Captulo 2

ESTTICA DOS FLUIDOS

A ausncia de movimento elimina os efeitos tangenciais e conseqentemente a presena de tenses de cisalhamento. A presena exclusiva de efeitos normais faz com que o objetivo deste captulo seja o estudo da presso. Nesse caso so vistas suas propriedades num fluido em repouso, suas unidades, as escalas para a medida, alguns instrumentos bsicos e a equao manomtrica, de grande utilidade. Estuda-se o clculo da resultante das presses em superfcies submersas, o clculo do empuxo, que tambm ter utilidade nos problemas do Captulo 9, a determinao da estabilidade de flutuantes e o equilbrio relativo. importante ressaltar, em todas as aplicaes, que o fluido est em repouso, para que o leitor no tente aplicar, indevidamente, alguns conceitos deste captulo em fluidos em movimento. Para que no haja confuso, quando a presso indicada na escala efetiva ou relativa, no se escreve nada aps a unidade, quando a escala for a absoluta, escreve-se (abs) aps a unidade. Exerccio 2.1 ( )

N13510101035,1G

Pa1035,1205104,5

AA

pp

Pa104,5210

5,21072,21010500AA

ApApp

ApGApAp

Pa1072,22000.136hp

ApAApAp

45

55

IV

III34

553

HII

II2I13

V4

IV4III3

5Hg2

II2HII3I1

==

===

==

==

====

+=

Exerccio 2.2

kN10N000.10525400

DD

FF

4DF

4D

F

N4001,02,0200F

1,0F2,0F

2

21

22

BO22

21

BO

BO

BOAO

==

=

==

===

Exerccio 2.3

mm3681000000.136

5000.10h

hh

Hg

OHOHHgHg 22

===

Exerccio 2.4

)abs(mmHg3400)abs(cmkgf62,4)abs(MPa453,0)abs(

mkgf200.46)abs(atm47,4p

mca10atm97,0MPa098,0Pa108,9cmkgf1

mkgf000.1074,0600.13hp

mca2,36000.1200.36ph

bar55,398,0cmkgf62,310

mkgf200.36p

MPa355,0108,9mkgf200.3666,2600.13hp

mmHg26601

5,3760p

patm5,3mmHg760atm1

22abs

422HgHgatm

O2HO2H

24

2

62HgHg

=====

======

===

===

====

==

Exerccio 2.5

kPa35,13Pa350.13025,0000.101,0000.136p

01,0025,0p

1

HgOH1 2===

=+

Exerccio 2.6

kPa1,132Pa100.1321000.13625,0000.108,0000.8pp

p8,0125,0p

BA

BOHgO2HA

====++

Exerccio 2.7

kPa6,794,20100p

kPa4,20Pa400.2015,0000.13615,0pp100p

m

HgA

Am

======

=

Exerccio 2.8

kPa55,36103,0500.834p

p3,0p)b

)abs(kPa13410034ppp

kPa100Pa000.10074,0000.136hpkPa34Pa000.348,0500.83,0000.136p

07,03,07,08,0p)a

3M

MOar

atmarabsar

HgHgatm

ar

O2HHgO2HOar

=+==+

=+=+===

====++

)abs(kPa55,13610055,36ppp atmMabsM =+=+= Exerccio 2.9

( )

( ) )abs(mca12,17000.10000.171ph

)abs(Pa200.171200.95000.76pppPa200.95000.1367,0p

Pa000.76p000.574

pp

000.57pp000.30p000.27p000.27pppap

000.30pp

p4p4AA

AA

AA

ApApAApApAp

2AA

kPa30pp

OH

absBOH

atmBB

atm

BB

B

ABAB

BCBC

AC

ABH

2

H

1

1

2

HB2AH1B1B2A

1

2

AC

22

efabs

===

=+=+===

====

==+=

=====

==

Exerccio 2.10

)abs(kPa991001pppkPa1Pa000.12,010500ghp

mkg500

2,01,0000.1

hhhh0ghp

0ghp

atm0abs0

AA0

3A

BBABBAABB0

AA0

=+=+=====

=====+=+

Exerccio 2.11

( ) ( )

( ) ( )

33243

o

OH

OHo

OHo

cm833.47m107833,41043,06

45,0xA6DV)c

m45,03,05,0000.8

6,04,0000.10x5,0x2y

D

m3,02

4,012

yyxyyx2

x2yx5,0D)b

m4,0000.10

5,0000.8y

y5,0)a

2

2

2

==+=+=

=+=+=

====++=++

===

Exerccio 2.12 ( )

