9.1 CENTER OF GRA~TY. CENTER OF MASS. ANO TliE CENTROIO OF A Booy 4 61

• FUNDAMENTAL PROBLEMS

F'l- I. Determine tile centroid (X. y) of tile slladed area.

.. -~I---

'm

.... , F9-2. Delermine Ille cenlroid Ix. ).) of Ille slladed area.

,

'm

~~-<---~-,

- "'~ 1'9- 3. Determine tile cenlroid y ofthe shaded area.

.,9-4. Lor;atc tile center mass X of tile straigllt rod if its mass pcrunitlenglll isg;"cn by /II - moil + .x:/L~) .

, I--l - -,

J ----L

~"'9-! . Locate tile C'emroid y of the homogeneous solid formed by re\'olving Ille shad<:d area about the yaxis.

'm- -, F>-S

~~ Locale Ille centroid: or Ille homogeneous solid formed by re\'olving the shaded area about the : axis.

462 C",APTER 9 CENtER OF GRAVI TY ANQ CENTROIQ

• PROBLEMS

·~I. Determine Ihe mass and Ihe location of the center of m~ss (i'. j') of the uniform parabolic·sh:lp<.'d rod. The mass p<.' r unit length of Ihe rod i~ 2 "glm.

r 'm---

Prill). 9-1

9-2. The uniform rod is bent inlo the shap<.' IIf a parabola and has a weight p<.'r unit length of n Ihj f!. Determine the r':'1ctions 3t the fixed support A.

,

r '"

I--- J ft---1

Prob. ~2

9-3. Delennine the d istance x to the center of mass of Ihe homogeneous rod bent into the shape shown. if the rod has a mass per un; t length of 0.5 kgfm. detcnn;ne the react ions :u the fixed support O.

'm -,-,

"roh. 9-3

. 9-4. Determine the mass and locate the center of mass (i. J) of the uniform rod . The mass per unit length of the rod is 3 kg/Ill.

,

frob. 9-4

9.1 CENTER OF GRA~TY. CENTER OF MASS. ANO TliE CENTROIO OF A Booy 463

· '1-5. Determine the mass and the 100ation of the cent~r of mass x of the rod if its mass peT unit length is /II - 1110( 1 + xIL).

'~I - ,.--1 "rob. 9-5

~ IXterminc the locatioo (.f. ).) of the centroid of the wire.

,

'"

I'roh. ~

'1-7. Locate the ~~ntroid:i of the circular rod. Express the answer in terms of the radius, and scmiarc angle R.

I?------t---f--II - • •

,~ Prob. 'J.-7

0<).41. ~tenninc the area and the centroid (i. Y) of the area.

~+----,

'm---I'rob. 9-8

. 9-'}. Detcmlinc the area and tbe centroid (x. y) of\hc area.

r 'm

'm I'rub.9-9

9-10. Detemlinc the area and the centroid (x. y) of the area.

---''' ---Proh. 9- l ll

464 CHAPfE~ 9 CEN TER Of G RAVITY AND CENlROID

9-11. I)clcrmi lll.' the area and lhe cenlroid ( .•• }'l oflhe area.

,

~--j _____ ---1_"

, _ J

Prob.9-11

' · 9-12. Locale Ihe cenlroid j of Ihe area.

0-9-13. Locale the centroid y of Ihe area.

,

Prob~ 9-IZlI3

9- 14. I)ctermine Ihe area and the cenlroid (x. y) oflhe area.

"

Prllb.9-14

9-15. o.:lennine the area and the centroid (:t, f) oflhe area.

, - ,

,

.- -f' rob. 'J- IS

' ')..16. Locale the centroid (x, jl) of the area.

,

,. , J-L-__ -+ ___ ~_, ,.- -

Prob. '1-16

9.1 CENTER Of' GRAVITY. CENTEII Of' MASS. ANO IltE CENT1!OCI Of' A 800Y 465

' 9-17. Dctcmunc the 3rea;md The cenTroid (.I. j') oithe area.

