5. ¥. "n<'l~il~~~'~LiiUflVl11.j"'~I'l~l.J1n~,!'" Fs ~:LlJt!ii"""'ln~U~~lIlHr1f ••1Il'LLH"uof.mn
,nuhl'~.l ".il1!l'~u11,lIlJ~"'~v.i'I'l~Ln'"01" tI~uumla.ltlil.1tl>ll6t,<'ln11n
3. ~Ul<'I'llv.llH,iiu.mlUffO..'1.,!"hwvhltl~.l.J1nn11'llUl"''ll".llH ,ifUVll'l1U~au ,h>li'lJ
Iuillri'lJ"",,,,v,.tu'!i1l1lJ~ffl<'l1 1l01.hn~llJ 011'"q>l~.Lfi5VU~.r,Ufil1lJ"'~'lJlnlJu;{u~l
-n1~ol1d. F, l.ii~ltll.lJ1ruLvhnlJ Fs uulia fls = fl,
8 . , 'x .r ...I'
4. '~flml~ul~a (Slipping) iiv<uihlllJ~ff'n"''llu 'lJtn",~a.ll1.L(fU"'l'l1Ufffi"'l'llJlnii1'(<'l
,nuri'~lhulVl""'HnlJ~Ul"'~Il.m.of.il1ntlnOi ,Tulia Fs = fls N ".fflJOl'~ 8-1
5. lriam,iiu'loa (SIipping) ~~uihlllJ~ffln"ifu 'llUl",~a.ll'.I(fU"'l'llu.au~'Lilull",
rilul",u"".nlJ~Ul"'~fl.'m.rNlntln~ ,Tufta F, = fl, N ".fflJm,vj 8-2
.,. . . ~ . ..: "" ... . ....
JlJI'll"."'1IJ'lf""'l'l1t' (Angle of FrIction) l",,"lUl1fflJOll1'1 8- 1 Lm: 8-2 lJ'llmnn". .lUlllli'll'llti'iftll>l1'lJfl. nl1lJL(fO"'VllU 1",0'In'll"a01'ti. 111. 1(fU"''VllUm"l'il~;{u~,t!lJ ~ff~111
... .,j..l ...... ~ ... .r ,1( ... ~ .!/ ....... t-
. MillO Fk = Jll.; N (llnl'l lflnal''YlfflJ'WVl1iLf)~'J'U 'j::'H1H'VfUN1l1>3a'V~ 'UDmnfitJ 011OlfJ'VlH'H'l.J
a"fllJ~ 'llUl<'l'llll.lll.,iiu"l'l1u F 11i~11'Ju.ramilnlJ fls N ".,ru F 0; fls N unrhHriillmi'~"
11i~01lLfiEau~ villll' F ii~lih,h~ln",1'.1'('" F = Fs = Jls N lI1'lnlloJ"dlfllJllOl',i'11;i.r.1Uifu
1",uVlmru1~lnnciil'~lllf".lultl~ 8.3 (n) ~'Qnm.1'i11",ulll' P lumrud P = Fs .,.,ru naa.
~,iilfln1ln1i,"m~~u'lo~ (Verge of Sliding) lil>li'lJm1lJlflJ~a~il.lll.of.Ulntln&i N lla"
uwilU"'l'llU Fs lrifl~llJnU ~"1'i1M,n,"~allV<,r R, l.llJ <1>, liltlljlJYi Rs l'i'fllJ N ol.,10011
~~,a.fl11lJ1(fo"l'l1tll'fii," (Angle of Static Friction) ,l1Jl1t1. .
<1>, ~ tan-'(~) ~ tan-' (11;;") = tan-' ~s
l~ancifl.acilucrm..11ilfi~atlyj"i7aao~. 1I~.1ullmnu1," 1 P < F ~,1'i1111,n",u~.ci'1,1l R. . ,
~,fi"Uln1ml.fi11uiif1fi1l.llJ <I> ~lmltJl~.· ,Tulia <I> 0; <1>, 0' P I'ilhtnaa.lfl~Du~mh.
l'fJ'Il'flJa ".,rtl P = F, lumrud m.ci'v<ll R, iillUln11m.vi,yjol>ltl"hw <1>, ".~tlyj 8-3 ('Il)
~lJil"ioni1 lJlJ'llD~fil11Jlfiv"mu~1u (Angle of Kinetic Friction) ,Tu~a. . .
w
I
f--~P
~ iF.
N~R.
ii160l{f~il::;lf1~ ()I~
(n)
~tl~ 8-3
•
w
-bI . ~p
I •i F
N~~L,
('ll)
375
6. 376
e,
N
(n) ('U)
nJii 8-4
•
~!JJ'UB~n11ll, (Angle of Repose) 'l1inllVl~~OHn~noH1mY1J1h=iiVl~'Uo~ml~
liiol'1'll'u,",1il~ifftJihIIlJ trll~'1l0,ii1l5tnOU.1,Vii".ncifl,iiiitl'",nJn W utJmnuiirll.1, U
"" l11>llinflrimi'~.llvi 8-4 (n) 1=U1U~tl1iU~~lJ 8, 1tJ~l'1iinaD."=ii1flnlll~tJ1oCl (Verge of
Sliding) ,).,ru Fs 0 ~sN tr.1~~~ll't'Ufl'm;fl.ii~l'1il'il.lIlll'1.1u.llii 8-4 ('U) 11'1vlilllJnn
lllJ"lCl'Ufl.U5' IIH,fNl nlln~ NoW cos 8, lla=ll5.liluVlI'llU Fs 0 W sin 8, I~e.~ln
Fs 0 ~sN ,)~,ru W sin 8, 0 ~s (W cos 8) 1110
8 = tan"' ", l"'S
~~ 8, cll'".'lIJ'Ua,nl;1b (Angle of Repose) UClt"lnnnnnuUliiuu ~tlJmll'llnuljlJ
'Ufl.fl1llJliiUI'1Vlll.HIOI'1 <I> I~Bjjnll11'1I1lJ1l1=iiVl~'U0.n"lllJliiuI'1YllU1l61 ~tMil U 0 tan8
ff~lO~'~"hnl1ftl'l.n(U1):a~uaff'j::'iJlnJllt,rn'UCJ-3naCl-3 ~~J'U ~l W 'illflnl'jll~~~:~-lhiGlO;
'YI'll1Jfll
8.2 "OryVl1lntJ:>nUA:>1UlafJfln1UllUUll~l (Problems Involving Dry Friction)
fi'111'1~lI;hIn~.0~11cr01Y<n"lllJcrlJ~ ClI~011'1q,)n, =uu'uO.m~m=y;,ii,"llJ ~Cl'U v.n"lllJ
IffUI'1I'llU,rUli ,=uuIll.il'vla,tI'hlllll'111JcrlJnml~"lollotnO~I~U"l"fl. r1UIIHIffUI'1YlltJ
'UUVl'UO.iJ!]pllIlHlaumlltJ (Types of Friction I'roblems) 1I'1uYh11l"dhjYl1ln~mlll'1f
crllJuuuii1110,,,0.r1Ufl"lllJliiul'1l'llllllUUIIlI. 'Ii.il: uti.110n1.1.10l~oiinln11'1tr'l1'1~~cr,:lIot
l:11~lmU')"l1111,hjmlulinr'l1lJl'1 ;"llJ~.ImauIiiaunU~,tnU'Uo.crlJnl1lllJ"la~mlJl'1iiiioti
'liUI'1'Uo.Uyl1lUI'!aeoU1."",)nvi1u1Ulla:On')"luU,.1l1: naumY<nm~~ n hiYJnm01,,:crUlJ9lil
yrnu~l'U{)>3~UVl'l-.ll'J'J1 fl WVl
" ."'UJff"'l" (Equilibrium) uyl1l1hulm1i,,"oylum'=fflJ~a 'Ii•.1u",n~,nU')"lLllb
111YI'J1U fh.J'-ll1lJ ~ llil rltJ';l'U111'U iH C'l'lJ 01 -n'1'lJ ~(U'r-ll1lJ ~Y1iif) ~ L~HU'J-llfi(J ~111'Uln1~~1 f1 01')
UnUyl1l OU1.'hnmlJ fil,)'la~il:'lnilcraullltJulll,"llJ F S; ~sN UIl:tru ~:I1il'1nl5Sulna
uae11'1q••llioylunl1: fflJVJa Uyl1l'lliil'1il'Uffl'1,hllllii 8- 5 (0) IIwiiumlluii ALia: C
7. ,~a""JlIflUil,jlU1nhml1J1IIJ'la~o.1U'lIllJl100Qhi til1U'iifi1'1JlIJ'llllJDIIB.lhmTnul>ia:
~ liD 100 N ,,~Ju rJ'~lVlQiill,"ullVl~lu~u~ 8-5 (~) ~~iiD~fllh:nDlJ~D~IIH11iY1l1mh
til~1I11J1,mllMJlnMnlllJnlmlJ,!a (1uuI>iB:lluci1UiillllJlIlJm,) l~a11l FA' NA
, Fe UB:
Ne vi'~Ju lU'1l:mD~lufll1:lIlJ'lauB:li'JulUVlllJ FA ~ 0.3 N" Ill!: Fe ~ 0.5 Nc
NA
B
(n)
('Il)
c
11;,. '" 0.3
B
,UB = 0.4
(n)
F8
('lJ )
nl>i'Ji"nla~."!fi~DI~ II 'lVl1Vl'l~M~~ (lmpendion Motion at All Points) ltJmliJU
'1U'U"" UU11lhlJllJfhvrm1J VlIvi1nlJ~'U1W)D.lIlJn" lIU'la"f~l1u~~jjDQnlJnu~'U11l'lJo.
ffum'~11milu",mllvr.l1lJ~~iiD~ F = ~N l",um'Vmmj"E~ tiljjlanifflii",nl1lfi~oll~~~",
tllJrJ'ff .r~,fll Fs = ~sN hl~ru"~1'lQrilil~~1I1oa F, nJ:lvi,nU 11,N Un""a"''''';ll >iJmlll
ifl)j"''lJa~m111l~lJlgn 1 8 ~1U'91~ii.Jl11un 100 N 11liui'i 8-6 (n) ~nlm;'~lr~ l~"U"flJln
m,~u1na rJ'~l~Q1ill'"Qnuff,,~llliU~ 8-6 (~) ~~ihh""uu,hjYmuril FA' NA
, FB
, N, lIa:
8 lunmu)'ify111if J:iillllJfflJmmlJ ~alIa:1inffO~fflJm'fi11lJlffu",mllffO~~Hnu~AfflJ"11
ri'~ao~ ",~JII F = 0.3 N ua: FB = 0.4 N. (1U'J:'1iD"1l1fm:~lii",nlll~~ou~"nrl'uA A ..
'ii1flliim'~1I1oa,f~~", A ua: B '"mll~"1nl)
nr;j IdtJ ~'H~ 01BtJ~ fl':ilJlfl ill il~i1:;lflgen!~~~fltll~~ V) (Tipping or 1m pend ing
Motion at Some Points) 111~il ';11l111"'1Uu,11ivm11Al,f."IJVlJ:iiounil~11l111~0~fflJnl1
~IJ~ ft~iifllinlJ n11~llnll'lJO.fflJ nl1'lJO~IIHIffU",Vllllvr."U>lM>0lIlJn111i01l1'lJfilM>11 n" lii"~
'1nNa",~ ntin iiAlllJIUII1u1"lJ1 n~,:In~all~ IIft:tf'l!'n~: 1~"1;;a~ nlJ mmlUHInl'l nl'
377
8. 378
'fifimJ~~'nl'l;1lJi'b On.J'Vci1~''liU W~lJru11m~mvlJ~Ul'flUl'fV~~U1UlU~ 8-7 (n) lUif!111
il'vlV~n1Jl1111H1UllU111lJ P ~vlv~mJl11,ii"f)1JlTlivu~'IJv~hmmvlJ t'111l'ia:~uri1uihhl1un
100 N i~tTlJ ~~l"'~1il'fJ:llM"'lJ~t1~ 8-7 ('11) ~~ii,~",i111tlJll.i'mlJfh NA
, FA' Nc' Fe' B,
B, liB: P luf11mti'if!l11iil1nl'flJf11J (1l~a:.fUl'huiii'lllJl'flJn1J) llB:iinl'flJn1Jm1lJ'iio",l'llU~
. ,1'IuluM ~~iifi11lJl1lJloi1'IJru:~ P iinlJlfilJ'lJlJl"'~:vi1111,1i"'OlJEuloa~ c ",tTu FA = 0.3
NA llB: Fe $ 0.5 Nc l1~aOlJEuloBln.,~~", C 'm:hjiiOlJEul0B~ A lumliiil Fe = 0.5 Nc
u,,::; FA ::; 0.3 NA 111~m',jru~lH'I'1lJl'j()1111V1unlHhtnruTh P ,hH-ruu~a::fl'HULUUaE{}nn'jdl~
p ii,hl!vu,!"' t'lyf'lV,mrul11~l P Mll1iiVW1U ~,'uvmuBlJ1itjlJ1UhjM .J~tTu OlJEul0B~
~",yf,i'lV'~:'n"luI1B1'~U1nu Juiie ii,~...J111uJll.im1lJ1il~,1'IUlt1."lJllUMlJn1> .i'-wciu~
i'la, W~lJru1nrie,~iimllJn·h, b, ~11lJ,!' h llB:,hmJn W 9},a~ii'lJlJ~ufh'IJ~'IJJ:i'jU~
8-8 (n) vla,nmmLJ' P ~vhl11,ii"'nlJlfifiau~ ~ln~~l.,qlii'lJ:llJJu~ 8-8 ('11) U'lJ~nii.r,
lluJhil'lnlJfil P, F, N llB: x ~ci"lJfimlJ iil'f1lJi'llJOlJi'llJ'lBllB:iiiimllJnlJfil1lJ,iiU"1'l1U
i'lii.,l1;alllJf11"~aul'IJ~vla, n1,111nri a~ll.i,iiu, lu~ilijm llJ,1'I"1U'I,,'11a, nlJ'fiEau~l'fe,lllJlJ
lUlU~ 8-8 ('11) nci~,iimJEuloa mUiil F = IlsN llB:ril~Ma'l1rlJ x fla 0 $ x $ b/2
l'f1UanlllJU ~e ncie,iinm1io",,~u~ 8-8 (n) mUiil x = b/2 llB:ll",liO"'l'llw1'lultJ""lJ
F $ IlsN ;i1"aUl:1"ri1~l!auYi,!,,'IJV' P ~'~:~lllumh:Lnl'l'lJij'nl"fiEauYi'IJa'nri~, 111
yf'lla,mruii,il P ~l11Mll1iiaunu 9l'il:ii1~n1l'f,ii",;l'Iltl'~ulJln i,tTu nl,EuloBllB:nlJ
,iiU'll:,nl'l;f'llul1ftl'~U1nil ,Tlliiv jjfi.J1llu,1l.im1lJ1i"uuhJ.l1lJ11'1flJ011
B
A
(n)
Ilc = 0.5
N,
JU~ 8-7
•
8,
B,
('U )
,.
