A= m

:=°

2 b

A



Chapter 3 Pressure and Fluid Statics3,4/ Solution Two parts of a water trou"h of trian"ular cross-section are hel to"ether !y ca!les place alon" the len"thof the trou"h. The tension 2 in each ca!le when the trou"h is fille to the rim is to !e etermine.



Analysis To e/pose the ca!le tension, we consier half of the trou"h whose cross-section is trian"ular. The water

hei"ht is "ien to !e h < '.: m at the misection of the trou"h, which is e7uialent to the with of the free surface b sincetan :=° < bh 4 +. The hyrostatic forces actin" on the ertical an hori8ontal plane surfaces as well as the wei"ht of the

li7ui !loc are etermine as follows$


( )

( ) ( ) ( )

a"

* &

&

&

+ B+''' "@m ; + m@s ' : & m ('.: m * m)

+ " m@s

&*=: B


. .

ρ ρ = = = =

= × ÷× =

The ertical force on the hori8ontal surface is 8ero since it coincies with the free surfaceof water. The wei"ht of flui !loc per *-m len"th is

B&*=:

m@s"+

B+m)@&Cm)('.:m)('.:*)D(m@s+.;)("@m+'''(

C&@D

&

&*

=

⋅=

×=== bhw g g W $ ' ρ ρ V

The istance of the centroi of a trian"le from a sie is +@* of the hei"ht of the trian"le for that sie. Tain" the momenta!out point A where the two parts are hin"e an settin" it e7ual to 8ero "ies

'* *

A H

b h 1 W $ 2h= → + =∑

5olin" for 2 an su!stitutin", an notin" that h 4 b, the tension in the ca!le is etermine to !e

( )&*=: &*=: B+=?; B

* *

H $ W 2

++= = = ≅ 1"! )

Discussion The tension force here is a factor of a!out *.= smaller than that of the preious pro!lem, een thou"h thetrou"h is more than half full.


*-:=

$ H

W

'.: m

:=°

2 b

A



Chapter 3 Pressure and Fluid Statics3,43 Solution 0 retainin" wall a"ainst mu slie is to !e constructe !y rectan"ular concrete !locs. The mu hei"ht atwhich the !locs will start sliin", an the !locs will tip oer are to !e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of the wall, an thus it can !e i"nore in calculations for conenience.

Properties The ensity is "ien to !e +'' "@m* for the mu, an &A'' "@m * for concrete !locs.

Analysis (a) The wei"ht of the concrete wall per unit len"th ( 3 4 + m) an the friction force !etween the wall an

the "roun are

B+&A+) B:&*(*.'

B:&*m@s"+

B+)m+.'&.')Dm@s+.;)("@m&A''(

!loc friction

&

*&* !loc

===

=

⋅××==

W $

g W

µ

ρ V

The hyrostatic force e/erte !y the mu to the wall is

( )

( ) ( ) ( )

a"

* &

&

&

&

+ B+'' "@m ; + m@s & (+ )

+ " m@s

&; B


. h h

h

ρ ρ = = = =

= × ÷×

=5ettin" the hyrostatic an friction forces e7ual to each other "ies

m !.%8=→=→= hh $ $ H +&A+&; &

friction

(b) The line of action of the hyrostatic force passes throu"h the pressure center, which is & h@* from the free surface. Theline of action of the wei"ht of the wall passes throu"h the miplane of the wall. Tain" the moment a!out point A ansettin" it e7ual to 8ero "ies

*@&;)&@( )*@()&@( ' * !loc !loc ht W h $ t W 1 H A =→=→=∑

5olin" for h an su!stitutin", the mu hei"ht for tip oer is etermine to !e

m!."$= ×××

=

×=

*@+*@+ !loc

&;&

&.':&**

&;&

* t W h

Discussion The concrete wall will slie !efore tippin". Therefore, sliin" is more critical than tippin" in this case.


*-:?

$ H

$ friction

'. mh

t <'.& m

A

W



Chapter 3 Pressure and Fluid Statics3,41 Solution 0 retainin" wall a"ainst mu slie is to !e constructe !y rectan"ular concrete !locs. The mu hei"ht atwhich the !locs will start sliin", an the !locs will tip oer are to !e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of the wall, an thus it can !e i"nore in calculations for conenience.

Properties The ensity is "ien to !e +'' "@m* for the mu, an &A'' "@m * for concrete !locs.

Analysis (a) The wei"ht of the concrete wall per unit len"th ( 3 4 + m) an the friction force !etween the wall an

the "roun are

B&=:*) B:A?(*.'

B:A?m@s"+

B+)m+.':.')Dm@s+.;)("@m&A''(

!loc friction

&

*&* !loc

===

=

⋅××==

W $

g W

µ

ρ V

The hyrostatic force e/erte !y the mu to the wall is

( )

( ) ( ) ( )

a"

* &

&

&

&

+ B+'' "@m ; + m@s & (+ )

+ " m@s

&; B


. h h

h

ρ ρ = = = =

= × ÷×

=5ettin" the hyrostatic an friction forces e7ual to each other "ies

m !."4=→=→= hh $ $ H &=:*&; &

friction

(b) The line of action of the hyrostatic force passes throu"h the pressure center, which is & h@* from the free surface. Theline of action of the wei"ht of the wall passes throu"h the miplane of the wall. Tain" the moment a!out point A ansettin" it e7ual to 8ero "ies

*@&;)&@( )*@()&@( ' * !loc !loc ht W h $ t W 1 H A =→=→=∑

5olin" for h an su!stitutin", the mu hei"ht for tip oer is etermine to !e

m!.&6= ×××

= ×=

*@+*@+ !loc

&;&

*.':A?*

&;&

* t W h

Discussion Bote that the concrete wall will slie !efore tippin". Therefore, sliin" is more critical than tippin" in thiscase.


*-:A

$ H

$ friction

'. mh

t <'.: m

A

W



Chapter 3 Pressure and Fluid Statics3,42 8 Also solved using !!# on enclosed &'&9

Solution 0 7uarter-circular "ate hin"e a!out its upper e"e controls the flow of water oer the le"e at - where the"ate is presse !y a sprin". The minimum sprin" force re7uire to eep the "ate close when the water leel rises to A atthe upper e"e of the "ate is to !e etermine.

Assumptions . The hin"e is frictionless. / 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !ei"nore in calculations for conenience. 3 The wei"ht of the "ate is ne"li"i!le.


Analysis e consier the free !oy ia"ram of the li7ui !loc enclose !y the circular surface of the "ate an itsertical an hori8ontal pro9ections. The hyrostatic forces actin" on the ertical an hori8ontal plane surfaces as well asthe wei"ht of the li7ui !loc are etermine as follows$

Hori5ontal orce on vertical surace$

B?.+A?

m@s"+'''

B+m)*mm)(:&@*)(m@s+.;)("@m+'''(

)&@(

&

&*

=

⋅×=

==== A % g A gh A P $ $ , ave . H ρ ρ

'ertical orce on hori5ontal surace (upwar)$

( ) ( ) ( )

a" !ottom

* &

&+ B+''' "@m ; + m@s * m (: m * m) *=* & B

+''' " m@s

y , $ P A gh A gh A

. .

ρ ρ = = =

= × = ÷×

2he weight o luid bloc6 per 78m length (ownwars)$

( ) ( ) ( )

&

* & &

&

:

+ B+''' "@m ; + m@s : m (* m) @: &AA : B

+''' " m@s

W g g w %

. .

ρ ρ π

π

= = ×

= = ÷ ×

V


B.A=:.&AA&.*=* =−=−= W $ $ y'

Then the ma"nitue an irection of the hyrostatic force actin" on the surface of the :-m lon" 7uarter-circular section of

the "ate !ecome

°=→===

=+=+=

&*.& :&;.'B?.+A?

B.A=tan

B&.+;&B).A=(B)?.+A?( &&&&

θ θ H

'

' H %

$

$

$ $ $

Therefore, the ma"nitue of the hyrostatic force actin" on the "ate is +;&.& B, an its line of action passes throu"h the

center of the 7uarter-circular "ate main" an an"le &*.& ° upwars from the hori8ontal.

The minimum sprin" force neee is etermine !y tain" a moment a!out the point A where the hin"e is, ansettin" it e7ual to 8ero,

');'sin( ' sprin" =−−→=∑ % $ % $ 1 % A θ

5olin" for $ sprin" an su!stitutin", the sprin" force is etermine to !e

&N.44=°−°== )&.&*;'sin(B)(+;&.&)-sin(;'sprin" θ % $ $

Discussion 5eeral ariations of this esi"n are possi!le. Can you thin of some of themJ


*-:

% < * m $

$ y

W

$ s

A

-



Chapter 3 Pressure and Fluid Statics3,4 Solution 0 7uarter-circular "ate hin"e a!out its upper e"e controls the flow of water oer the le"e at - where the"ate is presse !y a sprin". The minimum sprin" force re7uire to eep the "ate close when the water leel rises to A atthe upper e"e of the "ate is to !e etermine.