,

,

• " rob. 9- 17

9-11l lhe plate is made of steel ha"ng a dcnsity of 7850 kgfm). 1f the Thlclness of the platt IS 10 mm.dc:Tcrmine the horizontal and "erlical components of rcacllon 3TThe pin A and the Tension in c~blc He.

c

,.' - 20 ---,. --.

• m----·

"roh. !l-III

9-19. DeTcnninc Ihe locaTion :t to the cenlrold C of The opper ponTOn of the cardioid. r '" utI - (OS 8).

•

Prob. 9-19

' 9-20. Thc plalc has a thICkness of 0.5 in . and IS made of $Iel."l hal'lng a speClrtC .... clght of 490 lb/ ft '. Dctl',mme the horizontal and "cnkal componenTs of rC3cllon allhc pm A and The force m 11M: cord 318.

J ' " Ll!1_-,--.

j '"

Prob. 9-2il

' 9-21. Locale the centroid x of the ~adcd area.

" r a 2t (.r - l;;J

,.

-r--------------+~- , ~ •

PrOO. 9-2 1

466 C",APTER 9 CENtER OF GRAVI TY ANO CENTROIO

9-22. Locale Ille C"Clll roid "i of Ihe area.

9-l3. Locale tile centroid y of tile area.

Probs. 9-l!lB

. 9-2.1. Locate the centroid (.1'. y) of the art.'a.

'"

-'-+ ---1---- , 3 f1 ~ Prol!. 9-24

09-25. [Xlcrmine Ille area and tile centroid {X. 1) of tile ~rea.

'"

l'rob. 9-25

9-26. Locale the ccnnoid X of the area.

9-27. Locate the centroid y of the area.

I' robs. 9-26127

9.1 CENTER OF GRA~TY. CENTER OF MASS. ANO TliE CENTROIO OF A Booy 467

' '1-28. Locate tbe centroid).' of tbc ar~a.

·9-29. Locate tbe centroid y ortbe area.

,

9-30. The steel platc is 0.3 m tbick and bas a density of 7850 kg/m). Determine the location of its center of mass. /\!so determine the bori7.onwl and vertical reactions at the pin and the reaction attbe roller suppor\.lIim:The nomlal force at 8 is perpendicular to the tangent 3t lJ. wbicb is found fTonltan if ,. dy/d.l.

1 'm

2m--l

Pmb. '1-30

9-3 1. Locate tbe centroid of the area. Hill/: ChO(l$C clements of thickness <I)' and length [(2 - y) - il.

,

r 'm

1L-------'-------="'-I---t m~--! ~- I m---!

Prob. '1-31

-9-32. Locate the centroid x of the area.

' 9-.13. Locate the centroid yof the area.

"

2ft

1ft i

Prllhs. '1-32/33

468 C",APTER 9 CENtER OF GRAVI TY ANO CENTROIO

9-.l4. If the density 31 any point in the rcClangular plate is defined by p " (Joil + .fl u). where "., is a constant. dt.'lerminc the nla5S and locale Ihe cenler of mass j of tl'>c plale. l he plate It:Is a Ihiclme.!S I.

Prob. \1-34

9-35. Locale Ihe centroid y of the homogeneous solid formed by revoh~ng the shaded area about the y axis.

Y+ (: - Il)'·';'

Ao,,",,----j--'

Proh. \1-35

°9-36. Locate Ihe centroid ~ of 1he solid.

-I " 1 ,-;1'---,

/ , " rob. \I- .wi

0\1-37. Locate the centroid }' of the homogeneous solid formed hy re"oh';n!! the shaded arca about the y axis.

,-Zm

"rob. 9-37

11-38. Locate Ihe centroid -: of Ihe homogeneous solid frustum of the paraboloid formed by revolving the shaded area aboul the ~ axis.