9. F
(n)
w
(,,)
O ~ .r~"' 2
( F = ,1.1_. :VI
~u~ 8-8
w
I
- xlN
i
p
",,
-.-l
nl')lijtl~ (Tipping)
x = bO.
(F !!;' Ils N)
(fl )
fil11J<f1J"J~ti'U<flJfil'mllJlli~Amtl (Equilibrium Versus Frictional Equations)
Ul~liiu~mm.m. villuilF11'11~ OU'IllIJ OUnl'IfI gVu~elmi'VIfl11lfl'll,""""nmfl~vuyj"V.1Wq
uuifu1711elmlII 8th.l,n<'ll1J lI11J11m1IJIJ~ilF1V1l''IJ8.II,",ii"OlVllUluilll/111Yi"'8~n11V111Ufll F
~~I1'luIl1'lIlJ~"uim1'lu1tJ~11J F < flsN ilf(VIl.yj~n"'8'~.Vlnu11ii";llnnlmnlllJnlli'flJ~"
t11M1U F tJn~1atilWlf'W ell F lDUflltnmnfou Vif1Yll~'I.I()-l F ~::Vl)",yllJn'U~fflJlJwl1 9h~::
lI' ,"lnlunl'jiflJlJ~ilF1V1l'~fl' F ~IUUhJ'1"liiv.illnlIlJnlHI1J'1aurimilOlJ'!juu IIHU0 U'llfl'
I1tl1l'1sfm:vil1uilf(m>l~010U lumtiilflJn11f111lJlffUl'lVllU F = fiN ~oH'uonunilll/111
su"hiimlJ iI:llijjfl111J1f.mnlunlllflJlJ~ilflVll''IJS' F IiiVIflJnllA11lJliiul'1V1lUiifi111J
rl'lJWtnrl~fJ"J'lJU1~'lJi)"Jnnl~tl'f~~-3Ulnrlllfftl-ll,,) flU'lVf ~-:nrt F fl1'jUi'l'~~nlJn7::lhhrnffn1~li
Q""S'UU"'11'1qii1f1. liiViI.1i'lflJn11f111lJliivI'1mulil11rlJ nl1llnilll/111
55n1Scl1V1su5lAS1:1'i (Procedure for Analysis)
,Tu~OtH11l1i'tJ01';LUi'ifQJ111fflJ ~al~u1 nUfll1lJ 1*U Vl1'11U LlUUUl1-3ijY},rii
.r.ll'1~a<f" (Free- Body Diagrams) "1'1 ".1,"~iilfl.Yi~1IUuua'111~lu1U""llthlli
n"IJfi,11;S i'f1J fi11~hhmil'"VIJ 00rluluil]J111 ~Ul> liiU1'1I11U111u.1. 11111h imlIJ~1 !TuIi8
hil'fllJ1101f1J1J~ F = fiN v;.,:~ nli'nilJ Vl1uli'l U'lf1IJlIIJ0l; lflJ ~a~fl ~hmU1IJI~U1rlU~
illlJ1' miiuu1"ff1l1rIJ II';a. 11'1~ ",.J'u t11ii",,11,hllil'murillJ1n011<1IJfil1ClIJ'la 1i'~lIUU"fl.
hl'l'flJm,fi11lJli/u~Yl1U tilo"'1iItlU.11 V ~~ ell1"IfnIfllJ11millJlhl'i'lfHlIlJ nlllYilJl~lJ,ViD111
~1~DIJYi1f1J!;orll11U1..IiO.r1U
~
illJ fll 'H) 1 UUil'U VI HHAll {I ~ tf iJ p! rl (EqlIationc;; or Friet ion and Eq u ilib rlU Ill)
fillhi'lflJnnlflJI'1alIa:lIlJfi1;mllJliiu"l'l1t!Yi~lli'Ju (lf~8i'flJnmimll"t11iinmiiU>1Jlliivl'1l8') -
116' "nIfIJn11111"" IIIh'li1'1l1U fi1 t1li1ll/lfll~01rlm: IJIJIIHifllJ;;~ oil.~•rJ.U1n1t!filllf18.If
,hmvIJIIH11~flu'lJullJlIJuiii.hIUt! m,l'll'lIlJ0"lflJ?)alu;:lJu11nI..V{-Vi0,"m n (Cartesian
Vectors)
379
,
10. 380
fi:wthIJrl 8-1
nriD.~Ull>1l1Jl'lIlJDUlII>1.1u2ti~ 8-9 (n) iima 20 kg ('lUH P " 80 N m"lil
"'DnaM 'imllilnriD.vI.nrillO.D~hmll:lIlJ?la111Dhj ftlfflJtil:(lYlt'llD,mllllliuI>1Y1l11
1Ii'i1>1 flD 11, " 0.3
/
!P=80N
~ ..lti't'll
I i
r'~-- O.8 111 ~~ --I
~."iO.~.~~ -I
OAm
!
(n)
~tili 8- 9
('U) "· c
r-LnOl~iiI11: (Free-Body Diagram) 'lnFJli 8-9 ('II) llH.r.llntin~«"Vji Nc .:
nl:fil~l:U: x .1nmn(!lltlnal.'IIU,i1unriD'!l~D,rI>1'Ul1'-;;anmvu'~liiflhu p iil'fllJ.ll11lh
1UI'LnlJl'il F, Nc Ita: x li';D'nnl11i~uHlIllJlIlJmllllJ~a
I1IJnl>I1)J~lI (Equations of Equilibrium)
---'--.LF, ~ 0; 80 cos 30' N· F ~ 0
i + LF, ~ 0: ·80 sin 30' N+ Nc ·-t96.2 N = 0
C+ Dv!o = 0; 80 sin 10" NCOAm) - 80 cos 30' NCO.2m) + Nc (x) ~ 0
UnnlJfll'i
F= 69.3 N
Nc = 236 N
x =. 0.00908 m =·9.08 mm,
. "'<... ...
l'i~a~'iI1n x iifi1lthmu ~-3Ull IlHi-3111fltlfl~crYnffl~::;l'h«;1"'uil{J'UiNUl.nnfu~'Uuncn~'ltl-3
ncia, 1I.ijjnm51mii~ifllliiD'.ln ,$ 0.4 m "fln.lmr" ,,,,,i1Ul>1miLlJln~l'(1)1111lJl'(lLii,"li~1
ffll"" I'm,,: flsNc : 0.3(236 N) : 70.8 N. ,cta1'1n 1' : 69 .3 N < 70.8 N nriu,.':hni
nl'i~ulClu 0-3mjiTlI:::iirhlnalftU-lnUlJlflnV11lJ
1
11. I
,
. -'
Iii':JCllTlIJn 8-2
rifl.rlttl~ 8-Ia (n) Qnii~~lJl",'r:i1lmUr~" uUm",~fll11fllmu~lim"i'1.r'0nU.y'011~~~
C t1lThff"J1.h.i!Vlt"fllm1lJ,iiu~YllUll1i""'11'i1lmUr"UJJm:'~fll,,~.11fl ~fl ~ = 0.3
'1111~lJl.J1n~l!~"fll e Yi11flllllJ1~nii",~J1~utl'nff~1nnl~§u'n~ 1"'0l!iii..J,l1un"fll11fl
(il)
1tl~ 8 -10
v
O q •
lfilll
,,,
"'"'c
"~ ~}
""':;::: :;; 1
~ ~IS;
C ::.)1:::
Q gfEG - Je:
'Iillf, ' B
~!Tf
WC
( 'U )
N8
Hl1,,~lilI1' (Free-Body Diagram) ~lnitl~ 8-10 (,,) i:h1,.butl~hiYl>l1Jfi1 NA
, FA'
N., f. U~: e 11lM~lnll1lJlIlJnl~lllJ~~'ladimlfl1l(lJill~~11lJ'ffU"Ylluyj A ,m. RQml1lJll'1l'
'~fl11lml,ilu"Vl1U~n~:"il';l"~'" C ,viflumnlwmn~flUyjijfftj1il'lI'lJfl''';fl
alJfll~fll1lJlihIVlllllmil~alJ~a (Equations of Friction ..and Equilibrium) t1lJflll
n11)Hilu~Yllu lifl
FA= !-INA
FB = IlN,
1'1l'1<~.r1nri11ifua.hilllJnnl(lJ'la ,.1.ri,
(1)
~+LMo=O; - J.lNB(r)+ J.lNA(r) = O (2)
1 +1:F,= O; NASin(~)- J.lNACO{i)+NBSin(i)-J.lN.CO{~)=O (3)
381
12. 382
- . ~
Iil:Ja81IJn 8-3
, .1
~lnl!lJml~ (1) ll1B (2) YllJl1 NA~ N. 9i~l!llJ11Gll1,.,1"u""llnnlll!lJlJl"'tB~
J~~tlm~l<tim'lJ1fllilvHm",h11lrnm:vi1 Ul'IlJTi1lUl!lJnll~ (3) ~.1.ii1
Sin(~)-ft CO{~)= O
5io(6/2)
ft= 0.30
c05(6/2)
6 = 2 tao-l
0.3 = 33.4"
1l';~l!l.hll!lJDihh11lrn W un:m1lJu11 I ~mD'i'lJ~tln1u A un. B 9i~ijfl1rl'lJtlldil'lt
'lJlJ~m1lJli1U"l'IllJlI~" liB ft .r~llJltl~ 8- 11 (n) ~~11llJlJ~lJln~l!"'lJB' e lyjlJl1uYi~
••hj~lJlo0 1.,uhjii"fl11lJlllJ1'lJB~uYi~
B
w
/12
11'2 / G
A
(n) ('II)
.r~r"l!1illl. (Free-Body Diagram) ~ln4tl~ 8- 11 ('lJ) ij,·r,.r1utl,lijvmlJfh FA' NA,
FB' No uo. e 9i~11lM'lnl!llJl!lJnlll!lJ'lOuo:~nl!lJ~l!lJnllfl11lJliiu"l'IllJ~~" A uo: B
ItH tiiu.,1'IllJ.: 1utllJilffl'll~~.,1~nm]1lJn1JLIlJ11UlJ'lJlJ~nl" fl ~lJlJYl'lJlJ~II';~
fflJnllfl111JIiiu"l'Iluua:lIlJvm· (Equations of Friction and Equilibrium) t~Ull,
~'iJtll'J rl J llJ!~Umll U
13. h7fJaifua"1'li'1IlJn1H,(u~~ ~:,,ril
~LFJ. ==0; flN " +j.1Nocos e -Nasin 9 =0 (I)
+ l' l:Fy =0; NA - W + N, cos e +I1N, sin e= 0 (2)
(+ l:MG =0; N<~'o, +I1N,(~,;" ,) + N,(~) =0 (3)
llJLuu,rQm1ma1Jtlu6nm~~a~uYi~ G i~a~,r~ W 11UneUni'fUnl,ii (1) un: (3) 1"WJu
I1N, = N,(sin 8 -11 cos 8)
N, = N,(cos a - 11 sin a)
I1N, = N,(cos a - 11 sin a)(sin a - 11 cos a)
11 = sin a cos a -11 cos' a -11 sin' a + 11' sin ae os a
11 = (1 + 11') sin a cos e- 11(sin' a + cos' a)
a 2 2 v
tlHl.J"10 sin e + cos e = 1 UC'I:: sin 28 = 2 sin e cos e i -ltTu
(
1+ 11')211 = -2- sin 28
e I. _,( 4~l )=-Slll - -,
2 I + ~l- -
383
14. 384
fiotlthllll 8-4
nci5~I;im~Ulnui~JtI~ 8-12 (n) ihhmrn 20 lb UO"fl~ii,n1J~1JU~Ullim~jj
~, = 0.55 1l1111~lJlJln~l!~~o~nm5m 8 ~5~";uriou~ncio~1l"lfl~5U~
4I
(n) ('lJ )
~t1~ 8-12
fH1Vl'l6aj" (Free- Bodl':Diagram) 1llnJtJ~ 8- 12 (~) ii~ XQnH1JOn';lumi~~"~
UJ~.f"llntJn~tll,1f N, Mmio~ iJ~ollUtJ,1livlll1Jfi1 8, NB
, F8 Ho: x 11111'11l1mn~fflJnll
fflJ ~ cHla:-:ilf1 fflJ fll'J~'lA1"11fI fll'Jllf1'Jl:-;Ml~erw1'lJ,ill1i'1Jm'Jlfl'[j,ni1tl fl Yi~'U1(l a'llv~ fl civ -'l
OJ.mnamjil (Equations of Equilibrium) l'1icrlJnm'fl.J~o "<11'111
+ .(LF, = 0:
+ "LFy = 0;
C +:EMo=O;
20sin8lb - Fo =0
No - 20 cos eIb = 0
20 sin 8 Ib (4 in.) - 20 cos 8 Ib (x) = 0
ncia1jj1anlmfi~O"~ (Impending Molion of Block) HlIlJnnm1lJlcrO,"VllU
Fs = llsN;
U,hflJnl:i'li (1) ~,,({) ilehi'll
F B = O.S5NB
NB = l7.5 Ib Fo =9.641b 8 = 28.S" x =2.2 in.