Assumptions . The hin"e is frictionless. / 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !ei"nore in calculations for conenience. 3 The wei"ht of the "ate is ne"li"i!le.


Analysis e consier the free !oy ia"ram of the li7ui !loc enclose !y the circular surface of the "ate an itsertical an hori8ontal pro9ections. The hyrostatic forces actin" on the ertical an hori8ontal plane surfaces as well asthe wei"ht of the li7ui !loc are etermine as follows$

Hori5ontal orce on vertical surace$

B;.*+*

m@s"+'''

B+m):mm)(:&@:)(m@s+.;)("@m+'''(

)&@(

&

&*

=

⋅×=

==== A % g A gh A P $ $ , ave . H ρ ρ

'ertical orce on hori5ontal surace (upwar)$

B.?&A

m@s"+'''

B+

m):mm)(::)(m@s+.;)("@m+'''( &

&*

!ottom

=

⋅×=

=== A gh A gh A P $ , ave y ρ ρ

2he weight o luid bloc6 per 78m length (ownwars)$

B+.:;*

m@s"+'''

B+@:Cm)(:m):)D(m@s+.;)("@m+'''(

C:@D

&

&&*

&

=

⋅=

×==

π

π ρ ρ %w g g W V


BA.+*:+.:;*.?&A =−=−= W $ $ y'

Then the ma"nitue an irection of the hyrostatic force actin" on the surface of the :-m lon" 7uarter-circular section of

the "ate !ecome

°=→===

=+=+=

&*.& :&;.'B;.*+*

BA.+*:tan

B?.*:+B)A.+*:(B);.*+*( &&&&

θ θ H

'

' H %

$

$

$ $ $

Therefore, the ma"nitue of the hyrostatic force actin" on the "ate is *:+.? B, an its line of action passes throu"h the

center of the 7uarter-circular "ate main" an an"le &*.& ° upwars from the hori8ontal.

The minimum sprin" force neee is etermine !y tain" a moment a!out the point A where the hin"e is, ansettin" it e7ual to 8ero,

');'sin( ' sprin" =−−→=∑ % $ % $ 1 % A θ

5olin" for $ sprin" an su!stitutin", the sprin" force is etermine to !e

( ) ( )sprin" sin ;' (*:+.? B)sin ;' &* & % $ $ 8 .θ = = ° − ° = %14 k)

Discussion If the preious pro!lem is sole usin" a pro"ram lie 225, it is simple to repeat with ifferent alues.


*-:;

% < : m $

$ y

W

$ s

A

-




Buoyany

3,44C Solution e are to efine an iscuss the !uoyant force.

Analysis The up*ar #or$e a #lui e%erts on an i((erse bo! is calle the buoyant orce. The !uoyant force is$ause b! the in$rease o# pressure in a #lui *ith epth. The magnitude of the !uoyant force actin" on a su!mer"e

!oy whose olume is V is e/presse as V g $ ( - ρ = . The direction of the !uoyant force is up*ars, an its line o

action passes through the $entroi o# the ispla$e volu(e.

Discussion If the !uoyant force is "reater than the !oy6s wei"ht, it floats.

3,45C Solution e are to compare the !uoyant force on two spheres.

Analysis The ma"nitue of the !uoyant force actin" on a su!mer"e !oy whose olume is V is e/presse as

V g $ - ρ = , which is inepenent of epth. Therefore, the buo!ant #or$es a$ting on t*o ienti$al spheri$al balls

sub(erge in *ater at i##erent epths is the sa(e.

Discussion #uoyant force epens only on the olume of the o!9ect, not its ensity.

3,46C Solution e are to compare the !uoyant force on two spheres.

Analysis The ma"nitue of the !uoyant force actin" on a su!mer"e !oy whose olume is V is e/presse asV g $ - ρ = , which is inepenent of the ensity of the !oy ( ( ρ is the flui ensity). Therefore, the buo!ant #or$es

a$ting on the 2,$( ia(eter alu(inu( an iron balls sub(erge in *ater is the sa(e.

Discussion #uoyant force epens only on the olume of the o!9ect, not its ensity.

3,57C Solution e are to compare the !uoyant forces on a cu!e an a sphere.

Analysis The ma"nitue of the !uoyant force actin" on a su!mer"e !oy whose olume is V is e/presse asV g $ - ρ = , which is inepenent of the shape of the !oy. Therefore, the buo!ant #or$es a$ting on the $ube an

sphere (ae o# $opper sub(erge in *ater are the sa(e sin$e the! have the sa(e volu(e.

Discussion The two o!9ects hae the same olume !ecause they hae the same mass and ensity.


*-='



Chapter 3 Pressure and Fluid Statics3,5.C Solution e are to iscuss the sta!ility of a su!mer"e an a floatin" !oy.

Analysis 0 submerged body whose $enter o# gravit! G is above the $enter o# buo!an$! B, *hi$h is the $entroio# the ispla$e volu(e) is unstable. #ut a loating body (a! still be stable *hen G is above B sin$e the $entroi o# the ispla$e volu(e shi#ts to the sie to a point B’ uring a rotational isturban$e *hile the $enter o# gravit! G o# the bo! re(ains un$hange. If the point -’ is sufficiently far, these two forces create a restorin" moment, an return the

!oy to the ori"inal position.

Discussion 5ta!ility analysis lie this is critical in the esi"n of ship hulls, so that they are least liely to capsi8e.

3,5/ Solution The ensity of a li7ui is to !e etermine !y a hyrometer !y esta!lishin" iision mars in water an inthe li7ui, an measurin" the istance !etween these mars.

Properties e tae the ensity of pure water to !e +''' "@m*.

Analysis 0 hyrometer floatin" in water is in static e7uili!rium, an the !uoyant force $ - e/erte !y the li7ui mustalways !e e7ual to the wei"ht W of the hyrometer, $ - < W .

csu! ghA g $ - ρ ρ == V

where h is the hei"ht of the su!mer"e portion of the hyrometer an Ac is thecross-sectional area which is constant.

0n pure water $ cww A ghW ρ =

0n the li"uid $ c A ghW li7uili7ui ρ =

5ettin" the relations a!oe e7ual to each other (since !oth e7ual the wei"ht ofthe hyrometer) "ies

ccww A gh A gh li7uili7ui ρ ρ =

5olin" for the li7ui ensity an su!stitutin",

( )* *water

li7ui water

li7ui

+' cm(+''' "@m ) +'=* "@m

+' ' = cm

h

h . ρ ρ = = = ≅

−%

1!"! kg-m

Discussion Bote that for a "ien cylinrical hyrometer, the prouct of the flui ensity an the hei"ht of thesu!mer"e portion of the hyrometer is constant in any flui.


*-=+

i7ui

'.= cm

+' cm

mar for water

$ -

W



Chapter 3 Pressure and Fluid Statics3,53E Solution 0 concrete !loc is lowere into the sea. The tension in the rope is to !e etermine !efore an after the

!loc is immerse in water.

Assumptions . The !uoyancy force in air is ne"li"i!le. / The wei"ht of the rope is ne"li"i!le.

Properties The ensity of steel !loc is "ien to !e :;: l!m@ft*.

Analysis (a) The forces actin" on the concrete !loc in air are its ownwar wei"ht an the upwar pull action(tension) !y the rope. These two forces must !alance each other, an thus the tension in the rope must !e e7ual to the

wei"ht of the !loc$( )

( ) ( ) ( )

** *

concrete

* & *

&

: * : + = ft @* +: +*A ft

+ l!f :;: l!m@ft *& & ft@s +: +*A ft ?;: l!f

*&.& l!m ft@s

2

% . .

$ W g

. .

π π

ρ

= = =

= =

= = ≅ ÷×

V

V

6#8! lbf

(b) hen the !loc is immerse in water, there is the aitional force of !uoyancyactin" upwars. The force !alance in this case "ies

( ) ( ) ( )* & *

&

T,water

+ l!f ?& : l!m@ft *& & ft@s +: +*A ft & l!f

*&.& l!m ft@s

?;: & ?+'& l!f

- (

-

$ g . . .

$ W $

ρ = = = ÷×

= − = − = ≅

V

61!! lbf

Discussion Bote that the wei"ht of the concrete !loc an thus the tension of the ropeecreases !y (?;: 3 ?+'&)@?;: < +&.?K in water.

3,51 Solution 0n irre"ularly shape !oy is wei"he in air an then in water with a sprin" scale. The olume an theaera"e ensity of the !oy are to !e etermine.

Properties e tae the ensity of water to !e +''' "@m *.