( ) ( )

m105

5,11sen5,4

1000.8

10

senDd

pL0LsenDdLp

DdLH

4DH

4dL

Pa10001,010001,0p

0LsenHp

3

o22

x2

x

222

4O2Hx

x

=

+

=

+

==

+

+

=====

=++

Exerccio 2.13

( )( )

( )

( )( )

mca7,3000.10000.37p

Pa000.37000.17000.20000.17pp)b

absmmHg831684147p

mmHg147m147,0000.136000.20Pa000.20p

000.17p10331p104:)1(nadoSubstituin

p000.17p

p4,0000.104,0000.5005,0000.102p

m05,04,717,35

24,0

Dd

2hh

4d

2h

4Dh

phhh2p

1p10331p104

0357,00714,04

p3140714,0p

dD4

pF4Dp)a

2

12

abs1

1

1

21

21

2221

21

21

21

ar

arar

ar

ar

ar3

ar3

arar

arar

2222

arOHmOHar

ar3

ar3

22ar

2ar

22ar

2ar

===+=+=

=+=====

+=+=+

=+=

=

==

=++=+

=+=+

Exerccio 2.14

( )

1

2

11

22222

111

arar

21

ar

HgO2Har

TT

VpVpmRTVp

mRTVp)cPa050.12p0000.1361,0000.10155,0p

cm51

105,0hA.hA.y)b

Pa200.25000.10000.1362,0p

02,02,0p)a

===

==+===

===+

C44K31710095

200.125050.112373T

cm95105,01010V

050.112000.100050.12p

)abs(Pa200.125000.100200.25p

o2

32

abs2

abs1

=====

=+==+=

Exerccio 2.15

3A

AA

atmAAabs

atm

OH

AOH

A

2222A

212A

m

kg12,1293287

576.94RTp

)abs(Pa576.94200.95624pppPa200.95000.1367,0p)b

mca0624,0000.10

624ph

Pa6240015,02000.8600p

m0015,0404

23,0

Dd

2hh

4d

2h

4Dh

h2000.83,0000.103,0000.8p0hhh2p)a

22

====+=+=

==

=====

=

=

==

==++

Exerccio 2.16

3

1

2

2

112

1

2

11

22

absgsO2Hgs

O2Hgs

absgs

atm

gsO2HHggs

m16,2293333

100952

TT

ppVV

TT

VpVp

)abs(kPa1001090pkPa10Pa000.101000.10z.p)c

m5,0000.10000.5zz.p)b

)abs(kPa95590pkPa90Pa032.90662,0000.136p

Pa500016,0000.10025,0000.136p16,0025,0p)a

=====+=====

====+=

====+=+=

Exerccio 2.17

( ) ( ) 23222212332221221132

21

3,0p1,05,0p5,0p4DpDD

4p

4Dp

000.22,0000.10pp000.10pp

+=+===

=

( )( )

kPa5,43Pa500.43p3480p08,0180000.10p33,0p25,0

180p33,0p25,0000.2p09,0p24,0p25,0

11

11

21

221

====

=+=

Exerccio 2.18

3222

2

ct

c

tt

pGt

oG

p

22c

22p

22c

11p

mkg993.10

183,05,010950.34

LDgG4

L4Dg

GgVG)c

m183,05,0210

5,110005,0L

m0005,02

5,0501,02

DDDv

FLDLvF)b

N5,11FFFdesce196319755,0395030GsenF

cimaparaN196378549817F

N78544

5,0000.404DpF

N98174

5,0000.504DpF)a

=====

==

=====

==>===

=====

===

Exerccio 2.19

( ) ( )( ) ( )

cm8,127m278,1278,01L

m278,0ym0278,0x0600.36x10098,1x000.9080008002

600.552

0200.735,0x15000.10x98,0800AF2

x10yy2,0x2

0200.7330ysen30sen1y000.10y25,0x55,0000.81,0AF2

mN200.73

30sen1

8,0000.101,0000.82600.55

30Lsen

8,01,0AF

030Lsen8,01,0AF

6

oo

3oo

21

3

o321

==+====+++

=++++==

=++++++

=++

=++

=

=++

Exerccio 2.20

( ) ( )