( I)
(2)
(3)
(4)
15. x=2 in. (S)
ILti!'lJml~ (1) ii. (3) 1.whl'''lJml~ (5) Mil
Na = 17.9 1b F. = 8.94 Ib
a= 26.6 '
MlJ1Um~: 01"lJlJ~il0cia~liiu.1u~f~"'0ual ~ a'II<H F• •:ml~!'alJfllJ lq~lilo,"mu
1I"iiOl~,1I11hnv"lJ1n~l!I>l !Tufta
?
Fa = 8.941b <: (O.S5)(I7.9 Ib) = 9.84 1b -
I1ja~;nnlll~ii~l~luIV;'flU naa~.:I1iU~,l"tH~mia0Iiau~.:~uloa
. _ A:lClfho;l 8-5
mu AB ~nm:vilv"lU1rn1rnUlJU'ljll!'lJa'llUl1>l 200 N/m ua"~ma~i'lJ~ B l1>lOICfl
BC ""lUl,j~ 8 - 13 (n) 01ril«lJlh:liYlif'lla~m1lJliloVlmu1fi~~~," B Ita: C iia !l. 0 0.2
Ita: !lc 0 0.5 mlJ~l"'U .~M1Ul. P ~v"a.nlllvia,!m<llmin.ln1~mu l",oluiimhmrn'llu~
;uri1'UUi:l~fl11lJl1Ul'U(J.Jlffl
~ O.75m
! -' - - - 4 m ---~~.:::::...p
_ ~o.25 m
C
(0) ('11)
1,j~ 8-13
•
,
H'l,"~llal: (Free-Body Diagrams) .';;'i"",~ii!'l:'IIa~mu AB 1"',",lUl,j~ 8-13 ('II)
1'11 L~1 , - 0 u~1vr f; == 4()O N r.nntH1X~11t1~rn";:Cl5U'~';lJo.J~,(,,,,I-Jh,~,J~ 8 -1 3 (fl) 61~D~
n'uiiuriluif fi.hu,j,'lumllJ!l1 F., P, Fe "a" Ne ffllJl'OMl1.r"f}CfllJ<llJOll<llJ"lalla:
iin1i1'fflJnl,mllJlffo,"Yl1U~ B 1~a C 11lv"
385
(ft)
16. 386
"lJOl~"lJ~alla.fil'lJ!a~~ml (Equations of Equilibrium and Friction)
•
~l:F. ~ O;
i + l:Fy. ~ 0;
(+ l:Me ~ 0;
P - FB- Fc ~O
Ne-400N~0
-P(0.2S m) + F,,(I m) ~ 0
FH ~ 0.2(400 N) ~ 80 N
1ir.m""tfifU~.'lfr"lJnl~Yi (1) ii~ (3) ~.1.i'i,
P = 320 N
Fe =;240N
Ne = 400 N _
(I)
(2)
(3)
,iia~iIln Fe 0 240 N > ~cNc = 0.5(400 N) = 200 N 1ln;ruYi'R.,nmfl~mjYi~.
Olil~111~lU
'ft,ilnmfiajlG~IIl"':Yi e (Post Slips Only at C) 9l~ F. S ~BNB ,m:
Fe = ~cN6
Uf1ftlJonYi (1) 1i~ (4)" ~:1.i'i,
P = 267 N
Nc = 400 N
Fe = 20iTN
F, = 66.7 N
,l1I1.i''11'"ilmruif~:,fi"flr,um,i'ie",0"e'I11,111'fi, Pue0"'
(4)
'il~t11UH:-lfll mJn~~ f1J:;vll~flum.nr:n.H~(J Lnu "'ltJl-wJll1tJn 100 kg il1~t1~ 8 -14 en)
,v:a~ "0111';ijlJj..i,,1io~ 20· "'0fl"m~1"'Yi fil1i'lJ1h:iiVl~'IIa~m'lJ,lloflmlft~Wli":
- - --' -
(0) ('II)
nlYi 8-4
•
17. U~l(;]Qlil:n~ (Free-Body Diagram) l1n~uii' 8-14 ('I) iiililuu,11im11J~1 NA
, FA'
N u~: FB m:i19immujj"Ulvimnmnolv1 ~~1111"il1nff1lJfflJ01'fflJ"l~u~djnl1~~fflJ01'
fl~'1Hiio~Yl1uii~~ A l1~a B (hn1'~U'o~,fi",;luii B unujj''UlvillLn1Jff101Ylil:nii~;lu1Ju
~uliio~lu'Iru:vi0'1O1,~ul(~Iii",;lui! Aunujj''Ulvimn1Jff101v1il:l(l~;luiful~o~ ~~"a~111 NB
inlfll'ja'lJ~alla:fll1lJli1[Jvrrnl! (Equations of Equilibrium and Friction)
+1' LF, = 0;
+ '-.LF, = 0;
(+LMo =O:
-FA+'_NB - 981 sin 20' N = 0
NA - F,- 981 cos 20' N,= 0
FB
(400 mm) - FA
(400 hun) = 0
F, = 0.40 N,
(I)
(2)
(3)
(4)
Vifl'1H'I8~U,~,iiO""'llUii "B i): (l m:IJfl{h~ ( n"D~ lria~llnunujj',ulvi8Inmfl[;lvlii II"
nmi1;lulu,:u11J FB m:ri1111iifl*~~lvi8~8~n~01mlf~u1JIJl1ljUmlJl;;lJ'Jliin1'm~"nujj'-nJ
lvimn1Jff101v1.r~lmuii 8-14 ('I) UflfflJn)'Jii (1) 1i~ (4) i):hii1
NA = 1146 N N, = 559 N F, = 224N
,
224 N S O,18(1I46N) = 206N
.r~,rU nl,5tJ1omii",;fuii A U~IUlii",;flli! B 1ltmU'l8~nmflgDlIii~tJ ~ m'ilnmllo
umJjj',ulvimnuff101Wlon;lu~tJI5o~ (Spool Rolls up Incline) lumruil FB S; 0,45 N. un:
(5)
FA = 163N N, =498N F, = 163N
163 N < 0.45(498 N) = 224 N (mlilffDIJ)
8.3 au(Wedges)
• • v
U'j'U.Jl'lrlUtJ U
Wil",ru""ni18ciHfilJvillff",~lu,uii 8- 15 (n) ~~hl'onncia~iiihj',mrn W l",ohl'u,~ p
n,:rilii~lJ ",r",q1iff':'ID'~lJU,,:~a8~Uff""lu~ui! 8 - 15 ('I) hJviili):'1i~mr,mrn'lD'~lJ
l~a'HnflfirllUatJ1Jl,f1I~~ lmU,lJli1U1Jn1Jlhl1'umJD~nciv.J Ul.JlffO"1'VnW Fl. lie!:: F2 iHlf1'
"'Wll'hlJri'1J01"fl~!)1Jihm,5lJ luri11J8~liiolnll Ul<liiUI'lVl1U F 'liD"fnr"rim:!ll~a ncia~,
387
18. I
l
II
I
I
fonanun:Ju
8-1 "l1lllHl,.IIU1>1U PyjI>i'6.mll,.ml~1JOhll,,~auti'lv
.' ,iiunria. 300 Ib ~m:tnu liiafill1,.mll S = 0.3
8-2 ~.l1l'lh.~a.~lyjlll.l""U1nU P i1a.r1'U111lnria.
~lnm,~uln "".11~aiful'Ul.U1Ul~V. l;ja~;'111i~~1 ).Ls =0.1
p-...,,p-
29"
>,j~ 8-1/8-2
• ,
8- 3 llllmluUU111U P = 80 Ib ~.11l11'l.,,f.ll1n,jn"!lo:
U,)~La(lVl111m'i~l'il!ilf)'ihJncitH 300 Ib l~tHill1'U~Hi1 lis =
0.3 lLa:: ~k -= 0.2
8-4 fllI11.1,.IIU111lJ P = 140 Ib ~.l1lm.,f.ll1n,jn&hm.
11'.1~V~l'1lumnhviai'ilJncia. 300 Ib Iflarill1u~';l ).Ls =
0.3 U"::: Ilk = 0 .2
p- _?::
~,j,~ 8-3/8-4
8-5 ~.111~Ul~~a.lm P l>i'a. nm~lJl>i'uu~mm1unria.
40 kg nlJ',rJrlllI1Unnvhu'H1i~'lJU-1l1Hi.:jnlmJn~tfv~,r
m::";wicl11UfldD-l 1~trrVlVln~Vl A LrlBflTl1'lHlfll Jl.s = 0.3
8- 6 ~.J111U'j~Li1fJ~'Vll'UtJtn'hJrlcit).3 40 kg l1":;U'j-1yf~illfl
!ln~i1Yl1rIlO.';11l111i. x 1"'Vr~~ln~~ A allll. P = 300
NOll1U",fil ).Ls = 0.5 lm. ).L, = 0.2
405
8-7 1.loflnU"'IIUUIIUUl1"1lJ1 ~ma.'u1~uvia~jj~111'u
,jl.iiVltmllJliiu'll11UlYO~Yi~~l1'lJ"lY All". B ii~l ).Ls =
0.2 ~.'Hl~Ul~uauyjlY~ d lYiayhl11l1vi.ln"'nll~uln" lila
l.r'mlnnTnm:vh p •
d
B
d
A
L
P
>,j~ 8-7 ~
•
8-8 UH~llJUn'W T = 800 lb O'i:::'VI1vHHlvl-l thfhrrlJ
l.hdVlt~a.ml)JliiumllUlYfi",yj1.l1n~lJ C IW" D ii~l ).Ls =
0.5 ~.11lm.,f.llnuni1iyjuauYi,!~Yiffnlyj A !ii",;fu1JUviu
j:h~"'Jfl•..ra';ayj B 11<1. C lYiaf.l!Ti.hraQr1lJ~ l~u..ra';a
l~mJllUUl1lj~yj F 11"" G
~uyj 8-8
19. 406
8-9 nciD~III'llIh~nDll.r,0ll11Jr",I'iDL~D1IlLUll11l!"U~.