: . ~(".! _ }'l)

•

J'rob. \1-38

r h ,

9.1 CENTER OF GRA~TY. CENTER OF MASS. ANO TliE CENTROIO OF A Booy 46 9

~J9. Locate the centroid y of the homogeoeous solid formcd by rcvoh'IIlS thc shadcd area about the ), axis.

,

,

'" z! -.~ - 9

3 fl - '

J

", A---~-'!I-~-'

!'ruh. ~39

. 9-4(1. Locate the eeruer of mass y of the circular cone fomled by re\'ol~ing the shaded area about the ), axis. The density 8t nny point in Ihe cone is defined by I' " (p"j ll)'. where 1'0 is a constant.

"

t' roh.~

09-41. Delermine the mass aod locate the ccnter of mass y of Ihe Ilcmisphere formed by revolving Ihe shaded arCH about the), 3Kis. The density at any poiru in the hemisphere can 1M: defined by p - p..,(1 + y/a). where PI! is a constant.

I«--~.---,.

, /'

!'rob. \1-4 1

9-42. Determine Ihe volume and locate the centroid (y.:) of the homogeneous conical ,,·edge.

,

, I'rob. 9-42

9-43, The hemisphere of radius r is made from a slack of "cry thin plates such that the densi ty varies with height. p _ kz, where k is a constant . Determine its mass and the distance z to Ihe center of mass G.

Prob. ,*-_U

• FUNDAMENTAL PROBLEMS

t~7. Locale the centroid (x.,. Z) of the wire bi.-nl in the shape shown.

, 400mm

0'>-7

t--,...s. l./X~lc th.., centroid y of the beam"s crosHcclional area. ,

lSOmm

-'---'-H'--.'

2S .. ,"' 25 mm

PJ- 9. Locale the centroid )' of the beam's cross­sectional area.

9.2 COMPOSln BOOES 475

FY-1I1. Loc.llc the centroid (X. fJ of the cJ'OS/i·seclional area.

, 0.5;11.

3i". --I n - Io

0.5 ,no

tll- lI . Locale the center of ma5S (,t, y.:) of Ihe homogeneous solid block .

l ." 1

1>"9-11

P~- I l. Determine the center of mass {x.y,fJ of Ihe homogeneollS solid block.

476 C",APTER 9 CENtER OF GRAVITY ANO CENTROIO

• PROBLEMS

' 9-44. Locate the centroid (x. y) of the uniform wire b<!n\ in Ihc shape shown.

~ ,- iF===-I: ~mn'

100 nlm ---.

150mm

Yroh. 9-44

·9-15. Locate the centroid (:I'. y. :) of the ..... ire.

, ,.

Proh. 9-I5

?-46. Locale th~ C\':ntroid (:t. y.:) of the wire.

,

l' rob. 9-16

9-17. Locale the centroid (x.y."Z) of the wire which;s bent in the shape sho ..... n.

--2;n.- -

"-,. Prob. II-I7

. 9-48. The truss is made from 5C"en m~mbers. each having a mass per unit length of 6 kg/m. Locate the position (.r. y) of the center of mass. Neglect Ihe mass of Ihe gussel plales allhe join ts.

y

D ,-I '"' l,I!tiilooo-~,~~c '

3 m 'm

I·rob. ~

0'1-49. Locate Ihe cemroid (x. y) of Ihe wire. If Ihe wire is suspended from A determine Ihe angle segment AU m~tes wilh Ihe venical when the wire is in equi librium.

,

"

Il _lOOmm+2(X)n'm~-c--'

Prob. H9

9.2 COMPOSln BODIES 477

'J-541. Each of Ihe Ihree memb.)rs of th~ frame h~s a mass per unit lenglh of 6 kg/m. Locale Ihe position (:I'. y) of Ihe cenler of mass. Neglectlhe size of Ihe pins al Ihe joints and Ihe Ihickness of Ihe members. Also. ca leul31e Ihe reaclions allhe pin A and roller E.

y

~' "'-r ~ml ~ £

D

'm

f- H

'm

L A

"rob. 9-50

'J-SI. Locale Ihe cen!foid (.i. :n of Ihe er05s-5Cclionalarea oflhe channel.