Assumptions . The !uoyancy force in air is ne"li"i!le. / The !oy is completely su!mer"e in water.

Analysis The mass of the !oy is

";.A** B+m@s"+

m@s+.; BA&''

&

&air

=

⋅==

g W m

The ifference !etween the wei"hts in air an in water is ue to the !uoyancyforce in water,

B&:+':A;'A&''water air =−=−= W W $ -

Botin" that V g $ - water ρ = , the olume of the !oy is etermine to !e

( ) ( ) *

* &water

&:+' B' &:=A m

+''' "@m ; + m@s

- $ .

g . ρ = = = ≅V

%!.$46 m

Then the ensity of the !oy !ecomes

*

*

A** ; "&;A "@m

'.&:=A m

m . ρ = = = ≅

V

%$##! kg-m

Discussion The olume of the !oy can also !e measure !y o!serin" the chan"e in the olume of the container when the !oy is roppe in it (assumin" the !oy is not porous).


*-=&

W wir

<?'' BW water

< :A;' B

$ - ater 0ir

Mass,m+ '

W

$ -

$ 2



Chapter 3 Pressure and Fluid Statics3,52 Solution The hei"ht of the portion of a cu!ic ice !loc that e/tens a!oe the water surface is measure. The hei"htof the ice !loc !elow the surface is to !e etermine.

Assumptions . The !uoyancy force in air is ne"li"i!le. / The top surface of the ice !loc is parallel to the surface of thesea.

Properties The specific "raities of ice an seawater are "ien to !e '.;& an +.'&=, respectiely, an thus thecorresponin" ensities are ;&' "@m* an +'&= "@m*.

Analysis The wei"ht of a !oy floatin" in a flui is e7ual to the !uoyant force actin" on it (a conse7uence of erticalforce !alance from static e7uili!rium). Therefore, in this case the aera"e ensity of the !oy must !e e7ual to the ensityof the flui since

W 4 $ - → su!mer"efluitotal !oy V V g g ρ ρ =

flui

!oy

total

su!mer"e

ρ

ρ =

V

V

The cross-sectional of a cu!e is constant, an thus the “olume ratio” can !ereplace !y “hei"ht ratio”. Then,

'&=.+

;&.'

+'.'

+'.'

water

ice

flui

!oy

total

su!mer"e =+

→=+

→=h

h

h

h

h

h

ρ

ρ

ρ

ρ

where h is the hei"ht of the ice !loc !elow the surface. 5olin" for h "ies

h < '.A? m < 8.6 m

Discussion Bote that the '.;&@+.'&= < ;'K of the olume of an ice !loc remains uner water. >or symmetrical ice !locs this also represents the fraction of hei"ht that remains uner water.

3,5 Solution 0 man ies into a lae an tries to lift a lar"e roc. The force that the man nees to apply to lift it fromthe !ottom of the lae is to !e etermine.

Assumptions . The roc is c completely su!mer"e in water. / The !uoyancy force in air is ne"li"i!le.

Properties The ensity of "ranite roc is "ien to !e &A'' "@m *. e tae the ensity of water to !e +''' "@m *.

Analysis The wei"ht an olume of the roc are

*

*

&

&

m'.'?&;? "@m&A''

"+A'

B+??m@s"+

B+)m@s")(;.++A'(

===

=

⋅==

ρ

m

mg W

V

The !uoyancy force actin" on the roc is

( ) ( ) ( )* & *

water &

+ B+''' "@m ; + m@s ' '?&;? m ?+ B

+ " m@s - $ g . . ρ

= = = ÷×

V

The wei"ht of a !oy su!mer"e in water is e7ual to the wei"h of the !oy in air

minus the !uoyancy force,in water i n air +?? ?+ -W W $ = − = − = 1!"! )

Discussion This force correspons to a mass ofin water

& &

+'=' B + B+'A -"

; + m@s + -" m@s

W m

g .

= = = ÷×

. Therefore, a

person who can lift +'A " on earth can lift this roc in water.


*-=*

5ea

+' cm

h

$ -

W

Ice !loc

W

$ -

ater

$ net

<W - $ -



Chapter 3 Pressure and Fluid Statics3,54 Solution 0n irre"ularly shape crown is wei"he in air an then in water with a sprin" scale. It is to !e etermineif the crown is mae of pure "ol.

Assumptions . The !uoyancy force in air is ne"li"i!le. / The crown is completely su!mer"e in water.

Properties e tae the ensity of water to !e +''' "@m *. The ensity of "ol is "ien to !e +;*'' "@m*.

Analysis The mass of the crown is

"&'.* B+

m@s"+

m@s+.;

B:.*+ &

&air

=

⋅== g

W m

The ifference !etween the wei"hts in air an in water is ue to the !uoyancyforce in water, an thus

B='.&;.&:.*+water air =−=−= W W $ -

Botin" that V g $ - water ρ = , the olume of the crown is etermine to !e

*:

&*water

m+'=:.&)m@s+.;)("@m(+'''

B='.& −×=== g

$ -

ρ V

Then the ensity of the crown !ecomes

*

*:"@m=?',+&

m+'=:.&

"&'.* =

×==

−V

m ρ

which is consiera!ly less than the ensity of "ol. Therefore, the $ro*n is NOT (ae o# pure gol.

Discussion This pro!lem can also !e sole without oin" any uner-water wei"hin" as follows$ e woul wei"h a !ucet half-fille with water, an rop the crown into it. 0fter marin" the new water leel, we woul tae the crown out,an a water to the !ucet until the water leel rises to the mar. e woul wei"h the !ucet a"ain. Fiiin" the wei"htifference !y the ensity of water an g will "ie the olume of the crown. 4nowin" !oth the wei"ht an the olume of the crown, the ensity can easily !e etermine.


*-=:

W wir

< *.&' "f

0ir

Crown,m+ V

W water

< &.;= "f

$ -

ater



Chapter 3 Pressure and Fluid Statics3,55 Solution The aera"e ensity of a person is etermine !y wei"hin" the person in air an then in water. 0 relationis to !e o!taine for the olume fraction of !oy fat in terms of ensities.

Assumptions . The !uoyancy force in air is ne"li"i!le. / The !oy is consiere to consist of fat an muscle only. 3 The !oy is completely su!mer"e in water, an the air olume in the lun"s is ne"li"i!le.

Analysis The ifference !etween the wei"hts of the person in air an inwater is ue to the !uoyancy force in water. Therefore,

water air water water air W W g W W $ - −=→−= V ρ 4nowin" the wei"hts an the ensity of water, the relation a!oe "ies the olume ofthe person. Then the aera"e ensity of the person can !e etermine from

V V

g W m @air ae == ρ

ner assumption L&, the total mass of a person is e7ual to the sum of the masses of the fat an muscle tissues, an thetotal olume of a person is e7ual to the sum of the olumes of the fat an muscle tissues. The olume fraction of !oy fatis the ratio of the fat olume to the total olume of the person. Therefore,

)-(+anwhere

musclefat

fatmusclemusclefatfatmusclefat

mmm

+====+= V V V V V V V V

Botin" that mass is ensity times olume, the last relation can !e written as

V V V

V V V

)+( fatmusclefatfatae

musclemusclefatfatae

−+=+=

ρ ρ ρ

ρ ρ ρ

Cancelin" the V an solin" for fat "ies the esire relation,

muscle a"

fat

muscle fat

ρ ρ

ρ ρ

−=

−

Discussion ei"hin" a person in water in orer to etermine its olume is

not practical. 0 more practical way is to use a lar"e container, an measurin"the chan"e in olume when the person is completely su!mer"e in it .


*-==

W air

0ir

1erson,m+V

W water

ater



Chapter 3 Pressure and Fluid Statics3,56 Solution The olume of the hull of a !oat is "ien. The amounts of loa the !oat can carry in a lae an in the seaare to !e etermine.

Assumptions . The ynamic effects of the waes are isre"are. / The !uoyancy force in air is ne"li"i!le.

Properties The ensity of sea water is "ien to !e +.'* ×+''' < +'*' "@m*. e tae the ensity of water to !e +'''

"@m*.

Analysis The wei"ht of the unloae !oat is

-B:.'m@s-"+'''

-B+)m@s-")(;.+=?'(

&

& !oat =

⋅== mg W

The !uoyancy force !ecomes a ma/imum when the entire hull of the !oat is su!mer"ein water, an is etermine to !e

B+:A&m@s"+'''

B+)m+=')(m@s+.;)("@m+'''(

&

*&*laelae, =

⋅== V g $ - ρ

B+=+?m@s"+'''

B+)m+=')(m@s+.;)("@m+'*'(

&

*&*seasea, =

⋅== V g $ - ρ

The total wei"ht of a floatin" !oat (loa !oat itself) is e7ual to the !uoyancyforce. Therefore, the wei"ht of the ma/imum loa is

B+:*&:+=+?