n ciD~ lilo~Yl11J~ B Ihli'1Jlbdil'l~~D~Al1lJl1iom1l1ul'5~
"I1'i l~<lDlL~.mur"iilil 115 = 0.3 11~'n1JU~ 5 N m
nmhti'U<lD ll~I11'illhlU;All'l1J1,()f~<lbl1WQti'uvi djDm~
m.Yhti'umur~iifll (n) P = 3 N (~) P = 70 N
8-10 ll~l,r'llll'l(f<lrDvi 8-9 01n1,U" 5 N . m m,"hluiiff
't11H~lJ'l.nWnl
p
~tI~ 8-9/8-10
8 - 11 UU1Vl 15 ft 11';0 80 Ib LL":fl~ii-l-w.Jnu~n.r.J'jl1Jl'iUU~
B olcrlJ1.hdhl1j'IJD~mllJ !ilo"l'l1"ull'o"vi A fiD A = 0.4
)1111'i1uul"ln"nl,~ul~~111D']j I~Vri111U,,1i1 8 = 60·
8-12 ll~l1l1PJUflU~q'~ eYilJv
tl'lVl1,11'illUJ'lievi 8-11 'ffllJl':i()
11~fi,.rlU~D~ N1r,~',UllOU 1"mhlmInm,~u10"
j)15 ft 0 ~
I ·
/ :!I!A,,~(}
A
,,j~ 8 - 11/8-12
•
8-13 IA~D' n-J1UUm~U;,1'1 n0nH'on O~'J 0.1,j U1~1J'~
... ....... , ." Y V .. ' ~
u;'''ln m01 ~ 1 n ~1 1lJ1J1l11Jn 8500 Ib lL~:~~tlUom~1'l G
1l~1111UUHIAlil~vi.1D' n" (~lJ.1UUn l~DlilcrlJlJ,. iil'l~'IJD,
fl11lJliio",Yl1Ull'O>lii A u~. B iifil llA= 0.3 U~. llB= 0.2
I>11lJal.lU 1~ol~1i"mllJlJ''lJD'~1tliD~lU~ A
j,j~ 8 -1 3
•
8-14 'HltJUvlihna 1.6 Mg U"::~VlfJ''LHrmnnn(l~ G ouh
iiinh:iil'lt'IJij,m1lJlila",m,nl'ii~ ,:111,,'],1cim.l1lU,,:<lvii111
11-5
= 0.4. 1l,:nl1filfl11lJi'UlJln~,!,Vl-e ~'1'HamJ1JfI'llJl'Joij1~
1"mhlfl1l1nm,vi1111",0ou.l,r;~nl'~u1()~111fl 1'<1nAil01
'uU1J.luriu~111111rimlU~,111ul~'fl'~
j,j~ 8-14
•
20. 8-15 ";10flU'I1ii.,,"nnB.nrilmr~~ l"ou<ia: nrilmnTn 20
Jb tilfilfflJlhd iYlf'll B~f111~Iii~~YllUlI,1i ~P~i1~ u.ia:
nftV~ii~l !ls = 0.6 ua:l:'I1T~Wf.lLIa:~aB.'!!Ua1~ !1's ~
0.4 ~~'I11 111~lULlU1j1lJ,hn~iil!" P iill1lJl1()m:';lhw
~jlft~lnn1'j~ll()a'l1~m~~nfli1
8-16 'l11UfU'I1;:j~"annB~nriB~J~~ 1"ull';a:nri6~'I1,rn 20
lb til P ~ 10 Jb ~~1;I"ilno~nrio~,)~nri11Ifi~0l1~ul~a
1I~o115 nflil'111a1l.i Iiie,ilfflJl.il: llVlffl11)Hiiu~Yl1ll1rO.. '.
. , .. ' , x ,
l:lIiHLl..a:nafl~lJf1 !ls ·~0.6 Lla:l:'I111~YtULla:mw.
iUril~ !l's = 0.4
,,-I~ 8- 1 5/8-16
•
8-1 7 1~U111tl'HUn W C1nH'lJltl...r()Umntl)J~Yl A Ufl~
,,,,11IL1U111lJ P m:';l{ B Buml'l '1 t11~~U1111iimi'u
iiul()a~ A ,rio e~ 30 ~~I11';lfflJt.il:iiVlt'IJB~ml1J,iiu..Yl1U
ffii..,:1I1H1~ U'l11Uua:lIlJ"
8-18 1>lU'H1t..n,,,Jn W (lnu'J11Jt1{)ullnl1lJW't~ A u,,~
u'i~''lm'Ul'jl'IJ P f1<,i~vi1~ Bacil-'lGJ111 fllfila~tI'J::ff'Vlt'IJfl~
flll1JLiiu~mUlIOm:lIiln~Il'l11IH!a:'I1lJ." fiB !ls = 0.2
"milLiiBlllJ e = 30' 1~LL'I11U~:L~lJ.ru~ul~a'l1~Bll.i
""""'~~-+- p
B
~'ti~ 8-17/8-18
8- L9 'HlLlYl"JOLlilcl'fl1'un 4500 ib ij;i1~ff'Utf(h-'lii G nll.uu
Lft~m'/L~lJ~ci'B'I1a~ B h''IJru:vici'B'I1Ih'~ Aiiiilll:hIOll'l1lJU
t11i1H~iJlh:;ffVl t''UtJ~fll1lJ LaU ~vnuerCiW'i::tT":h-'llifl~ B LI;::
407
.... , ' ' ' ; ..
vht~'Uiifh Jls = 0.5 ~.3'111m~-rllnnVierVl P Vl'lOUVl'JOlVl[)";i
lIl1Jl1~~~1..'1"utlnll)1nm,iici'B B L~.,ml~"'o~'I1'oci'B, .'HtJl'Vl A Lfi~f11<wnvi'U1l1flY;'U~U
8-20 'HHl'n'lnlVltlfl1un 4500 Ib u~:;ii'i)~fl1.H1ri1~~ G fll')
.u1Jlf~Bl!LilJvici'BlI"~ B lu~ru:~ci'flmr;vi'A fl1illl:luOl1
'I1lJu t11,i1fflJt.i,:1tYlt'IJo~fl11lJLiiu"mulI1i"l:lI-jl~ci'o~ B
.4' .. "11 ' ,..; . ~ "' '''
ua. Ytu..ulJfI1 !ls = 0.5 ~~'I1111IlJBLLVll~L"B 1Qn ..~Yl P ~
1200 1b l..ut.imfinom,iici'B B lii"n1'~lioall~flci'amrlii
A Li"m'U0l1U~ln~u";u'I11Blu
-~ft~--
2.5 n
nJ~ 8-19/8-20
•
8-21 fi1J.;1ihrnlJlffV01l11'U1'l1,nn~'w~UCl ~.Jll1'lrnn,)~lh>ll'IJ
~U ~~'111 m,~~1I.1'liII~:fi111mh. iiVlt'lia~ ~11lJIfiu..m L
lIii..~t!BU~<I..~tlalu A ua: B 'IJo~lilJii';11111ilJlu~ul()a
L~V 1'11 fflJtJ'j;~Vlt'lJt)-3 fllllJ L58 VlYll 1.1 ero9ll ~1111-1 ~'j (),] uu::
~'Uiifh ~ls = 0.4
8-22 ';1fflJt.i,:iiVlt'llfl~fl11lJlfiu.,VllUlIOm:'I1iml~1vii A
L~" B 'IJB~1i1J!la"vi,a~iifil ,l's =0.5 ua:l:>lil~~,a'Lla:
wui:ith J..l . = 0.1 01u';i':)~-lhHl'l.nllu P = 300 N n'j~,hv)a:s , . ,
filJ )~'111lJ1 ~iilJl nVi'l"';]<11lJl1Oil.,~~1,)
~- {:; mill ~-- 75 mm
p
1t1ri 8 - 21/8- 22
•
21. 408
r v... d ...J. v
8-23 'FJ111n1UmJln'Vl,!~ W Yl'l51t1f'Ul1'U.:JfnlJl1t1Uf)fiI1U
ml~ill~ft~~I(i}v1'l7,"uman (!riaiiu":'liihan~ A) 'li1V~U
Juihhmlo 200 lb ll""~laiJlhdh'lif~a~fl11lJlllum'lTW
<Y1m:11'i1~lVl1'Ja~'li1U~UYUU":~'U~Ujjri1 Ils ~ 0.6
8-24 O',lh11lrom.li, W ~ 80 lb ~~111lm.r~U1mJn~
U":u'~liiu"mum:liwi()'li1UflUl1il~ 200 lb l'i'v~nl"a~i'u
tl,11Unm:li11uu~a:mru lila,j,fflJlh:ffl'liffll1lJl{iV(i}l'I1l.l
(0)
~t.I~ 8-23/8-24
8 -2 5 viUQnu~11.l"iJ vI"vli~1J 0'1li1ffmh:ffl'lif'lla~al1lJ
"" ",d do d.,:.d
mU(i}I'I1Ull~"1'l A llo: B IJm Il . ~~111'Jtn(i}l'll"nI'l1l~ b. , s •
li1mll1HviaVijj!ilUN'HlU!rO"'~mvlu d 111m,"Qnvn'll'i'
~u~ 8-25
8-26 l,ll,iitl,11un 90 lb llO,a~i'i~uui'luViiJ Ils ~ 0.25
O'1'lilVflUl1il~~ii'n1ii,luVi"umJ1u e ~ o· l~111'IJlJ1(i}iia"
~i!?l"Uu~u"j~ F ~a-lnl"jlfl~D'UU1U1w:: 'tHlfl'UlflJU t11'lJl(J
mnYuiitl111un 150 lb l~111rilfflJU"ffl'lif'Ja~al1lHffU(i}
1'l1U<YOf1~Uilu~llm'11'i" lVl1'Ja~'li1vf1UJUII":~lH~Uli1Ll1
'li1UAuJuili~uin11
8-27 1i.iitl,I1Un 90 lb 1111.il~ii~uu~tI~iJ Ils ~ 0.25
O'1'li1Vaul1il"Nii'nl,ll.luVi" 8 ~ 30' ~~I11'JU1"UilU~<Y"'Ja,
11>0 F l'i'a,nmfl5atlii1ul,ll. uaninntTu t11'li1Un~JtI~
,.hWUfl 150 lb 'il-l'Hlfi1«lJ1.h::ff'l~'IV-:)f111lJLaU?lYlltHY0Y1
UUv~llfl,.111"lVl1'IJa,'li1ufluJuuad1un'iiali11H'lil"flmrll
ili~uln"
1U~ 8-26/8-27
"
8-28 )1(l o01 illtJihnft 200 kg UU~V~U>lYi,'J~·nr,'JlUl::
u~uumu O'lrilfflJh:ffl'lt'Ja'fl111Hiivm'nuifi(i}~ A llo'
B ilri1 IlA ~ 0.4 ll11. Il, ~ 0.5 ~luci"ru ~~l11lmhl!!m~~
lJl n~,!W1 P ltlh:::~fl ~n1JlfHiJ"~i'hlHrHH)W1~lULnffl1JHlfU
8- 29 l1C1t:l?lvl1uihnCl 200 kg . llCl:::O~trVl'H,nr-Jna:::{)Quu
fll'U QlrhaiJ'lh::ff'Ylt'Ufl'lfll11JLihJ~Yll'U(I'fiVl~ B lirh IlB
= 0.3 lW::flu.HlI'UKfltJru:::11UL~i11J "-'.IM1LLi':)lii(Jl'll1111~ln?l;fti
~ B lrimmljuu1~<lh"~n,r~()!fl!iJ"iiril P ~ 800 N
- B
~u~ 8-28/8-29
1
22. II' "" 11'''':
8-30 ')ou'U~ A lJlJ1ft 1.4 Mg L1a:;~~fJ'UVfH'I1~lJ1aVl G
ti'nouulli' B 'n"Ul~IUUU1111J ~ A 2 leN ,ml1ilu~~il'
n';llJ10V"'~~"'''~6U'humuu9i' A 'H1fl11i ,rlVrillllJlh.
.i ... ... 11',... ... •
im~~ fl~ fi11lJll'W>ll'llUIi0""a.~ au~:mHaflua:ouUlJfi1fls
'" 0.5 Ui:'I::: U(lZ" ~t = 0.35 i:1lJlJArl'U'l1'U B ijcl'nfl(u:;g"u,,::;
B 1Iiliifin1~iiu1oa
nJ-ii 8-30v
8-31 1lJmuQflUflJlw,u,~~1.womiiu,'lJ;juua:~u'lo 11
11~~'llUL~UU A 'il1lJU"'J~~ G djfl11~hHl'I..nllUriTUril-l
C f'1JU~V1J~fl"'Ul~ii fls ~ 0.4 }''H:'~l).I''''lno-iii)1il d ~
}:~fl,flJ U,'0,:';, F1..uUnll}lnnmfu'l[la~8,1lJO,:fl1U
GlW~4 _ I
~ Topview
I i
c-;
0.75 m.
d
409
8-32 }~"m:u:1n" d ~~lU"Ul1;j'lIllJl,m~u6riH'1i'11
ifumlJ"""~UU" i't~10n1~iiu10a~fl,,,r-iu~u '~VrillllJU':_ 4 d _ ..l ... ,
IIYl~~6'mllJ'IIU"'1'll1IlYO"Y1 A ua. B lJfil fls ~ 0.3 ~lU
fiuJlIihhmTo 200 Ib lIa:~"'f,!u6ri,'yj G 1>lu1l.ifi..fil1lJ
11'1..11 LlH~Jl11un'UeN u~'U~''U
8-33 .,,,m:u.lna d ii'l11Uflll'H;j'lIllJllm~u"riwlfl1
;j'l'lllJU"U~1I 25 lb 1",ulim1}lnn1~~u'loa~fl,u"I~U
,d",;,lllJli~.1il'lt~fl'm'lJlliu"l'llull1i..ii A jjri1 flA~ 0.5
"":~U~l~ B iiafl"W"'lJ,'iUlJ 'I11UfiIlJuihJ,mYo 200
lb ua:~fif,!1l6ri"ii G lfiu1Iiiifi';,mllJ'HU'~"'U"U~U
30"
.