( ) ( )( ) ( )

( )kPa50109,39ppp)c

)abs(kPa1,60)abs(Pa100.6039908000.100p

Pa908.39103,50

150102013,50000.10100A

FAApGp

FApAApAApG

cm3,5048

4DA;cm201

416

4DA)b

N15005,008,016,0001,058,0DDvF

sm.N8,0

10000.810

g)a

abm

absb

4

4

2

t12ab

t2bH1aH2a

222

22

222

11

21t

2

3

=====+=

=+=+=

++=+======

=+=+=

===

l

Exerccio 2.21

2

3

p

pp

pp

p

2p

p

pp2

12

ms.N8,0

10000.810

g

m001,02998,01

2DD

DvL4pLv

4D

p

LD4D

p

pistonomdiapressopondephp000.10pp

===

=====

===+

=

Exerccio 2.22

N33933,04

2,1000.10b4RF

N160.23,02,16,0000.10AhF22

y

x

======

kPa23,25Pa230.25000.10230.15000.10ppmN230.152000.85,769hpp

Pa5,769998,0001,0

2,02,18,04p

21

2p2

p

=======

==

Exerccio 2.23

m4,02,06,0b

m2,06h

h2hAh

Ihh

N920.252,122,1000.30hhApF

m2,14,06,0000.30000.804,06,0h

6,0.4,0.h

2

124h

CGcp

22p

m

m

==

==

==

====

===

=+

N640.82,14,025920

hbFFbFhF pp ====

Exerccio 2.24

N948.59100.115,42,1F

N668.72

100.11100.5100.52,16,0F

N755.285,46,02

100.11100.55,46,02100.5FFF

Pa100.116,0000.10100.56,0ppPa100.56,0500.86,0p

f

B

21A

212

11

===

++=

=++=+==+=+=

===

Exerccio2.25

N500.225,121500.7AhFm0833,10833,01

m0833,05,124AhAh

Ihh

N102,15,124000.10AhApF

F2FF

2o2

1

12325,1

1

123bh

1

CG11CP

51O2H11

22B11

====+=

====

====+=

l

ll

m333,1333,01

m333,05,121Ah

hh

2

12325,1

2

123bh

22CP

=+====

l

N105F

333,1500.222F0833,1102,14

B

B5

=+=

F

Fp

h hcp

b

h

5m 2 m

A

B

1l 2l

3 m

F1 F2 FB

Exerccio 2.26

m736,0634.7680.42,1

FFyxxFyF

N634.73,04

8,1000.10b4RF

m2,18,132R

32y

N860.43,028,1000.10bR

2RF

y

xCPCPCPyCPx

22

y

c

2

x

====

===

===

===

Exerccio 2.27

m65,230cos75,02h

AhApFo =+=

==

kN4,991075,365,2000.10F

m75,35,25,1A3

2

====

Exerccio 2.28

( )

( )( ) ( )

3oO2H

2

O2H

22

oinfsup

2

2

O2Hinf

2

osup

mN000.35

6,05,2000.86,05,3000.10

6,0

h6,0h4D6,0h6,0

4D

4DhFGF

6,04DG

4D6,0hF

4DhF

=+=+=

+=+=+

=

+=

=

Exerccio 2.29

xCG CG

1

2

R R

O

Fx1 F2

Fy1

21 ll = 2

bRRb2RF

AhF

FxFF

21

11x

1111x

22CG1y11x

==

==+ ll

6R

Rb2RAh

Ihh 123bR

CG11CP ===

31

22

3R

2bR

3R4

4bR

3R

2bR

b4RVF

2bR

Rb2RAhF

3R

6R

2R

2

121

1

22

21

21

211y

22

22222

21

1

==+

=+

==

===

=== ll

Exerccio 2.30

( )

( )