,

tin. - ' r--.9 ,n.- I,n.

l' rob. '1-5 1

4 78 C",APTER 9 CENtER OF GRAVI TY ANO CENTROIO

-9-52. L.o;;alc the centroid y of the crOS$'~Cl ional area of the concrele beam.

12 ill

!'rob. 9-52

. 9-53. Locate the cenlroid y or the cross·sectional area of the buih-op beam.

'-J ill .

l,n. in.

!'rob.9- 53

9-54. Lo<:alc the centroid y of tbe channel's cross­sectional arCR.

lin.

!'rob. 9-54

9-55. Locate (he distancc y 10 the cenlroid of (he member's cross-sectional area.

6in.

, 0.5 111 .~ in

~

;---IS ,n

, ; . .ri----+--r=---­\- Jill. 1 3in. .j

Prob. 9-55

• 'I-~ Locale the centroid y of [he eross-sc"ional area of lhe buill-UP beam.

I'rob. 9-56

09-57. The gmvi t}· walt is made or concrete. Determine the locadon (:t. y) of lhe cenler ofmass G for lh .. wall.

" •um 1

--, 7.

O.6m 11.6 ".

I' rob. 'l-S7

9.2 COMPOSln BOOES 479

'l-SII. Locate the centroid .. ' of the composite area .

I'rob. '1-58

'1-59. Locate the centroid IX. YJ of the composite area.

3 In.

'---'-----'--~- ,

I'roo. 9-59

.~. Locate the centroid (:t. y) of th .. composite area.

,

l oS (. 311-t-3 ft --..

I'rob. 9-60

480 CHAPfE~ 9 CENTER Of GRAVITY AND CENlROID

- 'J-61. Divide th~ plate into parts.. and using the grid for mcasur~men1. det~rmin~ ~pproximatel)' the location (.t. y) of the «ntroid of the plate.

.. 200mm ,

200 mm ,

t>.

j" t ~ , 1'\ ,

Prob. 'J-6 1

'J-62. To det~rmin~ the location of the center of gr3lojt)' of the automohile it is first placed in a Inri pomiQu. with the twO wheels on one side resting on the scale platform I'. In th is position the scale records 3 reading of II' t. Then. one side is clcl'ntcd LO a ronl'cn;ent height (' as shown. The new readtng on the scale is IV!. If the automohile has a total weight of 11'. dctennine the location of its center of gm'it)' G{x.1).

"rob. 9-62

'J-6J. Locate the centroid}' of the cfoss·sectional area of the built-up beam.

..

]5Omm , ; mm /

~20m'n

""'1"----' 2On,m

!'rob. 9-6J

.~. Locale the cenlroid y of the cross·sectional arca of the built·up beam.

,

" mm ]1:::'~==1l~ SO mm I

t50mm

i ""mm

IOmm

20mm

IOmm

20mm

I ---'---'"'-"'------..

!'rob. \1-64

.~. The eomposile plate is made from bollt steel (A) and brass (B) segments. Determine Ihe mass and location (.t'. y.~) of its mass (Cnter G. T.lke p" _ 7.~ Mg/mJ and PI>< .. 8.7~ Mg,lm).

)Omm

.... 01t.9-65

9-66. The ~ar reSIS on four Kales and in tltis position tlte scale readings or botlt the front and rear tires arc shown by F" and fir- When tlte rcar wlteels!lre c!e\'aled 10 a height of 3 fl abo\"t' tlte fronl Kales. tlte ne'" rcadings of tlte fronl "lteels arc alro recorded. Usc tltis d~ l a 10 compute tlte Io<:alion X and r to the (Cnh.'r of gra\"ily G of tltc car. The lires eaelt Ita"e II diameter of 1.98 11.