~,j1i 8-32/8-33
~B
d ~ ~
8-34 m,ymm:1JflOlJlJlil m Ila:~ ..;;ruuO"Yl G ~'m
IllJlJlfl~i)1il e ~fl'~·l.Il1i'J1ii8 Qti"V:fl';lMfi~'m'm'Ufl 0
hifiu'to""'mlJ~u lij"ti1fflJ,j,.1i1'lt·Jfl' m llJ,liuOlI'llU
... , ¥ d ... . ::.
t'l'fl~'j ~1111-'JYnHLntflJ-l'VI'J-ln 'J!:iJ flfllJfll I1s = 0.3 'J1lJ'Yl-l't11
IllJ <I> iYll1f1J0~ru"";10
e
'nJli 8-34
v
23. ,
J,
t,
i
Ii
,"
p;-----
I
410
.r "'8-35 viofwun~" BOO lb f,)Oll~"'~lnl'lmOU;~lln 11W
viall~lu.i11111ti~.r.~IJ 01~1ffiJ1h.ffmr~a'fl11lJIffU~Yl1U
(l'fi~ii~"'IIll'~lU1a.i'u A Ul!. B ii~, !is "0.4 ~'111
.i11111U.~yjai~lJ~ul0" (~A I1Sll B I1lmf~~ A II". B)
lIJ~ 8-35
'" . . ..; ~ '" ~
8-36 l1tJn''O~ ()ylN1J'U." l ,:n .:.J lJlJ'H'I 500 kg U":;~~fJtW
n"" lJ1"~ G IIIall11lr'111Jr1'(l'''u.J'mru1,,~(l'.ifulJlllO:';,
1111IJ1:ffYlt~a~m111IffU"111U(l'11>11.wiHl1U~cia~II"."~~~
ii~, lls "0.4 5 ~'111I1'~lJ1nYill" P ~(l'11111oIJ1.~n",1..u
U'nf'l'ill fl f) 1'j Lfl ~t).rvl'U ()~lflJncifl~ ':11-3~-lhiihll'Hfl fifltUJl tI
t-O.) 111 1
~
0.4 III 0.3 m
:tl~ 8-36
8-37 J,UlJIJ,." AB oouunuuuhl'iJu<"lJomumh."
01rl1uuiilIJIIUU~miUI~U~IIUUl1lj"mul.r~IU"IJ1:<:l u,,;
';1IllJIJ,dh'lt'llu<m1lJlilU"Yl1u(l'fi""U~lliifi1 !is " 0. 5
~m1m111U1"llJ1oYi(l'" L YiJ,UUlI,lJ,'OUll.ilullJ1"lb",
Iii,,1;' l111ultiiimJ,mTn'llll<,;'UlJ •
uu
o
~IJ~ 8 - 37
8 - 38 Ifl~ll'llom"liull<nultil"t1011lJU ,iillt1ll0 nm.,
#I1tJnl'iU~ M = 5 N· m 51fh~inJ'i!:~~~1J[):Jfl11tJ~RM1'tr
(l'i11>11:l1iHnciu'l1J1flti",,:t1uiifi1 lls " O.B ~~111111<lulI'
~"rlluvil!" p nm11viviIlUll~UI1~"t1ll
8-39 ~,uri1~116,jllYi 8 - 38 01fl1,ij"iifi1 M 0 5 N . m
iil11>11lJloU1JU1Wnl
0.2 III
p
- 0.5 nl - --",---';.
--......... 0.15 m
~IJ~ 8-38/8-39
•
24. I
t
3-40 'lilJlumlnlum:~lU11-Ht1n~lUflriWil'11 ~'Ulhuuu
~fllM"I~flV'lum"muihrl11un 3 Iblft u~:M"I~auihh
lfUn 15 Ib 1I~"~~,:!uO(hO~ G HHlilM:I~au~:alilu
~ll1lfUO, ~ul(la l1~aVj1ififrhollun~:muQn rl<lnhJ<umlll'1
Ida d = 10 ft ~llfUlh:1il'l~'J"'mlUliiu~'YIlUll1i~UlI~l
llJ~t1
3-41 'lilo1u,,<lnll1fil:mulultUlnuaU11<U11 ~'UrilUUU
'Jol1~:I~vu 111m:~luiilhllun 3 Ib/ n lIa"lA"I~auii
J'l11rn 1 5 Ib l,a:~~!Uori11~ G ~Jl1lil1~"llivu~"ali
lu,iltt11UJ, Eul(la ll1vVj1in~ilOl'Ufil:~lUQn "<ln1t1<U1J
mIl I~a d = 14 ft ~llfllt1~"ill'l~'JVJftlllJliiu'll'llUll1i.,
uil~11u~t1
1------ 1 8ft-----~
1-- - --d-----1
i ,U =05
3 1"1
!
,u'=O.3 I'--,U' = 0.3
I ft--Ift
~tl~ 8-40/8 - 41
3-42
li!u~l'llUll1im:l1ilJlvi'l'Ja1'lilUmJ1j'Uflu;1uiiAl 11, = 0.5
~. tod . . . . . . . . "".
"11"I1I111U''JV1'l11UftUUUl'l1J~'''':11WmJ G l'l d tYWl'll
111ln.,IIHl1.l1I1.l111UlJln~'l'";iV1h:~ ~lU,f'11lIllJi1ncill
f11U
41 J
8-43 omu(juuihhllun w Ila"f1uu~1-Hiiu11",u1i~,j'u~
A olrillflJlh"ill'l~'Jfl1mllJliiu"l'llUll1i~~:l1i11vi,j'uua:
J/ ... ... . d u ...;
VjlJ1VU1Ufl1 11, = 0.5 ~111ll,!l1VflnIlUU 8 ·'JV11'1fttUYiV
vhll1011uli'iVlnlllftlivuulu hw'ufimhllun'Ja1~,j'u
1tl~ 8-43v
.., ....: ... ~ , "", d
8- 44 'il HTlll'].:J'tHI tJVl qVlVl"lfl tJ 'i'llJ'1'U.J'Yl l Yl tJ l~""W 1'lJ'f.1 flil'l D
... ~ ... , t:. d %
lfHl{)tlUl(Jl1'UOflfl-l 80 kg 'n1JVI-lflltClOn11'1.11J e 'YH1Hltl
0;,U';1lfutI'"ffl'1~'JV1ftll11IltU~l'll1.1lIfim:l1il1MUnciV1 110"
ifuiilil 11, = 0.3
'jtJ~ 8 - 44
•
25. 412
8-45 ",",11 1l~jjmUJ lffO"Vl11.1111illn1~~~n,flu~li"mi1.l1!~"
jjm~n;,U1l n1liiimhmTn ~'j:o!llJQn,WO!l:11'i 1~ mr~U~:
m~m:1J<ln 01 Ils = 0.3 Yi"ln~"lI1H1'i1 'il~mlllJ1l<lnILUU
e lYi<l1l1111'J1l11Inilil 1 W 'lJfl~m:IilllJ p ~nf~<lQM
8,- 46 iI::::'IJil~)jft 11lJlntl ~'Yll'UYil'ill nlfll-ln'.i auYiuYI m.iuHC1::
jjl'mn~,u1)nlli1ilil,hllrn m:mlJon1l1)Iil'j:Mil~mr~lm:YlH
m'U1ln 01 e= 20 il~m'hlllJ1.l~.iiYlt'IJ<l~fl111.Jliiulilmll
1Iii,"~,r1)U~<!" 11 ~"lf)~"lI1Hr1llvl<lyhl11'Jll1Uf)1O! 1 W
'lJfl~m:m1J p ~nf~1l~11'l
p
L~tl~ 8-45/8-46
8-47 mll AB 'H',J1l1Unt''':fl11lJMlIl~nm'1l11'l1mJ1Mun
f)'.i::vh UUU fl'J:::'ijlt1'ilh'l'1lJ Ll1~tllJ fllHCl n'H)-3i"J~tJ"lUvnl
J ... ... "tJ v .!... .. d ...
l1U-3'il1Ul1l.19UU'1 :;[If) f"f)'il l'Ul1U3Vl1EJHnVllJlJlCl 50 kg Llil ::::
1liii?lfl11lJl1111 il~mlLmr1)U~<!" P 011'l1)~nmfl~1llliilu
l1l1l~1llhlllJ1.h:frYlt'IJ1l~fl111.J,iiU"Yll11111i,,~ B"". C jjri1
IlB = 0.4 ,m: Ilc = 0.2 mlJcil,]U
8- 48 mll AB HJ1MUnt'''"fl11lJMll1Qnm:lllI'l1U1Jlmin
nl ::;yhUlJU f)';i Z'i! 1on.1f'llJ Ll1 ~VlJ fllli (l o'H)·:riuvi,J(11 VYllU
Mi1~')1UMII,"ll~di~tJ"lU')11111ii~')1m;ljjlJ1a 50 kg ,It,:
1l.i1i'"fl11IJM1I1 il~Mlril lImh:1iVlt'IJ1l~~11lJ,1IuIilYl11.1~ii'"~
B U":~ C lYi1l"h111'l~fl'IJ1I11il'IJ1l~U~~m:1l1'~IJil'1J5~ P =
150 N mllnV)nl'~~ll'()CI~ B UCI::: C l'Un"ll~tnn.l
1~~A==~±~;)b~I]~C[IJ:[18:~/mI~,- - - - - -- - 2 Hl - -
~,j~ 8-47/8 - 4.8
8-49 UN1Jtl,:nflU 45 kg 1l~;j~U1J~1Jt:i1~ijAlfflJtJ,"iiYlt
'lJfl~fl111Jliiulilm1JlIii,"jjri1 IlA = 0.2 'il~m1lJll.J1J.r~A11J
1J1n~<!~ M YillllJllmh"qn.r';1)uvi~ 1.,UtJ~lff'illnnll
n>"~i1111'tii.,O1,lfl ~fl1J~
8-50 ll~'UmlJ 45 kg lJ~ti-lu'U~'Uehijrhffin_h::ff1t.'Iil-l
f111lJl~(JVlVll'Ufffi~jjrh JlA
= 0.15 0'1 M = 50 N.m 1l-l"til
UHliiU~~l1l1Yi A
r-- 300mm ~
I C ~
'lt1~ 8-49/8- 50
•
400mm
8-51 "flMun 20 Ib U"""Q;jjl1J~U~1Yijj ~B = 0.2 lff1J
loii1lmi1l11'lJl"u>lluu,,:li~~.,nmh1Ju1J'lJ1l~ncifl~ 30 Ib ('1
~li1lJtb;1iVlt~ll~m11Jliiu,"1'l11!ff1i~Yi D jj~l Il D = 0.3
'il~mu;~11Jl,m~~~,'i1)u~ <!~~ fflUli mh. ~n~11Jl;1I1 iIlJ«11
tlun{J~vil1t11n~1Dnlf1'lumjlflgu'U~
p
"j,j~ 8-51
•
26. 8-52 1.mnlJihJ1~ 0.5 kg ua"e~ii.lJlJ~lJii1~e.M:
YlV'U'lJl~§elJ6'v1."111lJl",vii'uH.r~nmjn~ P ~'nih
~ ., 3 .. _ , . ~ ....
~enl"Vl1 O1llHlJl.JVlffVpmlJ~lJvn~,~~e~1~UmlJ 0 "'~
lU ~~m~lJl~~fl~ll7~l~fl1.U111lJflrillJ'1i1~~In<l011;lJ10~
~ A f;'ffl.JU1:ilVl~~a~mll.JliivilVl1~<I~l'1~ A uri, J.lA
=
0.2 ua:~ B jjf;'!1" =0.3
~tJii 8-52
•
8-53 ncio~"f.l'ffl~ A ua: B uvi~:ncia~jjl.J1a 6 kg ~n
~tl,1h)l1f11J.,j'tll"lai.:)~tl nl fi1tYlJlh::ffl1t9J CJ~ fllllJ lifu~VIIU
ffi11l~Y1lJii1ffl.J«ff~lil J.l, = 0.8 lla: J.lA
= 0.2 .~mul.llJ
ulJ1~~iil.Jl nii,!~ P ~m:l'hvifll1~il C l~o'hjvhll1ncio.
lii<lnm~ijuloaua"l!.i~~,r'l11rn~a~.uavia
p
c
413
. .