B+*:+:A&lae,

!oatsea,sealoa,

!oatlaeloa,

=−=−=

=−=−=

W $ W

W $ W

-

-

The corresponin" masses of loa are

&loa,la-e

loa,la-e &

+* -B +''' -" m@s+:+,='' -"

+ -B;.+ m@s

W m

g

×= = = ≅ ÷

14$,!!! kg

&loa,lsea

loa,sea &

+:*& -B +''' -" m@s+:= ;A' -"

+ -B;.+ m@s

W m +

g

×= = = ≅ ÷

146,!!! kg

Discussion Bote that this !oat can carry nearly :='' " more loa in the sea than it can in fresh water. >ully-loae !oats in sea water shoul e/pect to sin into water eeper when they enter fresh water, such as a rier where the port may !e.


*-=?

W loa

$ -

W !oat




5e/ie Problems

3,67 Solution Ene section of the uct of an air-conitionin" system is lai unerwater. The upwar force the water e/erts on the uct is to !e etermine.

Assumptions . The iameter "ien is the outer iameter of the uct (or, the thicness of the uct material is ne"li"i!le).

/ The wei"ht of the uct an the air in is ne"li"i!le. Properties The ensity of air is "ien to !e ρ < +.*' "@m*. e tae the ensity of water to !e +''' "@m*.

Analysis Botin" that the wei"ht of the uct an the air in it is ne"li"i!le, the net upwar force actin" on the uct isthe !uoyancy force e/erte !y water. The olume of the uner"roun section of the uct is

m '.*=*:<m)@:C(&'m)+=.'(D):@( *&&π π === 3 & A3V

Then the !uoyancy force !ecomes

k)%.4&=

⋅==

&

*&*

m@s"'''+

B+)m)('.*=*:m@s)(;.+"@m(+'''V g $ - ρ

Discussion The upwar force e/erte !y water on the uct is *.:A B, which is e7uialent to the wei"ht of a mass of

*=: ". Therefore, this force must !e treate seriously.

3,6. Solution 0 helium !alloon tie to the "roun carries & people. The acceleration of the !alloon when it is firstrelease is to !e etermine.

Assumptions The wei"ht of the ca"e an the ropes of the !alloon is ne"li"i!le.

Properties The ensity of air is "ien to !e ρ < +.+? "@m*. The ensity of helium "as is +@Ath of this.

Analysis The !uoyancy force actin" on the !alloon is

( ) ** *

!alloon

* & *

air !al loon &

: * : = m * =&*.? m

+ B(+.+? "@m )(;.+ m@s )(=&*.? m ) =;=.: B

+ " m@s -

9 r 9

$ g ρ

= = =

= = =

÷×

V

V

The total mass is

* *

He He

total He people

+.+?"@m (=&*.? m ) ?. "

A

?. & A' &&?. "

m

m m m

ρ = = = ÷

= + = + × =

V

The total wei"ht is

&

total &

+ B(&&?. ")(;.+ m@s ) &&&:.; B

+ " m@sW m g

= = = ÷×

Thus the net force actin" on the !alloon is

net =;=.? &&&:.; *A**.= B - $ $ W = − = − =

Then the acceleration !ecomes&

net

total

*A**.= B + " m@s

&&?. " + B

$ a

m

×= = = ÷

$16." m-s

Discussion This is almost twice the acceleration of "raity 3 aeroynamic ra" on the !alloon acts 7uicly to slowown the acceleration.


*-=A

Helium !alloon

m < +:' "

F <+= cm

< &' m

$ -




3,6/

Solution The preious pro!lem is reconsiere. The effect of the num!er of people carrie in the !alloon onacceleration is to !e inesti"ate. 0cceleration is to !e plotte a"ainst the num!er of people, an the results are to !eiscusse.


"ie 'ataE"r$o_air=%.%:"[&g/m^3]" "esit# o! air"g=9.807"[m/s^2]"_balloo=%0"[m]"m_%perso=70"[&g]"AoPeople = 2C "'ata sppie i Parametric *able"

"alclate alesE"r$o_He=r$o_air/7"[&g/m^3]" "esit# o! $elim"r_balloo=_balloo/2"[m]"_balloo=@pir_balloo^3/3"[m^3]"m_people=oPeoplem_%perso"[&g]"

m_He=r$o_He_balloo"[&g]"m_total=m_HeFm_people"[&g]""*$e total <eig$t o! balloo a people isE"D_total=m_totalg"[]""*$e bo#ac# !orce actig o t$e balloo6 B_b6 is eal to t$e <eig$t o! t$e air isplaceb# t$e balloo."B_b=r$o_air_balloog"[]""Brom t$e !ree bo# iagram o! t$e balloo6 t$e balacig ertical !orces mst eal t$eproct o! t$e total mass a t$e ertical acceleratioE"B_b- D_total=m_totala_p

Discussion 0s e/pecte, the more people, the slower the acceleration. In fact, if more than A people are on !oar, the !alloon oes not rise at all.


*-=

& 2 3 4 5 % 7 9 &0

5

0

5

&0

&5

20

25

30

)oPeole

a u

( m - s . $ *

Aup [m/s2] No. People

28.%9 %

%:.@: 2%0.2: 3:.@3@ @3.83% 5%.9@7 :

0.520@ 7-0.5973 8-%.@97 9-2.23: %0



Chapter 3 Pressure and Fluid Statics3,63 Solution 0 !alloon is fille with helium "as. The ma/imum amount of loa the !alloon can carry is to !eetermine.

Assumptions The wei"ht of the ca"e an the ropes of the !alloon is ne"li"i!le.

Properties The ensity of air is "ien to !e ρ < +.+? "@m*. The ensity of

helium "as is +@Ath of this.

Analysis In the limitin" case, the net force actin" on the !alloon will !e 8ero.

That is, the !uoyancy force an the wei"ht will !alance each other$

-"?'A.:m@s;.(+

B=;=(.:

& ===

==

g

$ m

$ mg W

-total

-

Thus,

people total He ?'A.: ?. =&' ? "m m m .= − = − = ≅ "$1 kg

Discussion hen the net wei"ht of the !alloon an its car"o e/cees the wei"ht of theair it isplaces, the !alloon@car"o is no lon"er “li"hter than air”, an therefore cannot rise.

3,61E Solution The pressure in a steam !oiler is "ien in "f@cm&. It is to !e e/presse in psi, 1a, atm, an !ars.

Analysis e note that + atm < +.'**&* "f@cm&, + atm < +:.?;? psi, + atm < +'+.*&= 1a, an + atm < +.'+*&= !ar (inner coer pa"e of te/t). Then the esire conersions !ecome$

In atm$ atm?.A&"f@cm+.'**&*

atm+)"f@cm(A=

&

& =

= P

In psi$

&

&

+ atm +: ?;? psi(A= "f@cm ) +'?A psi

+ atm+.'**&* "f@cm

. P

= = ≅

÷ ÷ .747 psi

In 1a$&

&

+ atm +'+ *&= 1a(A= "f@cm ) A*== 1a

+ atm+.'**&* "f@cm

. P

= = ≅ ÷ ÷

437 &Pa

In !ars$&

&

+ atm + '+*&= !ar (A= "f@cm ) A* == !ar

+ atm+.'**&* "f@cm

. P .

= = ≅ ÷ ÷

43' bar

Discussion Bote that the units atm, "f@cm&, an !ar are almost ientical to each other. 0ll final results are "ien tothree si"nificant i"its, !ut conersion ratios are typically precise to at least fie si"nificant i"its.


*-=;

Helium !alloon

m



Chapter 3 Pressure and Fluid Statics3,62 Solution 0 !arometer is use to measure the altitue of a plane relatie to the "roun. The !arometric reain"s atthe "roun an in the plane are "ien. The altitue of the plane is to !e etermine.

Assumptions The ariation of air ensity with altitue is ne"li"i!le.

Properties The ensities of air an mercury are "ien to !e ρ air < +.&' "@m* an ρ mercury < +*,?'' "@m*.

Analysis 0tmospheric pressures at the location of the plane an the "roun leel are

-1a+''.:? B@m+'''

-1a+

m@s-"+

B+m))('.A=*m@s+)(;.-"@m(+*,?''

)(

-1a;&.'? B@m+'''

-1a+

m@s-"+

B+m))('.?;'m@s)(;.+-"@m(+*,?''

)(

&&

&*

"roun"roun

&&

&*

plane plane

=

⋅=

=

=

⋅=

=

h g P

h g P

ρ

ρ

Tain" an air column !etween the airplane an the "roun an writin" aforce !alance per unit !ase area, we o!tain

1a;&.'?)(+''.:? B@m+'''

1a+

m@s"+

B+))(m@s+)(;."@m(+.&'

)(

@

&&

&*

plane"rounair

plane"rounair

−=

⋅

−=

−=

h

P P h g

P P AW

ρ

It yiels h < 4.1 () which is also the altitue of the airplane.