FA - 1129 tb + 1168 tb _ 2m tb f·._ 97S1b + 'I!I-Itb _ I9S91b

F~ .. 1269 Ib + 1307 Ih .. Z5761b

"'rob.~

9.2 COMPOSITE BOOtES 481

9-e7. UnIform blocks ha"ing a lengllt I. and mass III arc stacked one on top oflht' other. willt each block O\'erhanging tile other by 11 dislance d. lIS sho"-n. If Ihe blocks are glued together. ro thaI they "i]] nOI lopple o\"cr. dClemline the localion .r of tile cenler of mass of a pite of" blo<:ks.

o'J-6IL Uniform blocks hll'ing a length L and mass iii ate stacked one on lOp of the olher. " ';Ih eaelt block overhanging Ihe olher by II dist~nce II. as shown. Show Ihal the maximum number of blocks whiclt can be stacked in tltis manner is" <: 1./11.

~C-______ ~ __________ •

r , j I'.ub!'. k716X

·49. Locale the cenler of IIra'ity (:I'. :) of Ihe sheet­metal bmcket if the material is homogeneous and has a ronswn t Ihkkness. lf the bracket is resting on Ihe horizontal ... _y p1:l.ne shown. dctermine the maximum angle of lil l 0 which ;t can hal'e before it ralls o\·er. i.e .• begins 10 rotale aOOU1 Ihe y axis.

, ",mm

20mm

10 mm dla. holes

I-'roll. 9--60

4 8 2 CHAPfE~ 9 CENTER Of GRAVITY AND CENlROID

')...711. Lorale lite ('ellte r of milS$ for Ihe comprcssor 35SCmbl)'.l'he loclI1ioll§ of lite rentel'$of mU5 of tlte "arious components ~Ild their masses arc illdicated and labul~ted in Ihe figure. Whal Dre lite \'Cnical reacliollS al blocks A and 8 needed 10 suppon lite plalform~

o In.<lrumcnl ~nd

e Fihc. ')"I~m o Pil'inlllU<'nlhty

o LIquid Storage

o SUIKtur.ol (r:II"''''ork

I'.ob. ')...10

23(HI

183 kl lZOkg ,n, "" "

m

')...1 1. Major Ooor loadings in a shop arc caused by Ihe weighlS of the objects sltown. Each fore<: aCIS through ils rcspccti"e centcr of gravity G. Locale lhe center of gra,'ity (1',),) ofalltltcse components.

, 450lh

G ,

611 ....

,

l' rub.9-7 1

-9-n. Locale the ceiller of mass (1'. y.:) of the homogeneolls block a5SCmbly.

l00mm

I'rolt. 'l-n

"" mm

----

0'1-13. Locate the rente r of mass z of tlte assembly. The hemi~pltere and the cone arc made from materials having dcnsiliC$ of II Mg,lmJ and.j Mg,lmJ. rcspcclh·c\)·.

,

, I'.ob. \1-13

~74. Locate the IXnter of mass "f of the assembly. The t')' linder and the rone nrc made from mate rial§ having densities of S Mynl! nnd 9 M!!/ml. respectively.

l'roh.9-74

9-75. Locate the cent .. r of gral1ty 0', y,:) of the homogeneous block assembly having a hemispheriC3.1 hole.

"9-76. Locate the center of gr~\"ity (X.y.:!) of the assembly. The triangubr and the rectangular blocks are made from materials hal'ing spccirlC weights of O.25lb/ inl

and 0.] ]b/ inl. resp«lively,

" 13 in.

lin . , l .15 In.

, '< l,n. W ""',.

9.2 COMPOSITE BOOtES 4 83

' 9-77, Dclennine the distance .r 10 the centroid of the solid which ronsists of a cylinder wilh a hole of length /J - SO mm bored imo its bas.:.