8-54 ualU C ~[)~.uijviaul'i~vr~<lfl~a~ii~lJlJ~~~lJrrn~H
~fl~ri1lJlJlJ~a.VlHm"lJijn 50 kg thrilri'l.JtJl"iiVl~~a~
fl111H«U",Vl1tl<l~l'1ii C U~: E uril J.lc = 0.6 ua: J.l, =
0.3 ••muHllJulJ1~.l.Jln~<I~ P iillllmotJ1:~n~il B
lilV1.h,ffil,n011ln~nlllflEe~ii 1~v1!.iiiillJ1~~O~ul'i~
p
i-loo mmJ
v~t
AI ~B
I·
I
c
"
200 mill
" I.. i
•
50 mm50 mOl
~'iJ~ 8 -5 4
•
8-55 nci{h1 C iiln" 50 kg 1!f1::~ml1nV'lnl'HfI~m.rYiYllV
r·.n.r~J-1ffD-11V'1tJcfvIEmri11Jl~tJU 01ncifl-ltl ciii~'LItfff'ltJ'U'U'JtH
l1~a~",V 40 kg '~11lu7.mulwfuiJaueu~,!" P ill."o.n"
lfl~ijlJiil1aij~",ulflliia~mimrlJl~u,auumlllJ~a.l1~a","'u
Iilri'lJ1h:iiVlt~a~mll.Jliiu~l'1llJ<l1il1l~ A u~: B Uf;' J.lA
=
0.3 u~: J.l, = 0.6
• p
"j,j~ 8-55
•
27. ,.
414
8- 56 nov, C ihn n 50 kg un:~n~lnm:wil,mr,,r,
ffv,1",uam~vunUl1uu tllnriv,vriU,uuri1uuu'Uv,,,,nv,,
ii'lU40 kg ~,mril«lJlh:ii'Ylt'Uv,m;lJ,iiu"'YllUl!~"~ A ,laO
B ,Yivrilhi",nv".rlu~lllon~ A un: B '~V'llUl~'IIV'
.~.¥.,
u"m:m,fjlJ'UUllJU P = 300 N
~'l1~ 8-56
8 - 57 nciv,ih1111un W ~n?h;ju1u~:U1Uliiu,~jjmllJ";U
a l"u1,,"5, P ell P m:rill~u~ <I> ",.t1 ""cr..,ilth",'u
m,~ulon~'n1l;ju p ~ W sin (a + 8)/cos (<I> - 8 ) ,~v 8
iiv IjlJ'UV,fl11lJ,iiu1l'Yllu e= lan-' )l
8-58 ~~mlJlJ <I> ~ P m:riwivncivm~vrilhr'llU1"'IIV~ p
I!VU~ll"'vh~,i'lu1U'l,r nlJ,r,11lAl P ~<!v"m'v~"'u
,jjvnriv,l1un W ua:1l11lJ'lfuiiril a
,t1~ 8-57/8 - 58
•
8-59 UmlJTl11lJ0l1 I un:lJ1n m Qmv" ul"u,vu~v
~nnnlJUn:l1rl~ A elluvi,vQ~~:U:'Yll' a ~lnt-lU, ~"1l
1l11lJl!,~l!vU~1f" h al"" umnl,Yi,~u,~hiuvlJl11Uvi,
,ij"n;~gu10a •'~Vfil«lJtI~:iY'Ylt'llV,fl11lJ,iium'llu<!a,,~
B iiril)l
B
~t1~ 8-59
8- 60 u~1-!n nlJihll I1un W un:vQ1Hn:ullJ~~fil«lJh:
fi'Ylt'llV~mllJliiuAYllUffa" )l ~'11lmllJ l'lJln~,!" h ~
~.UlUffllJll0lnuoiful"u11.i,;,111",.;" nnlJ~u10a
•/'
1,j~ 8-60
•
28. 8-61 H~~=ncia~iiJ111tTn 50 Ib (i'1rilffiJ1h:iil1t'IJV~
fi11lJl1iU~111Hj1I'IYi A iiril 11, = 0.2 Uo"':WilJU<>io:
nria.iifil 11', ~ 0 .4 ~'~llilU1UnciflJii~:nvJMriauii~:
nlJfl11
ltJ~ 8-61
8-62 uc;in::ncv-3ih}',lnrn 400 Ib 'Il-3'Hlrll UH P ~ffllJl'H)
ti'~ffm.~um:,rJnciaJ B "iYlnu~uloouuncia, A 51lJ
~ OJ ..; ... ¥ ....
~~,:u:l1l'1m'IJaJffmJl1lnl'l'IJllmU
nJ~ 8 - 02
•
415
Uo: C ii,il 11,~ 0.4 ~'11111'J P iio1fl.m,unfilJ B uo:m.
fflJ~a P' ,"aJm5f'~lJ AMa~ri'uii ,na;;uihfflJr<ff':l1il'
A um: D iiKn1:Jru::~11H~UU
8-64 Iffl D onm:lh;i',uJlmrnnm';lluuul~' 8000 Ib
'ff1oma,fu;i'; ufilJ.r'M' A ua: B iiiiri1ii'lJIh:iil1t'IJa.
~11~'llUl'll11Uff1l'Iii~u;;1ii'lJr<ff':l1ilJ AUo: B nlJ.r. B
uo: C iiril 11,= 0.4 t1lllH P ua: p' ~m~~flj(Y1U1tJu
ci'nlJ01::nl~U~H'l1'W1Pi'1l()-3'1liH~lJ'H5tl1,j l~tlif'U~1crlJ ~n'7::1111-3
A Uo: D ii.fnlJru:51lJI1UlJ
8000 lb
p
Jt1~ 8 -6 3/8 -64
8-65 t11fftR~111'1 60 mm 1l,,:ri1ii'lJlh:iil'lt'lia'fl11lJ,iiul'l
mUffOl'lm11lJffl Ullalu<'iul S uO:,J1511!uo A Uril 11,,=
0,5 ~'111m,lu"UlnlJ P o1a,m'lfi§flWJ1UI'11Euloo1ll'li'lJ
11111 ,hull olUIlulu1iff,:luO1" ~Eau"1u1.lUll>1ll~l n
m1miiul'lmu01ulutlaanliYl~I'I"tiu C ,iiafllfflJ1.hdil1t
1JOUnl1JlfftlV11'll 'Utfn,",':;l1'h-:J A uC1:::Vl'Wi~h B ijfil I1D=
0.4 1I'1u'hjiiI'lJ'11tTn'IJa'1'11~uloouo:fflutlcnUl,ul .
8-63 Iffl D ()nn,:>il';luJll1tTnm:lhlu"Ul~' 8000 Ib
lffl~ma'flJ,";u~)J~'lYa~ A ua: B iiihilii'lJtll:iil1tya~
mllJiiiUI'Il'llUffiilili'i~uihii'lJr<ffm'ril' A ua: BnUlr~ B lU~ 8-65
29. 9
~ . ~
i)(51RU£Jfl'JuIla~ mUnS8£J61• u
:Center of Gravity and Centroid)
luul'lUo ci111i~'ill0n~Hm";l11mi~~u~il"flUU01~ nO"il"fltlU0" N1J1"~u~,:UUUUf11~• OJ • " •
llJ~~0m1l1tll'linu-i'., ~~uj,.~1. 1 01,'11ft,1:111"oi iiii'il:HmfJuu00Hl'Ul'IH I)'U 1ftIii..111u
.ume0"''IJu~IlItI, ~u~ 1l":iJ1lJl.,, l~um1u1'i1Ul1ti~~a~l'Jjlll'l,ao",ucllil: Uff"~Ol,m~u~
C1:mlJl~'J'UlI>liffu r:haurcttJ-Hll n fll1'l1lfU'H.l1JUnUUU: 111 ~(1 tlV1n'U(},JJnnrflf)'j::~l1ulllJU ~1-3 '1
1.1 ~fIlJu6ri:l)lla~~~1I;!u6na1JU:lac{-1Vlsus~uuuaJa4Jl1A
(Center of Gravity and Center of Mass of a System of Particles)
~"'Ju,hh' (Center of Gravity) ~;mum:uu'lJu. n iiu~molu'lJUUI'IJ"'IJU~U1!ll1
511!llUff,,~lu~u~ 9- 1 (0) J111,rO'IJu~a'imm.h:n uU.ll0,:UU'IJa. "'~~'IJU1U 91.~mmu~
. r .... u ~ . co . . . . ..: .. of", , ~ , _
10U111Uf1av.UmIVIa1 (moulm) IW:~"l'I",:m ~.,tlI' 0n11 ~'lfJuom~ (Center of Gravlt)',
;) J111,rflcl"njil:lvil nUlhl1,rnvf.l1lJ"'IJD~D'if1lfl n m!mflvf~l1lJ" JtI~u
WR
= 'f.W
W I
! ,,-
I
Vn
IG iS'
,
I/o
/x/x
(n) ('IJ)
431 ;3
30. 432
~~>1lJ'la~llJllJu.r'la~tl111ll'n'lfl~alin1fll,fmlJ"1aULLnu x, y U~. z ll'hO'u1lJllJU.r'la~
tl111umrv<1haULLnl"~ncil1 ,,~trU "t11"1"O'., x 'Ifl~ G 1"u~~11lJ1lJllJu.r1autmu y ~:lo1'il
1i~u,ril,Y,mrn~:h;riahr,ii.,1lJllJu"ha1JllnU z ".101,,1,,0'., Z 'Ia~ G ~~u"uln111m:u1J
"n.,~alimflaQ l.,ul1lju 90 a~,ma1Junu X l1~a y .1~.U~ 9- 1 ('I) ~~11lJllJllJu.r1auunu X
):1",il
dlu
x, )', Z
i, y, Z
uWX=--
EW
_ ~zW
} ' = --
. ElV
_ ~W
z=--
LW (9-1)
UYl'ln~ n~'UtJ-l~~i;JUU~1:J G 'U ()~j i;;lJ1J 'U()~ O'lfllfl lvlUhii'iltl:JO-l~ f'lVll-l'Utl~
l~ntJ. x, y u,, ~ z
lmu~n"'la~ul>i~:olimfl1m.uu
'ill'lfl1.lVflilH:JJ1ll (Center or Mass) hwnffn~lifru'l111ri(J1nUfn~LfI~()Ul~'U().:jffffl'l, .mul"1iI'1~"'~'Ia~'l1~ ,tulia "'MI1l!~f ( Dynamics) ~~~ld'Ju"fl~ii~m;'umj~~l~onil ~.,
'.!u6nal~ma (Cenrer of Mass) n1llJd~'lu~mllJlii1J01~ g <il"t1fuYjn1 alimfl fl~Vi .1~UU
W = mg Ul'1'U~1tH"lJf)l'~ 9-1 LLa::-fh,r~ g 'U~1Y111
_ Lim
X=--
Lm
_ }:z:m
Y=--
Ull
_ Lzm
z=--
Lm
(9-2)
;)1 nfll 'HU'1(Jud;uu vil LLl1U-1'U tJ-1'il~ft'l-Hrn1-J 'il~~1lJ'ill'l L&i til nu ~1 Ul1U,HI ~ ftUtJfl al·Una. .. . ...
ucil~'hfi.,llJ 0lim~":iitll"t1UmrlooQillUM1i1'1 ~~a'Ia~lt1'>i~ ~.,lUlJolmilti'u 111'1(U:~~.,
,:!Utlnal~lJ1a.:'hj;f1l0~nufll1lJluIJci" un.1,0ci"l';11 .:luiifll1lJ"t1lJlulun11ilulJJ~.,'.!Utl0"
'U fJ'n::uu'Ufl-l O~fl1f1~ uVlulC'1n1....'l::UlJ '1~(J,r0'111" h-l'Uru::~~Vl tI'WUf) cn.3lJ1 ft'lJ a~j::;tJuiif111 lJ
,hi'lqJlJlfl
31. 9.2 )flfluEiri:lIJ, )flfluEina1IJU:>a lla~({junsmJfitJElIJ5~n'I '" 'I V ..
(Center of Gravity, Center of Mass, and Centroid for a Body)
;)"i1t1~ri1~ (Center or Gravity) l~flmi'nOlll'll'l1llllJnn~ 9~ I lJl1h: un.,1'linm:uu• v •
~fl~flljn l~~th:nau"1U1A ~u~~m5~ 'ilo~:ii.1nuru:~.110 nu unrj'uluu~ ~:1J1j0l~ii';l1111140
1JQ~ U, ji, Z) 1I~:iilhl1umin 1 dW .ro,ru ~:"flol'limliilllhmmvlfll1l~anlJ'lJflOlllJml
_ L,dW
x:--
!dW
_ !ji dW
y: !dW
~ !ZdW
z=--
jdW
...-
(9·3)
lYifll11nlll'll'lIlJn1l;1l11lJl:lIlJ Jl11UnLiln 1 dW ~:fl~lt1(YJijlJ'lJflomlJwniillfl"flCfl~ dV
01 Y 1I1'1uJll1un~1IV'/1: (Specific Weight) 'lJfl01"'~ 'il01"li'JuJll1uII~fll1~~11i1umlJl"'l
,,~,ru dW = ydV II:M1l
_ !iydV
x = -i!:'--y-'---cd
-
V
_ !,yydV
y:! dV, y
_ !. zydV
z:
!. y dV
(9·4)
1u;jifnn uuyj1m'"~:m:Yi1..afl'"mlJl..5'IJfl01"Qvl'ol1lJ..