Discussion E!iously, a mercury !arometer is not practical on an airplane 3 an electronic !arometer is use instea.

3,6 Solution The pressure of a "as containe in a ertical piston-cyliner eice is measure to !e ='' 1a. The mass of the piston is to !e etermine.

Assumptions There is no friction !etween the piston an the cyliner.

Analysis Frawin" the free !oy ia"ram of the piston an !alancin" the erticalforces yiel

( )

( ) ( ) ( ) ( )

atm

atm&

& : & +''' "@m s;.+ m@s ='' +'' 1a *' +' m

+ 1a

W PA P A

mg P P A

m −

= −

= − ×

= − × ÷

5olution of the a!oe e7uation yiels m < 1$$ kg.

Discussion The "as cannot istin"uish !etween pressure ue to the piston wei"ht an atmospheric pressure 3 !oth“feel” lie a hi"her pressure actin" on the top of the "as in the cyliner.


*-?'

W < mg

P atm

P

h

' 5ea leel



Chapter 3 Pressure and Fluid Statics3,64 Solution The "a"e pressure in a pressure cooer is maintaine constant at +'' 1a !y a petcoc. The mass of the

petcoc is to !e etermine.

Assumptions There is no !loca"e of the pressure release ale.

Analysis 0tmospheric pressure is actin" on all surfaces of the petcoc, which !alances itself out. Therefore, it can

!e isre"are in calculations if we use the "a"e pressure as the cooer pressure. 0 force !alance on the petcoc ( Σ $ y < ')

yiels

? & &

&

(+'' 1a)(: +' m ) +''' "@m s

+ 1a;.+ m@s

gage

gage

W P A

P Am

g

−

=

× ×= = ÷

= =!.!4!8 kg 4!.8 g

Discussion The hi"her pressure causes water in the cooer to !oil at a hi"her temperature.

3,65 Solution The air pressure in a uct is measure !y an incline manometer. >or a "ien ertical leel ifference, the"a"e pressure in the uct an the len"th of the ifferential flui column are to !e etermine.

Assumptions The manometer flui is an incompressi!le su!stance.

Properties The ensity of the li7ui is "ien to !e ρ < '.+ "@ < +' "@m

*

. Analysis The "a"e pressure in the uct is etermine from

"a"e a!s atm

* &

& &

+ B + 1a(+'"@m )(;.+ m@s )('.'m)

+ " m@s +B@m

P P P gh ρ = − =

= ÷ ÷× = 3 Pa

The len"th of the ifferential flui column is

( )sin cm sin*= 3 h θ = = ° = .3'6 $(

Discussion Bote that the len"th of the ifferential flui column is e/tene consiera!ly !y inclinin" the manometerarm for !etter reaa!ility (an therefore hi"her precision).

3,66E Solution 27ual olumes of water an oil are poure into a -tu!e from ifferent arms, an the oil sie is

pressuri8e until the contact surface of the two fluis moes to the !ottom an the li7ui leels in !oth arms !ecome thesame. The e/cess pressure applie on the oil sie is to !e etermine.

Assumptions . #oth water an oil are incompressi!le su!stances. / Eil oes not mi/ with water. 3 The cross-sectionalarea of the -tu!e is constant.

Properties The ensity of oil is "ien to !e ρ oil < :;.* l!m@ft*. e tae the ensity of water to !e ρ w < ?&.: l!m@ft*.

Analysis Botin" that the pressure of !oth the water an the oil is thesame at the contact surface, the pressure at this surface can !e e/presse as

wwatmaa !lowcontact gh P gh P P ρ ρ +=+=

Botin" that ha < hw an rearran"in",

( )

( ) ( ) ( )

"a"e, !low !low atm oil&

* &

& &

+ l!f + ft?&.:- :;.* l!m@ft *& & ft@s *'@+& ft

*&.& l!m ft@s +:: in

w P P P gh

.

ρ ρ = − = −

= ÷ ÷× = !.$$ si

Discussion hen the person stops !lowin", the oil rises an some water flows into the ri"ht arm. It can !e shown thatwhen the curature effects of the tu!e are isre"are, the ifferential hei"ht of water is &*.A in to !alance *'-in of oil.


*-?+

W < mg

P atm

P

0ir

*=°

cm 3

*' in

#lownair

ater

Eil



Chapter 3 Pressure and Fluid Statics3,.77 Solution It is "ien that an I flui an the !loo pressures !alance each other when the !ottle is at a certainhei"ht, an a certain "a"e pressure at the arm leel is neee for sufficient flow rate. The "a"e pressure of the !loo aneleation of the !ottle re7uire to maintain flow at the esire rate are to !e etermine.

Assumptions . The I flui is incompressi!le. / The I !ottle is open to the atmosphere.

Properties The ensity of the I flui is "ien to !e ρ < +'&' "@m*.

Analysis (a) Botin" that the I flui an the !loo pressures !alance each other when the !ottle is +.& m a!oe the

arm leel, the "a"e pressure of the !loo in the arm is simply e7ual to the "a"e pressure of the I flui at a epth of +.&m,

Pak 1$.!=

⋅=

=−=

&&

&*

!ottle-armatma!sarm"a"e,

B@m+

1a+

m@s"'''+

B+m))(+.&'m@s)(;.+"@m(+'&'

gh P P P ρ

(b) To proie a "a"e pressure of &' 1a at the arm leel, the hei"ht of the !ottle from

the arm leel is a"ain etermine from !ottle-armarm"a"e, gh P ρ = to !e

m$.!=

⋅=

=

1a+

B@m+

B+

m@s"'''+

)m@s)(;.+"@m(+'&'

1a&'&&

&*

arm"a"e,

!ottle-arm g

P h

ρ

Discussion Bote that the hei"ht of the reseroir can !e use to control flow rates in "raity rien flows. hen thereis flow, the pressure rop in the tu!e ue to friction shoul also !e consiere. This will result in raisin" the !ottle a little

hi"her to oercome pressure rop.


*-?&

+.& m

P atm

I#ottle



Chapter 3 Pressure and Fluid Statics3,.7.E Solution 0 water pipe is connecte to a ou!le- manometer whose free arm is open to the atmosphere. Thea!solute pressure at the center of the pipe is to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / The solu!ility of the li7uis in each other is ne"li"i!le.

Properties The specific "raities of mercury an oil are "ien to !e +*.? an '.', respectiely. e tae the ensity of

water to !e ρ w < ?&.: l!m@ft*.

Analysis 5tartin" with the pressure at the center of the water pipe, an moin" alon" the tu!e !y ain" (as we "o

own) or su!tractin" (as we "o up) the gh ρ terms until we reach the free surface of oil where the oil tu!e is e/pose tothe atmosphere, an settin" the result e7ual to P atm "ies

atm P gh gh gh gh P =−−+− oiloilH"H"alcoholalcoholwater water pipewater ρ ρ ρ ρ

5olin" for P water pipe,

)( oiloilH"H"alcoholoilwater water water pipe h#)h#)h#)h g P P atm ++−+= ρ

5u!stitutin",

sia$$.%=

⋅×+

+−+=

&

&

&

&* pipewater

in+::

ft+

ft@sl!m*&.&

l!f +

ft)C(:'@+&.'

ft(+=@+&?.+*ft) (?'@+&'.'ft))D(*=@+&ft@s&.*&()l!m@ft(?&.: psia+:.& P

Therefore, the a!solute pressure in the water pipe is &&.* psia.



*-?*

*= in

Eil

:' in

Mercury

+= in

Eil

?' in

ater



Chapter 3 Pressure and Fluid Statics3,.7/ Solution The pressure of water flowin" throu"h a pipe is measure !y an arran"ement that inoles !oth a pressure"a"e an a manometer. >or the alues "ien, the pressure in the pipe is to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / The effect of air column on pressure is ne"li"i!le.

Properties The specific "raity of "a"e flui is "ien to !e &.:. e tae the stanar ensity of water to !e ρ w < +'''

"@m*.

Analysis 5tartin" with the pressure inicate !y the pressure "a"e an moin" alon" the tu!e !y ain" (as we "o

own) or su!tractin" (as we "o up) the gh ρ terms until we reach the water pipe, an settin" the result e7ual to P water

"ie

water w&w"a"e"a"e+w"a"e P gh gh gh P w =−−+ ρ ρ ρ

earran"in",

( ) ( )water "a"e w + "a"e "a"e w& "a"e w & "a"e + &5G 5G sin sinw P P g h h h P g h 3 3 ρ ρ θ θ = + − − = + − −

Botin" that ???A.'+&@sin ==θ an su!stitutin",

kPa %%.6=

⋅×

−−+=

&&

&*water

B@m+

1a+

m@s"+'''

B+

m)'.???AC'?.'(m)'.???A'?.'(:.&m))D('.='m@s(;.+)"@m(+'''1a*' P

Therefore, the pressure in the "asoline pipe is *.? 1a oer the reain" of the pressure "a"e.