9-711. DClennine Ihe distance Ir 10 which a hole must be bored into the cylinder so thnl the IXnter of mass of the assembly is loc:lIed 31 j ' • 64 mm. 111e material has a densilrof8 Mg/ml.

l20mm

Probs. 9-77ni1

9-79. 'Ille assembly is made from a steel hemisphere. p" - 1.00 Myml. and an aluminum t')·linder. P. - 2.70 Mynt' . Delcnninc the mass cenler of Ihe asscmbly if the height of Ihe C)'linder is II _ 200 mm.

*9-80. The asscmhly is made from n steel hemisphere. P .. _ 7,SO MgJml. and nn aluminum cylinder. p. - 2.70 Mg/ml. Determine the height /, of Ihe C)'llnder so that the mass center of Ihe assembly is located 3\

: . l60mm.

" , 160mm

r

Prulls. 9-7Y76 I'rob" 9-7'J/lIO

484 CHAPfE~ 9 CEN TER Of GRAVITY AND CENlROID

The an>O\lnl <If rooIing.m~1<cr;31 usedon this .torage huild,ng can be c'''",3tcd by ""ing tbe first theo",m 0/ J>~J>J>u, and Guldin"" 10 determine ;a ~urb", arca,

*9.3 Theorems of Pappus and Guldinus

The IWO IIIt'OUl/1S of PI/pp/lS 1/1111 GllidillllS arc used to Find Ihe surface area and volume of any body of revolulion, They were Firsl de\'eloped by Pappus of AlexllOdria during the founh cetllury A,I), and Ihen reslated;1I a later time by the Swiss mathematician I)aul Guldin or Guldinus (1577- 1643),

, .. Surface Area , Ir we re\'olve a I,itllll' CI/n'l' about ,III :txis Ihat does nm inlersect the curve we will generate a s"rftlce (orell (If rt" 'Oll/film. For example, the surface arca in Fig, 9- 19 is formed by re\'olving Ihe cun'c of length L about Ihe horizontal axis. To delermine Ihis surface area. we will first consider Ihe differenlialline clement of length dL. If this clemenl is revoh'ed 211' radians about the axis. a ring having a surface area of dA = 21fr IlL will be generated,Thus, the surface area of Ihe enti re !xxIy is 1\ "" 211' frilL. Since J r dL :: n (Eq, 9--5), Ihcn A "" 211'1'/.. If the cUl"\'e is revoh'ed only through an angle 0 (radians). then

where

(9-7)

A = Surf:lce area or re\'olulion

0 = angleofre\'olulion measured in rndians,O S 211'

r = pcrpendicul:lr dislance from the axis of re\'OIUllon to Ihe een!roid of Ihe generating curve

t = length of Ihe generating eurve

Therefore the first Iheon:m of Pappus and Guldinus stales that Iht'

/lrf/l of II s"rflll:r: of re,'o/",im/ eqllilis flit' pmdllfl l'f III" Ifllglll of Ille genl.'raling fltrl 'e /11llllh~ dislIllice Irlll'e/t'li by Ihl.' cell/Will of Iht, CUrl'/, ill g('lIl'rlllil/~ Ihe sllrfi'C/' IIrl.'lI,

502 C",APTER 9 CENtER OF GRAVI TY ANO CENTROIO

• PROBLEMS

·'-10." The tank is used to store a liquid hll"ing a $pecifie "-eight of 8OIh/ ftl. If it is filled to the top. determine the magnitude of the force the liquid ~~erls on e~ch of its two sides AIJDC and 8DFE.

!'rob. 9- HM

. 9-105. The conere!e "gravity- dam is held in place by its own weigh!. If the densit), of concrete is 1'< _ 2.5 My ml• and water has a densil)' of P. , - 1.0 Mg/ml. determine the smallest dimension II that will pre"em the dam from overturning about its end A.