N3,4651

579,0300.14583,0000.15BA

brFbrFFBAFMM

m579,0079,05,0br

m079,05,106,1

125,0Ay

Iyy

m06,156,05,0y

m56,0000.9032.5ph

N300.145,11532.9ApFPa532.92

032.14032.52

ppp

Pa032.141000.950321pp

Pa032.5037,0000.136037,0ppm583,0083,05,0br

m083,05,11

125,0yy

m125,012

15,112bI

AyIyy)b

N000.155,11000.10ApFPa000.102

000.15000.52

ppp

Pa000.155,1000.105,1p

Pa000.55,0000.105,0p)a

esqesqdirdirBBesqdir

esq

esq

CGesqCP

esq

o

arareq

esqesqesqBesqA

esq

oesqAesqB

HgaresqA

dir

dirCP

433

CGCG

CP

dirdirdirBdirA

dir

O2HdirB

O2HdirA

===+==+=

====+=

===

===+=+=

=+=+=====

=+===

====

====+=+====

===

l

Exerccio 2.31

( ) ( ) N6363,06,04

3,0000.103,0D4

hApF

N107,14

6,06,0000.104D

hApF

2222MMMMM

322

FFFFF

====

====

Exerccio 2.32

N230.762

083,1000.1205,0000.45F083,1F5,0F2F

m083,0412

2y12by

12/bAy

Iyy

N000.1205,12000.40ApFPa000.402

000.50000.30p

Pa000.505000.105p

m3000.10000.30ph

N000.455,11000.30ApF

Pa000.304,0000.1025,0000.1364,025,0p

BCAB

223CG

CP

BCBCBCBC

O2HC

O2H

AB

ABABAB

O2HHgAB

=+=+=

=====

====+====

======

===

ll

l

Exerccio 2.33

Exerccio 2.34

m1CBMM2CBbCB3M

33b3

23M

BCAB

BCAB

====

F1 F2

1l 2l ( )

( ) ( )( ) ( )

m27,6z5,1108,225,6z5,2

5,115,2z

08,25,25,2z

5,2106,45,2z

08,25,25,2z10

m5,2N106,4251046pAF

5,2z08,25,2

55

2

532

1

==+

=

+

=

+

====

+=

l

l

Exerccio 2.35

21

hxh

3x6h

3x

2xhxb

3xb

2x

2x

hxbF3x

xb2xAhF

FF

2

1

22

1

2

22

1

1111

2211

===

=

==

=

===

l

l

ll

Exerccio 2.36

kN204H880.218015H

m.kN1805,1120MkN120000.1

134000.10V

m.kN880.2000.1

41126000.10M

V

x

=+=====

==

Exerccio 2.37 O ferro estar totalmente submerso.

N2183,04

3,0300.10h4DVE

22flfl ====

A madeira ficar imersa na posio em que o peso seja igual ao empuxo.

sub2

fl

22mad

h4DE

N1593,04

3,0500.7h4DGE

=

====

m218,03,0300.10

1594

D

E4h22

flsub =

=

=

Exerccio 2.38

N625023,0000.25500VGG conconcil =+=+=

F1

F2

1l

2l

( )m3,02,05,0h

m5,01

23,0000.10

62504

DV/G4H

H4DVGEG

22con

2

con

===

=

=

+==

Exerccio 2.39

( ) m7,29,08,1BAx:Logom9,0

2706,0080.13,0350.1

FGE

m6,038,1

3BA

m3,039,0

3IH

N270080.1350.1GEF:Logo

N080.112

6,08,1000.2b2

CBBAVG

N350.112

9,03,0000.10b2

IHCHVE

2BAIH

FGEEGF

2F

213

2

1

ccc

OHsubOH

321

22

======

===

======

====

====

=+=

=+

l

lll

l

l

lll

A fora dever ser aplicada direita do ponto B, fora da plataforma AB. Exerccio 2.40

( )( )( )( ) 22dd444

3

odo

3

m1036,3A02,0A3,031055103,002,01012

6,0

AARhGRA26

D

=++=++=

A B

C

I H

E

G

F

1l 2l

3l

Exerccio 2.41 Supondo o empuxo do ar desprezvel:

3cccc

3

flfl

ap

m

N670.2603,0

800VGVG

m03,0000.10

300EVVE

N300500800EEGG

====

======+=

Exerccio 2.42

mm2,7m102,7005,0

104,14dV4hh

4dV

m104,11068,21082,2V

m1068,2200.8102,2GVVEG

m1082,2800.7102,2GVVEG

32

7

2

2

3766

362

22222

362

11111

===

====

=====

=====

Exerccio 2.43 ( )

( )