Discussion Bote that een without a manometer, the reain" of a pressure "a"e can !e in error if it is not place at the

same leel as the pipe when the flui is a li7ui.


*-?:

P '<*' 1a

ater ater

0ir

Ga"e flui5G<&.:

h& < =' cm

3+<? cm

3&<? cm

h+< cm

1ipe

+='C

θ



Chapter 3 Pressure and Fluid Statics3,.73 Solution 0 pressure transucer is use to measure pressure !y "eneratin" analo"ue si"nals, an it is to !e cali!rate

!y measurin" !oth the pressure an the electric current simultaneously for arious settin"s, an the results are to !eta!ulate. 0 cali!ration cure in the form of P < a0 b is to !e o!taine, an the pressure corresponin" to a si"nal of +'m0 is to !e calculate.

Assumptions Mercury is an incompressi!le li7ui.

Properties The specific "raity of mercury is "ien to !e +*.=?, an thus its ensity is +*,=?' "@m *.

Analysis >or a "ien ifferential hei"ht, the pressure can !e calculate fromh g P ∆= ρ

>or ∆h < &.' mm < '.'&' m, for e/ample,

1aA&.*B@m+

1a+

m@s"+'''

B+m))('.'&'m@s(;.+)"@m(+'''=?.+*

&&

&* =

⋅= P

epeatin" the calculations an ta!ulatin", we hae

∆h(mm

)

&.' ++.= &;A. :+*.+ A?=.; +'&A ++:; +*?& +:= +=*?

P 7kPa %.$ $4.14 %#.61 "4.#" 1!1.# 1%6.6 1"$.8 181.$ 1#%.# $!4.%

0 (m0) :.&+ =.A ?.;A .+= ++.A? +:.:* +=.? +A.? +.: +;.?:

0 plot of P ersus 0 is "ien !elow. It is clear that the pressure aries linearly with the current, an usin" 225, the !estcure fit is o!taine to !e

P < +*.'' 0 8 =+.'' (1a) for ?:.+;&+.: ≤≤ 0 .

>or 0 < +' m0, for e/ample, we woul "et P < #.! kPa.

Discussion Bote that the cali!ration relation is ali in the specifie ran"e of currents or pressures.


*-?=

4 % &0 &2 &4 &% & 200

45

90

&35

&0

225

I , m9

P ,

k P a

Multimeter

∆h

1ressuri8e0ir, P

1ressuretransucer

ale

i"i container Manometer

Mercury5G<+*.=?



Chapter 3 Pressure and Fluid Statics3,.71 Solution 0n oil pipeline an a ri"i air tan are connecte to each other !y a manometer. The pressure in the

pipeline an the chan"e in the leel of manometer flui ue to a air temperature rop are to !e etermine.

Assumptions . 0ll the li7uis are incompressi!le. / The effect of air column on pressure is ne"li"i!le. 3 The air olumein the manometer is ne"li"i!le compare with the olume of the tan.

Properties The specific "raities are "ien to !e &.? for oil an +*.? for mercury. e tae the stanar ensity of

water to !e ρ w < +''' "@m*. The "as constant of air is '.&A 1a⋅m*@"⋅4.

Analysis (a) 5tartin" with the oil pipe an moin" alon" the tu!e !y ain" (as we "o own) or su!tractin" (as we"o up) the gh ρ terms until we reach the air tan, an settin" the result e7ual to P air "ie

air H"H"oiloil P gh gh P oil =++ ρ ρ

The a!solute pressure in the air tan is etermine from the ieal-"as relation P V < m%2 to !e

1a++?;m*.+

&A*)4 4)('@"m1a")('.&A+=(*

*

=+⋅⋅

==V

m%2 P air

Then the a!solute pressure in the oil pipe !ecomes

( ) ( )

oil air oil H" H"

* &

& &

+ B + 1a++?; 1a +''' "@m (;.+ m@s ) &.?('.A= m) +*.? ' &' m

+''' " m@s + B@m

++&* 1a

oil P P gh gh

.

ρ ρ = − −

= − + ÷ ÷ × = ≅ 11$! kPa

(b) The pressure in the air tan when the temperature rops to &' °C !ecomes

1a;A'm*.+

&A*)4 4)(&'@"m1a")('.&A+=(*

*

=+⋅⋅

==V

m%2 P air

hen the mercury leel in the left arm rops a istance , the rise in the mercury leel in the ri"ht arm y !ecomes

ri"htleft V V = → yd .d &&)*( π π = → . y ;= an °= ='sin; yvert

an the mercury flui hei"ht will chan"e !y °+ ='sin; . . or A.;: . Then,

air H"H"oiloil );:.A()( P .h g .h g P oil =−+++ ρ ρ → ( ) ( ) air oil

oil oil H" H"5G 5G A ;:w

P P h h .

g ρ

−+ + − =

5u!stitutin",

⋅−=−++

1a+

B@m+

B+

m@s"+'''

)m@s)(;.+"@m+'''(

1a)++&*;A'();:.A&'.'(?.+*)A=.'(?.&

&&

&* . .

which yiels m1#.4m!.1#4 == . . Therefore, the oil-mercury

interface will rop +;.: cm as a result of the temperature rop of air.

Discussion Bote that the pressure in constant-olume "as cham!ers is ery sensitie to temperature chan"es.


*-??

0, Eil5G<&.?

#

0ir, ' 'C

hH"

< ∆h < &' cm

Mercury5G<+*.?

*d

d < : mm

hoil

< A=

cm

=''



Chapter 3 Pressure and Fluid Statics3,.72 Solution The ensity of a woo lo" is to !e measure !y tyin" lea wei"hts to it until !oth the lo" an the wei"htsare completely su!mer"e, an then wei"hin" them separately in air. The aera"e ensity of a "ien lo" is to !eetermine !y this approach.

Properties The ensity of lea wei"hts is "ien to !e ++,*'' "@m*. e tae the ensity of water to !e +''' "@m *.

Analysis The wei"ht of a !oy is e7ual to the !uoyant force when the !oy is floatin" in a flui while !ein"completely su!mer"e in it (a conse7uence of ertical force !alance from static e7uili!rium). In this case the aera"eensity of the !oy must !e e7ual to the ensity of the flui since

flui !oyflui !oy ρ ρ ρ ρ =→=→= V V g g $ W -

Therefore,

water

lo"lea

lealo"water lo"lea

lo"lea

total

total

ρ ρ ρ

mmmmmave

++=→=

+

+== V V

V V V

where

"'.+=A B+

m@s"+

m@s+.;

B+=:'

m+''+.* "@m*'',++

"*:

&

&

lo"

lo"

**

*lea

lealea

=

⋅==

×=== −

g

W m

m

ρ V

5u!stitutin", the olume an ensity of the lo" are etermine to !e

*

*

**

water

lo"lea

lealo""@m+'''

")+=A*:(m+''+.* m!.1#4=

++×=

++= −

ρ

mmV V

%kg-m8!#===

*lo"

lo"

lo"m+;:.'

"+=A

V

m ρ

Discussion Bote that the lo" must !e completely su!mer"e for this analysis to !e ali. Ieally, the lea wei"hts mustalso !e completely su!mer"e, !ut this is not ery critical !ecause of the small olume of the lea wei"hts.


*-?A

ea, *: "

o", +=:' B

$ -

ater



Chapter 3 Pressure and Fluid Statics3,.7 8 Also solved using !!# on enclosed &'&9

Solution 0 rectan"ular "ate that leans a"ainst the floor with an an"le of := ° with the hori8ontal is to !e opene

from its lower e"e !y applyin" a normal force at its center. The minimum force $ re7uire to open the water "ate is to !eetermine.

Assumptions . 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !e i"nore in calculations for conenience. / >riction at the hin"e is ne"li"i!le.


Analysis The len"th of the "ate an the istance of the upper e"e of the "ate (point -) from the free surface in the plane of the "ate are

mA'A+.':=sin

m=.' an m&:*.:

:=sin

m*=

°==

°= sb

The aera"e pressure on a surface is the pressure at the centroi (mipoint) ofthe surface, an multiplyin" it !y the plate area "ies the resultant hyrostaticon the surface,

( ) ( ) ( ) ( )a"

* & &

&

+ B+''' "@m ; + m@s & m = :.&:* m

+''' " m@s

:+? B

% , $ P A gh A

.

ρ = =

= × ÷×

=The istance of the pressure center from the free surface of water alon" the planeof the "ate is

m*=;.*)&@&:*.:A'A+.'(+&

&:*.:

&

&:*.:A'A+.'