I'rob. '1- 1115

9-106. The symmetric ronerete "gravity" dam is held in plaa: by its own weight. If the den5ity of concrete is p< - 2.5 MgJmJ

• and water has a density of I'. - 1.0 Mg/ml. determine the smallest distance tI al its base 1hal will prc"cnt the dam from Dvertuming 300ut ilS end A. The dam has a width of 8 m.

" rob. '.1-106

9-1117. The lank is llsed 10 store a liquid having 11 specific weight of 6OIh/ fIJ • If Ihe tank is full. determine Ihe magnitude of the hydrostatic force on plates CVEF and AHVC.

,l 1.5 ii' . :t

Uh LS 11"-.......:

I.H'l

!'roh. 9-1117

E

>

· 9- 1011. lhe circular slcd plale A IS used to scal the opemng on Ihe "'-Mer Jloragc lank. Determine !he magnitude oIllle r('SuLl:!.n! hydrostatic fmn: Iha1 M1S on it. Thcdcnsuyofwalcrisp. _ I Mg,lmJ,

' 9- 109. The dliplical lied plale B IS us.ed 10 K,d Ihe openmg on !he WIlIer sloragc lank. DctctnllMe the maplltudc of the rcsul1~nl hydrlKlallC force Ih:)! XI.!i on il. The dCllSIt ),Or"'''lu I5p. _ I My ml.

9-110. l)elcrTninc the magmtudc of the h)"drOSlalic force Kling on the &lass ,,"'Indo ... , If 11 IS circular. A.1bc 5peCIfic "ClghlofKa'I.-alcrl5y. _ 63.6Ib/fll.

9-111. Determme 1M: ma&nnudc arK! location of 'he Tcsullanl h)"drlKlanc force Khn, on the gl:1.S5 ... indQ.l· ifit is dhplical. 8 . ~ $p«ifK l'Ie'ghl of seawater is r. - 63.6Ibj rrJ.

· .. ~ f1

O_~ h

A O-S h··· wB IIl lh

I' rum. V- IIIII I II

50 3

"'}-Ill. Determine Ihe magnnudc of the l'Iydroslatie forox leling per foot of length on 11K: su .... ll. Y. _ 62.41b/ f11•

l' rob.9-I U

09-1 U. If segment JI8 of gate JIHC is long enough. the gate .. ,11 M 01\ thc ,·crgc of openln," Octcrminc thc length I. of this sc:gmenl In order for this 10 O«'ur. ~ gatc IS hlngl:d at H and has a ",·Klth of I m. lhc denSIty of w;lIcr is p. _ I Mg/ml.

9-1 14. If L . 2 m.dctcmul1oC liM: foru thc gatc ABC cxcns on the SIIlOOIh slOJIpi!r a\ C. The pIC IS hinged al H. free al A. and is I m WIde. 1M densll)' orwater is p. _ I M&lml.

,11---_ •• ,

A

" ,. C •

I' rol». Y- II .II I I ..

50 4 CHAPTER 9 CENTE~ OF GRAYITY ANO CENlR010

~II S. Determine the mass of the eoumerv.'el!ht A if the I"m·,.-;\k !a.c is on the \"erge of opening ",hen the "'OIlel is at .hc le\ (I shovon. The gate is hinged a. B and held by thc smooth s.op al C.llIe dens.ly of .... ater IS p _ _ I Mg/mJ•

- '1-116, If the mas:s of the countcrv.·cight at A is 6500 kg. determine the forcc Ihc gale exerts on the smooth SlOp al C. 11\e gate is hinged at 8 and is I·m wide, 111c dcnslty of "'aIel is p_. _ I MgJmJ ,

,. ,.-

"~. 'l- II Sl I16

0'1-111. "ll1e COOCICte sm\;ty dam ,$ dcsiSJ'll'd 50 Ihon It IS held III po$'llion by lIS O\\n ""righ!. [),::.erminc lhe f:l('tor of safet)' against O\"Cllulnlng about poult A if ;$ _ 2 m. lhc factor of sMet)' ISdcfincd as .he I"ilOO oflhe Sl3bilu.mg monlol'nt dl\'!dcd h)' the O\"cnurning moment, llIe densities of c:orocn'te and "OItel arc p~ - 2.40 ~ 1 g/mJ Md (1 _ _ I Mg/mJ. IC$pC(lI\'dy.Assume Ihat lite dam docs nOi slide.