( )( )

m8,0hh000.16000.40h000.6000.32h5,2000.16h000.6000.32

h5,14hpm

N000.324000.8p4AApGAp

2Situaom

N000.1622A4A

EG1Situao

ooo

oo

ooobase

2basebasecbasebasebasebase

3cbbc

=+=+=

+=====

====

l

lll

Exerccio 2.44

m6000.61009,22105,4x

N1009,212

21026

DE

N105,4135,110AhFGE2FxxE3

32FxG

4

4

43

43

44

==

===

=====+

E

G F

Exerccio 2.45 ( )

( )

( )

3B

B

BAbase

2b

bc

bbase

bbase

3cAbAbc

m

N000.25

4,02,0000.15000.132,06,02,0p

m

N000.131

000.1016,0000.5A

FA6,0A

FGp

FGAp2Situao

m

N000.15000.5332,0A6,0AEG

1Situao

=+=+=

=+=+=+=+=

=====

Exerccio 2.46

( ) ( ) N171.106121085,7132,110

6DgG

1085,7293400.41

200.95TR

pmkg132,1

293287200.95

TRp

Pa200.957,0000.1367,0p

33

3

2Har

3

2H

2H2H

3ar

arar

Hgatm

===

===

======

Exerccio 2.47

79,0x21,0x

6216466x:Razes

01x6x6

02x

21

x121xFazendo0

221

12

02b

2b

b

2b

2b0

V

Ir

bhbhbEG

2

2

cc

c

c3

c

12b

c

c

y

csub

2sub

3c

4

==

=

>+

>+=>

+

>

=>

====

lll

ll

ll

ll

ll

179,021,00 cc

Exerccio 2.48

estvel0m037,00467,05,2

103,083.2000.10r

cm3,083.212

102512bLI0

GI

r

cm67,433,05cm5yCG

cm33,05,032yCC

cm5,010

5,2L

Vh

hL2

bh2V

m105,2000.105,2GV

GVEG

8

433

yyf

im

2im

imim

34

fim

imf

>==

===>====

==

===

==

=====

l

l

l

Exerccio 2.49

( )( )

+

>+

>

====

>==

==

ll

l

ll

ll

ll

l

ll

l

l

l

l

12

1RH

x1x21

RH

01x2x2R

H0R

H2.xR

H2x1

:RportudodividindoexFazendo

0H2H2R02H

2H

H4R

0HH21

HR4

R

HH21

2h

2H

4RIHRG

0G

Ir

Hh

HRhR

GE

22

2

2

2

2

2

2

2222

2

4

sub4

y2

y

sub

2sub

2

CG

CC 0,5cm

Exerccio 2.50

z6gg51z

ga

1zp yz =

+=

=

Exerccio 2.51

hkm2,646,3

sm83,17557,3tav)b

s

m57,320tg8,9a20tggag

axz)a

x

2o

xo

xx

====

====

Exerccio 2.52

ooo

x 4130tg30cos8,9

45,2tgcosgatg =+=+=

Exerccio 2.53

( )

2x

3x

3

Hg

sm72,1

5,1257,010

xzga

m257,0000.136

10140175zg

axz)b

m29,1000.13610175ph)a

===

===

===

Exerccio 2.54

)abs(kPa10610

6,010000.1100ghpp

)abs(kPa7,12510

6,010000.17,119ghpp

)abs(kPa7,119100106,025,10000.1p

srd5,10

601002n2pr

2p

3atmC

3AB

322

A

atm2

2

A

=+=+=

=+=+=

=+

=

===+=

Exerccio 2.55

2xx

sm78,2

106,3

100

tva

gatg)a ====

140

175 Pa

z

( ) ( )( ) ( ) Pa600.314,05,0000.10h5,0p

Pa400.614,05,0000.10h5,0p

m14,0278,05,0h5,0htg)b

5,15278,01078,2tg

O2HB

O2HA

o

====+=+=

===

===

Exerccio 2.56

2o

xxo

ooo

4

3dir

dir

4

3esq

esq

sm8,530tg10a

ga30tg

m73,130tg1

30tghL

Lh30tg

m11011hm1110

10110ph

m1010

10100ph

===

====

====

===

Exerccio 2.57

s54

6,372

avt

tva

s

m45,02,010a

ga

tg

xx

2x

x

====

==

=

Exerccio 2.58

( ) kN6,13N600.131010006,31000GmaFmaGFsm6,31

000.10200.27600.13g1

zppa

ga

1zpp

Pa600.131,0000.1361,0p

Pa200.272,0000.1362,0p

212

yy

12

Hg2

Hg1

=====+=

+=

+

=

+=

======