)&@(+&&

&&

=+

++=+

++=b s

bb s y P

The istance of the pressure center from the hin"e at point - is

m?=&.&A'A+.'*=;.* =−=−= s y 3 P P

Tain" the moment a!out point - an settin" it e7ual to 8ero "ies

&@ ' $b 3 $ 1 P % - =→=∑5olin" for $ an su!stitutin", the re7uire force is etermine to !e

k)"$!===m:.&:*

m)B)(&.?=&:+?(&&

b

3 $ $ P %

Discussion The applie force is inersely proportional to the istance of the point of application from the hin"e, anthe re7uire force can !e reuce !y applyin" the force at a lower point on the "ate.


*-?

$ %

$:=°

#

'.= m

* m

A



Chapter 3 Pressure and Fluid Statics3,.74

Solution 0 rectan"ular "ate that leans a"ainst the floor with an an"le of := ° with the hori8ontal is to !e opene

from its lower e"e !y applyin" a normal force at its center. The minimum force $ re7uire to open the water "ate is to !eetermine.



Analysis The len"th of the "ate an the istance of the upper e"e of the "ate(point -) from the free surface in the plane of the "ate are

m?;A.+:=sin

m&.+anm&:*.:

:=sin

m*=

°==

°= sb

The aera"e pressure on a surface is the pressure at the centroi (mipoint) of thesurface, an multiplyin" it !y the plate area "ies the resultant hyrostatic on thesurface,

( ) ( ) ( ) ( )a"

* & &

&

+ -B+''' -"@m ; + m@s & A m = :.&:* m

+''' -" m@s

=?& -B

% , $ P A gh A

. .

ρ = =

= × ÷×

=The istance of the pressure center from the free surface of water alon" the planeof the "ate is

m &++.:)&@&:*.:?;A.+(+&

&:*.:

&

&:*.:?;A.+

)&@(+&&

&&

=+

++=+

++=b s

bb s y P

The istance of the pressure center from the hin"e at point - is

m=+:.&?;A.+&++.: =−=−= s y 3 P P

Tain" the moment a!out point - an settin" it e7ual to 8ero "ies

&@ ' $b 3 $ 1 P % - =→=∑5olin" for $ an su!stitutin", the re7uire force is etermine to !e

k) 666===m:.&:*

m) B)(&.=+:=?&(&&

b

3 $ $ P %

Discussion The applie force is inersely proportional to the istance of the point of application from the hin"e, anthe re7uire force can !e reuce !y applyin" the force at a lower point on the "ate.


*-?;

$ %

$:=°

#

+.& m

* m

A



Chapter 3 Pressure and Fluid Statics3,.75 Solution 0 rectan"ular "ate hin"e a!out a hori8ontal a/is alon" its upper e"e is restraine !y a fi/e ri"e at

point -. The force e/erte to the plate !y the ri"e is to !e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !e i"nore in calculations for conenience.



(mipoint) of the surface, an multiplyin" it !y the plate area "ies the resultanthyrostatic force on the "ate,

( ) ( ) ( ) ( )a"

* & &

&

+ B+''' "@m ; + m@s * = m * ? m

+''' " m@s

% , $ P A gh A

. .

ρ = =

= × ÷× = 618 k)

The ertical istance of the pressure center from the free surface of water is

m %.&1=+

++=+

++=)&@*&(+&

*

&

*&

)&@(+&&

&&

b s

bb s y P

Discussion Nou can calculate the force at point - re7uire to hol !ac the "ate !y settin" the net moment aroun

hin"e point A to 8ero.

3,.76 Solution 0 rectan"ular "ate hin"e a!out a hori8ontal a/is alon" its upper e"e is restraine !y a fi/e ri"e at

point -. The force e/erte to the plate !y the ri"e is to !e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of the "ate, an thus it can !e i"nore in calculations for conenience.


Analysis The aera"e pressure on a surface is the pressure at the centroi(mipoint) of the surface, an multiplyin" it !y the wette plate area "ies theresultant hyrostatic force on the "ate,

k) 118=

⋅×=

==

&

&&*

m@s"+'''

B+Cm?m)D&+)(m@s+.;)("@m+'''(

A gh A P $ , ave % ρ

The ertical istance of the pressure center from the free surface of water is

m 1.%%===*

)m&(&

*

&h y P

Discussion Compare to the preious pro!lem (with hi"her water epth), the force is much smaller, as e/pecte. 0lso,the center of pressure on the "ate is much lower (closer to the "roun) for the case with the lower water epth.


*-A'

$ %

* m

y1

h < & m

A

$ %

* m

A

& m

y p



Chapter 3 Pressure and Fluid Statics3,..7E Solution 0 semicircular tunnel is to !e !uilt uner a lae. The total hyrostatic force actin" on the roof of the tunnelis to !e etermine.

Assumptions 0tmospheric pressure acts on !oth sies of thetunnel, an thus it can !e i"nore in calculations for conenience.

Properties e tae the ensity of water to !e ?&.: l!m@ft*

throu"hout.

Analysis e consier the free !oy ia"ram of the li7ui !loc enclose !y the circular surface of the tunnel an its ertical(on !oth sies) an hori8ontal pro9ections. The hyrostatic forcesactin" on the ertical an hori8ontal plane surfaces as well as thewei"ht of the li7ui !loc are etermine as follows$

Hori8ontal force on ertical surface (each sie)$

( )

( ) ( ) ( ) ( )

a"

* &

&

&

+ l!f ?& : l!m@ft *& & ft@s +*= += & ft += ft '' ft

*&.& l!m ft@s

+ '?A +' l!f (on each sie of the tunnel)

H , $ $ P A gh A g s % A

. .

.

ρ ρ = = = = +

= + × ÷× = ×

ertical force on hori8ontal surface (downward )$

( ) ( ) ( ) ( )

a" top

* &

&

+ l!f ?& : l!m@ft *& & ft@s +*= ft *' ft '' ft

*&.& l!m ft@s

& '&& +' l!f

y , $ P A gh A gh A

. .

.

ρ ρ = = =

= × ÷× = ×

ei"ht of flui !loc on each sie within the control olume (downward)/

sie)each(on l!f +':+'.&

ft@sl!m*&.&

l!f +ft)@:)(''-(+ft)+=)(ft@s&.*&)(l!m@ft:.?&(

ft)&''')(:@(

?

&

&&*

&&

×=

⋅=

−===

π

π ρ ρ % % g g mg W V

Therefore, the net ownwar ertical force is ?

& & '&& +' & ' '&:+' +'' y $ $ W . .= + = × + × × = × 8$.! 1! lbf

This is also the net #or$e actin" on the tunnel since the hori8ontal forces actin" on the ri"ht an left sie of the tunnelcancel each other since they are e7ual an opposite.

Discussion The wei"ht of the two water !ocs on the sies represents only a!out &.:K of the total ertical force on thetunnel. Therefore, to o!tain a reasona!le first appro/imation for eep tunnels, these olumes can !e ne"lecte, yielin" $ '

< &.'& × +' l!f. 0 more conseratie appro/imation woul !e to estimate the force on the bottom of the lae if the tunnel

were not there. This yiels $ ' < &.&= × +' l!f. The actual force is !etween these two estimates, as e/pecte.


*-A+

$ y

% < += ft

$ $

W



Chapter 3 Pressure and Fluid Statics3,... Solution 0 hemispherical ome on a leel surface fille with water is to !e lifte !y attachin" a lon" tu!e to the topan fillin" it with water. The re7uire hei"ht of water in the tu!e to lift the ome is to !e etermine.

Assumptions . 0tmospheric pressure acts on !oth sies of the ome, an thus it can !e i"nore in calculations for conenience. / The wei"ht of the tu!e an the water in it is ne"li"i!le.


Analysis e tae the ome an the water in it as the system. hen the ome is a!out to rise, the reaction force

!etween the ome an the "roun !ecomes 8ero. Then the free !oy ia"ram of this system inoles the wei"hts of theome an the water, !alance !y the hyrostatic pressure force from !elow. 5ettin" these forces e7ual to each other "ies

g m g m % %h g

W W $ $

water dome

water dome' y

+=+

+==∑&)(

$'

π ρ

5olin" for h "ies

% %

%m %

%

mmh

domewater dome −+

=−+

=&

*

&

C?@:D

ρπ

π ρ

ρπ

5u!stitutin",

m!.&&=−+= m)*(m)*()"@m+'''(

?@m)*)("@m+'''(:")''',='(&*

**

π

π h

Therefore, this ome can !e lifte !y attachin" a tu!e which is AA cm lon".

Discussion Bote that the water pressure in the ome can !e chan"e "reatly !y a small amount of water in the ertical tu!e.