;)"i1t1~na1..na (Center or Mass) fl11lJl1Ul11UU p 11;ijlJ1MiflmlJl"l11~mU1U• v
jj~11lJfflJWll1jnu y 1~ullJm5 y = pg I~fl g tjufl11lJdo'IJ1)~m1lJl,rlJri10 UYlUiim1lJ
fflJwurl'iflulllJm,;j 9 ~4 """on",,, g 1I:1,;'lIlJnml1UflUnU l!llJllOH111~"''.1Urrna10lJla'Jflo
1"'qM
l'IItime u~ (Centroid ) 1'II1.l1n1Ju.rli'Ju~"'lJu6nalOltimm'IJ1Ali",'lJ flO1",q ';1Ul1U~il'
l1lM~lIIl1lJnnYi1'li111~"''.lUffri)011;fl1"''.1tl6na10lJ1~'IJ1)'1>1q l.,ulllvn: flu1.li. 01111,)ti,: nflU
~ltrr~~~ljFll1lJ rn,r,lC'flJ tll17tl Ltl'U~'Ul~U1 nu fll11J'HU1ll1.l'LnI1flthHUfl~1l'Vn::1"I ::A-3~~av~l'~ c:!
".,ru !I'lfllJdll: Qn.r'"afln'ltiIIInoniiuyj1m...r'lflJOllYi 9 ~4 lflJom.a<Iv<rl'1I:UUllJI'llumflU~
~a-l1~f1 L~()~"nfli 1LUtti) ffJ::~ tltillnrfl'lltl-3l~ fI U~::~U () citllJ'jthn-lL"i'Ulfl ruYl'U ().:) 1Y1f) il::. . .,,, ~
l"ti'tn1j{)tI~mlJ1~" (Volume Centroid) t9l'tm'w[J~yfw~ (Arca Centroid)
~ti~ 9~2 ~,j~ 9~3
433
32. rI
434
IqfUmijEl~HrU (Line Centroid)
nJli 9-4v
mmm (Volume) ti'Yr~~~nlLti~L'Ju;'uriluliumlJlm elV "~llili 9-2 I'illLllU~~V~" .
L9iUmvo,f C (",. .Y, z) alll-j1JmlJl""'lV~1~~l'fllJn(l11l1~"'0';1l'lUllJLlJU.r'l8~;urilU'81J
_ 1idV
x=-'--
I, dV
_ f. ydV .
Y =- !. dV
(9-5)
rf'U~ (Area) 1tniltW-3L~hnn'U Leu'U'rl'HHJvfril'Hi"'u.yhf¥i~1'U(}31A~ t1til-3L"lh.J U~UU1~
11~vl"~LL1J1JLll1i8nJ1~vl~illvi 9-3 l'flUlmH1'[,n..un"LLti~>i"uliL'JuiurilUL~fl1 dA lLa:
11l1uLlJU,f'lV~;'urilu,tuvhD1Junu6H5~ lTuIin
_ J, idA
x = - -
tdA
- UdAy=---
I, dA
(9-6)
Iffll (Line) olitlmm'llmU"''la~1''~ vow'liu LL1i~1Jl~111v"i'lIal",lijj~lljl~'lV~"1I1LffU
vi'-3lU~ 9 - 4 mll1WlfUYl'HlfJ'ii'ij11jn"H~hnnU<UT3~'U 'il::l~"h
(9-7)
mru'lll~"lnr~l1lJ'"I'ilUI1U~'lD~ C 1,j1i1lulIIiia~aQn1olui'"~ (lljJII"ul'ilLLllU~aQuan
1.,QlL>iuoia~lllalul1J'L1U~a~1'" Q L9iUmDo.r'lD~illl1mmDOHtfllJn m"~1~.,Ol'liL~Dlj'l'l
nmllJlJl." (Symmetry) 'lL>il1jmojdja~tl'm;:;LLnutfUUlm L9iu1'l,aV"''lD~211vmJ.tlQmuLm~
l'fUlJ1~' VnvlltlUw'liu L9iumvtJll C alll-jULffll"~LLl'f..~lu~tlli 9-5 J".rv~vQ1Juunu)
L~tl~Jlnftl1Imll 'lnillrilu dl li,.o: +x ltlm~"'U'llljjV'la~lLnll y J<LI1iiaUt11J%ufflUV
,:0: - x ltl1'l1~"'W1ilOjjV llJLlJu.hfmu"'lV~{ufflllvfmlJmV1JlLnlltflJlJ1"lJ:oflL~n tTuii!
fidL = 0 "~il1Jfll,li 9-7 vl~JU "' = 0lun,ruli,llvlHjjilD~111atflUlLnUilmrljL' L9iUYnOU,f"
fltiii~OliYl'Uin(LO'Uvi'''n al1i-.l~u~ 9 - 6
~"J~-HiJIJ~L'mI1J X, )' Llu~ :: h-rCl'Jnl')i~ 9-4. ~-1 9-7 "tlthHl'l.Jl!llJl'IJ'U~r;'7uV;;tl>liJd'
~?lfJu6fil~111m9iuYnvu,f'la~{UfflUL~n '1 iih1uilunn 61Luultl1,,{mi111L1ln 1 if ftl,,<i8i
lM~'lUl"111vmlUI1UlLa n1 vQltliiIlVll~L~Olt1l1 il<1O~1LviLn;~~f,,~ ~lLL":rnVU~~lJ'la1
l'U~.,.r-1'H1JVl
33. !'
"
5smSa'lIllSu5IflS'l:vi (Procedure for Analysis)
iiiomn~I'llJtI{rri1~l1l019lumlw.1'IJ0~1~qlijoltlYml1'lvHn1~§uii,ml'l"f'l~v1ihl,d
;mhul~o 1 (Difi'erential Element) ~"1!llnU(h~il'l1a"'~on{uri1u'~n 'llMll1lJ1. fflJ
........ .,. ~ .... v.::. 3 .!' . v ~ ...... %..;,:' ,
ff11,mJf1nm'(Vllml'l ff1mUlffU 'lfUff1UU dL , "UVlll'lfUffllUff"'on 'l, ff1mtl"UVl 'lIUff111 dA
i~u"j'1'ltl'"ltJu1i'"~VlJ~ur<lYiiim1lJU11~10'1'"o"iim1lJ011,la0'lila"a111lu~ri1uYiiitBlJ1m
dV '"ltJlll~mm nolJiliilffii,h0'1'",,'1'1111)1114ll~ n'1 1170111,111JUltl~0nUl~Yijjn111J011,
fffjj~lO'l' Uo""'11J11U1Ian 'l oi11111ti~.furi11J':"I'llU'lJD1JI'IJ"'IJ"'~t1m~Yi~l'lf;'ml" (x, y, z)
'lJI1Vl""''''lJl111J1IJU~ (Size and MOlnen! Arms) Ut'f",,,11lJ011 dL, ~uYi dA 1110
mmI'll dV 'IJ D~%U1htli14lvWIJ'lJ1"'-;O'"Yil'll'lviOUV11J'IJOU''IJI'l'IJo,~t1m~ 111'-;O'''I1;UI1'UU
llJllJtJvf i, )' L1(1:: Z all1i'lH"lItIVl·HJU~111U~VlfJu6th~'IlfJ.J9fllrilU
m,au,hmVl (Integrations) um,Yi'li'D1;!oYi1i1nIUM'Ii'1~viulufflJn1'Yi 9-4 1, 9-7
ll<l,1iui'itmOl l~flii.htltlmliiouO'u liD f1111J11U11an 'l 'UD~'il'whu ~"~lO'I'l'IJO~n1~iiuiilml'l
- .. , ... " . ~ ~ . ... ~ ... ... ,
."UV1lJ.1n"'l U11U~'IlI'll!"ffO'''' 11111U''U0'm11Jl1U1la n'l 'IJ 0 ~'lJ tlff1U I'l~UtI IlW'lfWI1UQm11J
l'ili)vilnl'JVuiiln'i~llfl::;)::fl'imJfl~lJ'UmJl'lV1J.Jl11JVI
435
,-
34. 436
••
1m x = '~ I
" 1
I"'''""" ,.
--'--o~----;------'- ,
y =y
I
.l'=x
Q~ •
1'fjl'l1
'lilllhiu~n 1 (Differential Element) %urilw~n 1 "l'f.,.lu,l1~ 9-7 "li';l~~1lj.lJU
,illfi"~~"riil1u" (~. y)
'Uul1'lmnWilI!U" I!'UlillJllJlIyj (Length and Moment Arms) ""1)Wll~an 1
'U1l.%urilU dL C'lllJllmiiulJ1uFJ'U1l'1ltp,un dx ,,~" dy l.wHVlq)j~Vifil1n1auu (Pythagorean
Thorem)
dL = ~((L')' + (d»' = I(dx)' + I til'
I dy
dL = ~(2 )!! + I ely
_ 1, .telL 1:, x~4y'+1 dl'
x - - - - :..::--?~~-.:...
- I dL - J" I~ ., + 1 dJ'
I "...... ~
= 0.746 = 0.504 In
1.479
_ l ,y dL I:, y~ til'
y~--=
f, dL i:,,!4;'+ I dl'
1:./141"+1 dy
I:,J4/ + I dy
0.848 = 0.573 In
1.479
'Wi
35. fi:m81Url 9-2
~~1f11>i1lll1t.i~L9IUmDU"~D~~U~~UnllJlI1~UlJ')~iU!U~ 9-8 ( n)
"
y
,
--"'~----''-- x ...L_+'--4-'-'~'--+__--'_ x
3m-!!- dx 1
J m - ",
LlffYl~Ilf)U (Slrip) 'lJU11mnU y H(f'il~1l0U (Strip) 'Ut.ntmntJ. x
(n)
""CS 0 ""
Utlllli
9l1lcilulan 1 (Differential Element) ifurillH~n '1 1JD'ltJ~II1~OlJ"u'h~iifl11lJllUl
dx '),uMl,lulU~ 9-8 (n) liiD,,'nifUiilu,)~nulU1Ii'~~~~ti'llU" (x, y) iifl11lJ!' y
'JU1Vlvfl~u":II'U!1:iH:IJUA (Area and Moment Arms) '~'U~'U{)~~urilU fio dA =
ydx 1H':;lt)11H'I'iUUlPffl~~lUl1t.i-3~ t = X, }' = ~ y
m,illlnm,.. (integrations) HlIlJnl'~ 9-6 UlI!aUVilm..,V;uunu x _:'.ii,
">d • d.
x =_1,_x_d_A ='-;!;c-X)-,-'_d,-x
J, dA f; y dx
f:x 2(l<.£ x) dx
J; 2(3 - x)dx
J' I J' 4 ,J -'dA ,, (- )y dx ,,-(3 - x) dx
y=_,_Y_ = 2 =",,~2-;-::--:--:-
J,dA J;y dx f; 2(3 - x) dx
l~mll 2
2 m
2 ,I'G;. - ~ x -
2 ",
9lual1ullo 1 (Differential Element) ;'HillHan '1 ~D~lU~II1~UlJ"U,;,~jjml1J11Ul
dy ",ull~,1ulU~ 9-8 ( 'I) ifUrilU"~lU1Ii"~~~ril11U" (x, y) UfllllJOll x
437
36. 438
A:J<Jf.hlln 9-3
~Uli!l~~II'IJu11HJJU~ (Area and Moment Arms) tluVi~V~~UlY1U iiv dA = xdy
uft:I'b'umv~.rvci~";lmnl~ -" =1 x 11ft: y- =y
• 2
nnlluiilOlOl (Integrations) 1'll'l'IlJnl~~ 9-6 uftdiuVi,mVllii~1Jf)tJ y ,0Mil
I, I I d- (3 -- y) y
' 2 2
I 3 , I 'I'-(9y - - y+ - y
2 2 12 ,
_ , I' (3 - -'- )' d
_ 1 y dA J yx dy , y 2 Y Y
Y = - '- - = -"-- = --~~--
y
I
l, dA J;x dy I: (3 - ~Y) dy
X- j
(.t.y) --....'
1 m
. --~
y
I I
1----- 1In - - --1
(n)
I 'I'3y -- y
4 "
l l- -'-/ )"2 6 II 2 m
I ,".3y - - y'
4 ,
y
ex. y ) '" .~•.-~.