Prom. 9-111

'l-II IJ. The tGncn:tc gr.t\'lIy dam IS designed 50 thaI illS held in posIlioTt by us o·, .. n " 'Clght. Dctcrmine the minimum dllnension.l so Ihat thc factor of safety againsl Q\'crtuming about pomt II of thc dam is 1.lbe factor of safe t)' is defincd as .hc ratIo of tile stabilizing momcnt divided by Ihe ovelturnmg momcn!. ' ll1e densities of concrelC and 'Io"Jh:1

ale (1,,_ - 2.40 Mg,/mJ and P ... - I Mg,/ml, Tcspeetlvely. Assume Illal Ihe darn docs not slide. ,

A

2", .f - -

" rol».9- 118

'J.-119. TlIc: undc""'OIter tunnel in the aquall" Cl:ntcr IS

fabncalcd from I lransparent polyc:::tToonate malenal formed III the shape of I parabola. Delermine Ihc magnilude of Ihe h)'drostallc force Ihal acts per meIer length ~Iong thc surface AD 0( the tunnel. The density or Ihe \\'lIter is (I ... - 1000 kglmJ.

,.

- 'm 'm

PTob, '1- 119

508 C",APTER 9 CENtER OF GRAVITY ANO CENTROIO

• REVIEW PROBLEMS

"9-120. Locate tile centroid x of tile shaded area. ' 9-121. Locate tile centroid yof tile 5haded area.

,

I tin.

1 in. - 1--1 in. --I

l'robs .9-I.201 I.21

9- 122. loc31c the centroid y of the beam's cross·sectional area.

,

rSO

Plm - 7S 111m

~ mm ~--I --.1..1--1-l .­, 100 Plm

4 , 25 nlm

L_---"L =------'-_, -. , 25mm

Prub.9- 1.22

9-12.1. locate the centroid I of the solid.

Proll.9- I.B

°9-IU. llIc sleel plJ1e is 0.3 m thick and has a dellsi1), of 7850 kg/m' • 0\.1cnnmc Ihe IOX311011 of itS ccnler ofm;tSS, AIS<) oompute lhe rcacllOns atlhc pm and roller suppon.

,

fj,-- --++-,

I' roh ';1. 1.24

· 9-125. Locate the centroid ("'. y) of the area .

"

I'roll. 9-I2.5

9-1,26. Determine the location (x. J') of the centroid for the structural ~hapc. Neglecllhe thickness of the member.

,

J in'

l I L-_~~---,-----_.

-1.5;n._ u;n.1,_. I.5 ln. 1 1.5 ," .-In. l In.

Prob. 9- ll6

~1l7. Locale the centroid ). orlhe ~haded area,

1

(-+------''1--.

I. • , , J r T-rT i

,

P,oll. 'l- 117

509

~1 2lI.. The load over lhe plale I'aries linearly alOllg tbe sidCl; of lhe plme such thai II '" i [x(4 - y)[ kPa. Delermine Ihe resultant force alld il s position (:t. y) on lhc plate.

, 8 kPa 1

JWffr rrl~:ZT' / 4," ;

•

Prob. 9-I28

.9-129. The pressure loading on Ihe plate is described by lhe function II '" t - Z-W/ (x + I) ... J40f Pa. Determine lhe magnilude of lhe rcsulwnt force and coordinalC5 of lhe point where the line of aClion of lhe force in lersects the plate.

, ,.",

"roh. 9-12~