*-A&

$ '

% < * m

h

W



Chapter 3 Pressure and Fluid Statics3,../ Solution The water in a reseroir is restraine !y a trian"ular wall. The total force (hyrostatic atmospheric)actin" on the inner surface of the wall an the hori8ontal component of this force are to !e etermine.


Properties e tae the ensity of water to !e +''' "@m*

throu"hout.

AnalysisThe len"th of the wall surface unerwater is

mA.&?'sin

m&==

°=b

The aera"e pressure on a surface is the pressure at the centroi(mipoint) of the surface, an multiplyin" it !y the plate area"ies the resultant hyrostatic force on the surface,

( )

( ) ( ) ( ) ( )a" atm

& * & &

&

+ B+'' ''' B@m +''' -"@m ; + m@s +& = m +=' & A m

+ -" m@s

% , $ P A P gh A

+ . . .

ρ = = +

= + × ÷ × = × 8#.64 1! )

Botin" that

mAA.++ B+

m@s"+

?'sin)m@s+.;)("@m+'''(

B@m''',+''

?'sin

&

&*

&' =

⋅

°=

° g

P

ρ

the istance of the pressure center from the free surface of water alon" the wall surface is

m1&.1=

++

++=

++

++=mAA.++

&

mA.&'+&

m)A.&(

&

mA.&'

sin&+&

&

&

'

&

θ ρ g

P b s

bb s y p

The ma"nitue of the hori8ontal component of the hyrostatic force is simply $ %sin θ,

)1!8.%" 8×=°×== B)sin?'+'?:.;(sinθ % H $ $

Discussion 0tmospheric pressure is usually i"nore in the analysis for conenience since it acts on !oth sies of thewalls.


*-A*

$ %

h < &=m

y p



Chapter 3 Pressure and Fluid Statics3,..3 Solution 0n elastic air !alloon su!mer"e in water is attache to the !ase of the tan. The chan"e in the tensionforce of the ca!le is to !e etermine when the tan pressure is increase an the !alloon iameter is ecrease inaccorance with the relation P 4 ,&-&.

Assumptions . 0tmospheric pressure acts on all surfaces, an thus it can !ei"nore in calculations for conenience. / ater is an incompressi!le flui. 3The wei"ht of the !alloon an the air in it is ne"li"i!le.

Properties e tae the ensity of water to !e +''' "@m *.

Analysis The tension force on the ca!le holin" the !alloon isetermine from a force !alance on the !alloon to !e

-balloon -cable $ W $ $ ≅−=

The !uoyancy force actin" on the !alloon initially is

BA.+*m@s"+

B+

?

m)('.*')m@s(;.+) "@m(+'''

?

&

*&*

*+

w+,w+, =

⋅=== π π

ρ ρ &

g g $ balloon - V

The ariation of pressure with iameter is "ien as&

−= ,& P , which is e7uialent to P , & @= . Then the finaliameter of the !all !ecomes

m'A=.'M1a?.+

M1a+.'m)*'.'(

@

@

&

++&

&

+

+

&

+

& ===→== P

P & &

P

P

P ,

P ,

&

&

The !uoyancy force actin" on the !alloon in this case is

B&.&m@s"+

B+

?

m)('.'A=)m@s(;.+)"@m (+'''

?

&

*&*

*&

w&,w&, =

⋅=== π π

ρ ρ &

g g $ balloon - V

Then the percent chan"e in the ca!le for !ecomes

#8.4:=−

=−

=−

= +''OA.+*

&.&A.+*+''O+''OK

+,

&,+,

+,

&,+,

-

- -

cable

cablecable

$

$ $

$

$ $ ,hange .

Therefore, increasin" the tan pressure in this case results in ;.:K reuction in ca!le tension.

Discussion e can o!tain a relation for the chan"e in ca!le tension as follows$

−=

−=

−=

−=

−=

&@*

&

+

*+

*&

!alloon,+

!alloon,&

!alloon,+w

!alloon,&w !alloon,+w

+,

&,+,

++''++''++''

+''O+''OK

P

P

&

&

g

g g

$

$ $ ,hange

-

- -

V

V

V

V V

ρ

ρ ρ

.


*-A:

ater

P +<+'' 1a

&+<*' cm




3,..1

Solution The preious pro!lem is reconsiere. The effect of the air pressure a!oe the water on the ca!le force asthe pressure aries from '.+ M1a to +' M1a is to !e inesti"ate.


P%=0.% "?Pa"

$age=%00,%-,P%/P2^%.5

Tan pressure P &, M1a

KChan"e inca!le tension

'.+'.&'.*'.:'.?'.+&*:

=?A;

+'

'.'?:.?'.A.=;*.&;=.?;?.;.;;;.:;;.?

;;.A;;.;;.;;.;;;.;;;.;

Discussion The chan"e in ca!le tension isat first ery rapi, !ut leels off as the !alloon shrins to nearly 8ero iameter at hi"h pressure.

3,..2 Solution 0n ice!er" floatin" in seawater is consiere. The olume fraction of the ice!er" su!mer"e in seawater is

to !e etermine, an the reason for their turnoer is to !e e/plaine. Assumptions . The !uoyancy force in air is ne"li"i!le. / The ensity of ice!er" an seawater are uniform.

Properties The ensities of ice!er" an seawater are "ien to !e ;+A "@m * an +':& "@m*, respectiely.

Analysis (a) The wei"ht of a !oy floatin" in a flui is e7ual to the !uoyant force actin" on it (a conse7uence of ertical force !alance from static e7uili!rium). Therefore,

W 4 $ -

su!mer"efluitotal !oy V V g g ρ ρ =

88:or'.'+':&

;+A

seawater

ice!er"

flui

!oy

total

su!mer"e ==== ρ

ρ

ρ

ρ

V

V

Therefore, K of the olume of the ice!er" is su!mer"e in this case.

(b) Heat transfer to the ice!er" ue to the temperature ifference !etween theseawater an an ice!er" causes uneen meltin" of the irre"ularly shape ice!er".The resultin" shi#t in the $enter o# (ass $auses the i$eberg to turn over.

Discussion The su!mer"e fraction epens on the ensity of seawater, an this fraction can iffer in ifferent seas.


*-A=

5ea

$ -

W

Ice!er"

0 2 4 % &00

20

40

%0

0

&00

P$, MPa

C a n g e i n ' B , :



Chapter 3 Pressure and Fluid Statics3,.. Solution 0 cylinrical container e7uippe with a manometer is inerte an presse into water. The ifferentialhei"ht of the manometer an the force neee to hol the container in place are to !e etermine.

Assumptions . 0tmospheric pressure acts on all surfaces, an thus it can !e i"nore in calculations for conenience. /The ariation of air pressure insie cyliner is ne"li"i!le.

Properties e tae the ensity of water to !e +''' "@m *. The ensity of the manometer flui is

( )* *

mano 5G & + +''' "@m &+'' "@mw . ρ ρ = × = =

Analysis The pressures at point A an - must !e the same since they are on the same hori8ontal line in the sameflui. Then the "a"e pressure in the cyliner !ecomes

* & &

air, "a"e w w &

+ B (+''' "@m (;.+ m@s )('.&' m) +;?& B@m +;?& 1a

+ " m@s P gh ) ρ

= = = = ÷×

The manometer also inicates the "a"e pressure in the cyliner. Therefore,

( )& &

air, "a"e

air, "a"e * & &mano

mano

+;?& B@m + " m@s '.';=' m

(&+'' "@m )(;.+ m@s ) + B@m

P P gh h

g ρ

ρ

×= → = = = = ÷

#."! m

0 force !alance on the cyliner in the ertical irection yiels

air, "a"e c $ W P A+ =

5olin" for $ an su!stitutin",

( ) ( ) &&

&

air, "a"e

'.*' m+;?& B@m A; B

: :

& $ P W

π π = − = − = "#. )

Discussion e coul also sole this pro!lem !y consierin" the atmospheric pressure, !ut we woul o!tain the sameresult since atmospheric pressure woul cancel out.


*-A?

0ir

h Manometer flui 5G<&.+

&' cm

& < *' cm

$

ater A B• •




0esign and ;ssay Problems

3,..4

Solution e are to iscuss the esi"n of shoes that ena!le people to wal on water.

Discussion 5tuents6 iscussions shoul !e uni7ue an will iffer from each other.

3,..5

Solution e are to iscuss how to measure the olume of a roc without usin" any olume measurement eices.

Analysis The olume of a roc can !e etermine without usin" any olume measurement eices as follows$ ewei"h the roc in the air an then in the water. The ifference !etween the two wei"hts is ue to the !uoyancy force,

which is e7ual to y - g $ !owater V ρ = . 5olin" this relation for V !oy "ies the olume of the roc.

Discussion 5ince this is an open-ene esi"n pro!lem, stuents may come up with ifferent, !ut e7ually alitechni7ues.

ch03 solutions

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