(X, y)
-.I-~)~
i
y
x
1m
(I - x)...-J
( 'IJ)
ltl~ 9- lr
•
",,4:1 • d -
l1i't11T1 1
1m
dy
..L 1m
~
OlBtJ
OlatJ
x
'ihuhwiin9 ( Differential EI~ment) ~Ul:i1Ul~n 1 ~jjmllJl1Ul dx ul'l~~lUltl~ 9-,
(11) ;u1i111"w2Jri'~~~Ylnlm'~ (x, y) lIa:,ift111J!'. y
~J~"".II'IJul1HIJJ~ (Area and Moment Arms) ~u~~a~~UlY1U 1'1a dA = ydx 11ft:
.. . .: 0 •
19:n.tVl·H)(J Yl tl~Vl ~ l LLl1J.:J i = xu,,:: 51 = Y 12
37. nnlhlihm>l (Integrations) 1';:<!lJn1l;j 9- 6 ~1ft.UuYiL01~~iiU1JfrlJ x ••1.ril
x ~ I,.i dA ) ; xy dx I:, x' dx 0.250 ~ 0.75 In
, I, dA I; y <Ix I; x' dx 0.333
""'''' 0 .d
lfillltl 2
_ I, ydA I;(yl2)y dx 1;(x'l2)x' dx o. LOO
y ~ I, dA ~ I; y dx I; x' dx 0.333
0.3 m >l0ll
~Ilhulii01 (Differential Element) .furi1u~~n 1 ~jjA11lJ'fIUl dy ..r~U(I",~1u~tJ;j
9-9 ('IJ) ~u'i1u..r"'i1Jir'~;j~..til'flU", (x, y) U~.jjA11lJU11 (l-X)
Vlunua~II'1ullJllJu.j (Area and Moment Arms) ~u~'IJ"A'uri1'W~" dA = (l-X)
.. .." . .dy u~" ~9fUm"u"WQYl",lLL'fIU~
_ (I-X) l +xx=x+ - - ~--
2 2
y ~ y
0116Unlll1>1 (Integrations) 1"'<!lJnll~ 9- 6 ua.uuYi~m..~iiUllnlJ y ••Mil
I I'2:" (l - y) dy_ IJ<lA I:, [(I+x)/2J(I-x)dy
x ~ - - ~ '-"--"----,;'---""---'--'-
l,dA ):y dy I; (1 -.Jy) dy
_ i,ydA 1:, y(l-x)dy
y- - - -
- I, dA - I ; (I -x)dy
t.?~::JElthJn 9-4
f'( ". d, y-y-)y
f; (1-.JY) dy
0.100
0.333
0.250
~ 0.75 111
0.333
0.3 m >lOll
.,m.hu'flth i ~9fUm"O';'lJil~~U~uml~ilQ1u'IJillJl'IJ",'IJil~lU1r'~<!il~ltJ'lJ8~ y = x ua.
y = x ..r~1UltJ~ 9-10
.,.
t
0 '2- YI) 'f'
~17",,-'j-·'_··.~';liO' ~--(:-x.-y:-')-yrl, 'I
,
I
='-~'-------"-'- x
(0)
~tJ~ 9-10
r
I-- - If, · 1
y=x
1 f'
!
"""'--'--------'-x
('I)
439
,
38. ,J
•
.,,
;r~.
..
440
illlcillililn 1 (Differential Element) il'Uli1u'~n 1 ~jjmllJl1U1 dx ui!'l,ll1~t1;;i 9-10
( n) ~UrilU.r'l~tI1lb;;i~>lrill1U~ (x, y) Ua" (x, Y,) jjfll1lJ~' (Y, - y)"
ffu~uii'l:;u"UullJnJ1'~ (Area and Moment Arms) ~u~'Ufl~~ui'h'U fit)
dA =(y, - y) dx ua'I'liumvu;i'a~;;i.ilUl11h i =x
Ol"l6Ul1lO:m (Integrations) li'alJnl'fvi 9 - 6 :;)::'~':h
_ J,i dA J;x(y, -y,ldx
x ::: - - - ; -
1" dA J;(y,- y,ldx
""~ • d
lfitlll1 2
J; x(x - x' )dx 12
I;(x-x ' ldx ; - 1- ;
6
0.5 ft VlolJ
'li'lIihlllilo 1 (Dinrential Element) ';ufflu(an1 ~iimllJl1Ul dy .r~llll~~iu~li~
9-10 ('Ill ~UfflU.rVlltl1fi" ~~~fhl1uYl (x" y) ua" (x" y) iimllwll (x, - x,)
~l!~lIa::!I"1..Il:IJ!:IJ'UR (Area and Moment Arms) vtUVi"il-l.iUri1U fi ll
dA = (x,- x,) dy Ua"I'liUmau;i'aQ~.ilUl1ti~
_ X I- X , X1 + X ,
X=X , + - -- =---
- 2 2
fll~!illihmVl (Integrations) i'll'i!lJn11~ 9-6 ~"1.-i'il
_ 1,.rdA !: [(x, +x,)/2](x ,- x _) dy
x= - - = -
l, dA i:(x,- x , )dy
1 I' , ..,Zo(y - y)dy 12
I
r.: - 0.5 [I
;(-vY-y) dy 1
6
i: [(,fY + ),)12)(.,[; - v) dy
i:(,fY -y) dy _
39. fi"::l<lthun 9-5
~.lmhul1u. yl'liUl'lJaa,rall1i'UW1'illu~au,r'IJa.nl'j11~u 9i~lii~l~umm~u~ui1mm
.r~lu~tl~ 9-11 (n) ,aULlnU y
dy
x
oQ •
)'Ii'fll
""'" Q .d
UIll'l11
( n)
---- ~,j~ 9-11
x
( 100 - Y)-----1
i---100mm-- --1
('II)
:imbuilln 1 (Differential Element) il'u1hljjJ!ll'm'lo~Ll~unalJuH~.Qm~aml~lu
ltl~ 9-11 (n) ~tHh"ifjjm llJl1"l dy l"'i~nl,'ilmmfll~unalJif il'"rilu'la~~"i11"
':lIllJl~"lnU dA ~:.i'~llnnUllnU'lO~nlm~" il'uril,,;l~~illfi'.~,n~iiu~~~ri"'lu~ (0, y,
z) lla:iif!'1JJ r = z
,1i,nA'iJln~II'111Jll11l1W;) (Vol um'.! and Moment Arms) mUl;;]71Ju,J~'u,hw JV =
(1tz2) dy tla::lqftfV1'HJU~otiVi~lHl1,i-3 5' = y
441
lOOmm
- .,--'---- )'
40. 442
~:J€JthJn 9-6
.'I
"I'
I,
~
fll11lUnmlOl (Integrations) fll11.h"~n,fillJfmih"o~·m~!!lJnn~ 9-5 1I~"1iui'hm~
lViolJnlJ y ~.1;i'il
.Q .."~ 0 .os
lti'fllYl 2
_ Jv y dV
Y = Jv
dV
16'" y(1Iz' )dy
!6"''(1IZ')dy.
1001IJ~OO /dy
1001[1:00
y dy --
66.7 mm
:Uu;:i,ul~n 9 (Differential Element) ~lnltl~ 9-11 ('II) ~UlhU1fllJl""!!llJl,nlaM
1u,tl'IJa"tl~an~111i'~lJmtlV1,~nmJUn~jjmllJ11ln tia dz "Ui~fl1,'ilml'!1ltl~anihlli'olJl~. .
.{ul'ilu'IJa~~u~1u"U1lJliiolnU dA ~.'JulUnlJlInU'IJaofl1'''~tj ;j'ul'ilU~•.J"'ltl1Ii'~~lnYl~
~'" (0, y, z) lI~.ffljj'IJaOI1J~anlhlli'~lJl~ r = Z
. mmmll~~II'Jl!llJ!lJu;j (Volume and Moment Arms) mlJlm'Ja~;j'ul'ilu dV =
21(r dA =21(Z(100-y) dz """I"'U-mflu"a~~.i1mni~ )' =y + ( 100-y)/2=(100+y)/2
OnflUflUl'lij (Integrations) li,nJnl'j~n'{)~'J()-.ln'llnll~ 9-5 lL(l:::~'Ulhn'j~lYitJ1JnU z
~.1;iil
..._ J,ydV J;OO[(JOO+y)12j21Iz(lOO -y)dz
y---- -
- J. dV - J::" 21Iz(100 - y)dz
1[ J~{W) z(l 0.1
- 10-1 z~ )ciz
21I J;oo z(100 . 10" z' )dz
66.7 mm
~~m~11111li''Ja'~''''.1u6cil~'Jaom'm.uan.J'ltl~ 9-12 (n) IhmllJ"UlI1Uu,tl~uulllh"
ril1",u"'lJnm.~.iJln~lU Julia p = 200z (b /ft'
0.5
(0,0, oj
d,
I
y y
x x
(n) ('II)
~,J~ 9-12
41. ...'l!illl
.1n11~ fj~'IV1 m11flllJ1~,'ID1ii1'~
i=~=O ~~
~u,hlIlio 1 (Differential Element) <l1tHi1Ul!';Un~lJjji'ffjj 0.5 ft ll~.m1lJl1U1 dz
OnL~onlu01,iiuihm"",~uii 9-[2 (0) liio.'lnfl11JmimltiU'l0'~Uli1U~'l1lJOl~1'iirill1i'tJ
~1~0. z iimnJ <l1U('flU••flQ';lll11U.mlJUnu z ii~..n111UOl (0, 0, z)
tfh nmua<U'lIllhJlm"; (Volume and Moment Arms) llilJl"''IJV.~Uri," dV =
2 *' ..,; 0 ,
71(0.5) dz 11~'I'1!Um ou..v~mnUl1U1 Z = z
011iill,hOl~ (Integrations) 1'1l'fflJnl~iiff1lJ~fl1fflJ01'~ 9-4 Uil.fluihmOllYiutJOIJ Z
Z = '-il'C-. Zecp---::d.,.V I~ z(2ooz)1< (0.5)' dz
I, p dV.,.; 1;(200z)1< (0.5)' dz
= I:,z' dz -~.667 fI
I, z dz
, . ,
l1lJ1Ull1'l : ti'Jll1tl'hi·lilii••l'1l'~uri'Ulul1flni111ri1lJ11rill1i'lJnl,fluiiIm"".lulUVi 9-[2 ('I)
l~0"1nmllJl1U1utiU'lI011ff~u••nOlJ'IJfl.IUa0nn,1ri.1J1."IIU,~10l1lJ~11lJ!''IJv1lul1on~,
hi'.lJ1111~'';l11mi.'IJV1 Z rill1i'lJ~UlilUlliffllJl'mnll1UOl1il
9.3 :Jflqus:nau (Composite Bodies)
10lqlh.nvlJ
luii'l1~UlJ ~w1'1,
(Composite Body) IUll'll~'IJa11..qlu,.mlltJtJ.1U ~ lJl';fll~alJOll 9l,lilllrl
.ul'lllJl11EulJ, .U~~.,. n~lJ Ii'JU';ll 10lnffllJ1JOuU10amulHhlllh:natJ,. . .r v . ' .. , • • • ...: "" _
1111UU1'M1J.il UU :: ~1 Ul1U~'l ()-l~~ TIlW cn-l'U v-l UVl" ::fflU ffllJl'S ()" if) fllllJ q~ til r1'YH1:; UUl'llO"iVl
l~(lM1 ~Yl i:J'utJcll-1'UV-ll~(pr..'ll1lJYl inn1'j~1'l1nuu9i Cl:: ffTUll1ij fl'lJv'llDIf! Uft::hin"1111'ml")1u
I11<UO'ii 9-[ fflJnl'l11iiallfflJnm~ 9-[ liiuulmi1il'i1
Li~{1
liW
x=--
DV
_ LYW
y=--
LW
_ LiW
z= ~W
x, y. Z uVl'W~n~'J OJ~ ~,!!lHj'(h~ G 'lJC),rr~l(l1.h::n{l1J
.t, y, Z llVlUiio~~0~~~!utf~1~'lJfl~ll~~:ril11U..nutJ~0110lq
(9-8)
L l-V !U UH!;l'i )iJ1J fhl t11111:i.'rl'llfhJrl 11 J I hznm IJ-ll11 I01'11tl,rl101 ~~~mflmrm,f,""1J(Ol'Ut) 6 'fl Q
1~o-r~flijfnllJ'HHllLl.hn1'tm.f1l1trn~lrvn:::f1~~ ~V1fJUuri1-31l::til'U~Vll;iu1nlJl9:1Ul1'HlU~'UtH
-r~~ rill1i'lJl'1!llmvu,r~u."iI, iu~ u~.IfllJ1m'IJV110lqU,"nOlJffllmnl1lhi1Olv1'1l'fflJm1~
9-8 l"vUVlll~ W';lV L, A ua: V OlllJiil.i'lJ
443
46. I
~.
>.1,
448
wunliiJ (Surface Area)
~u~~o~ r:;1~o~nm1ljULl'hfllJ ~a~tU~0"11llJ011'11O~~illfl~~lIfl~il'UflU1:0"~ITl~au~o~
l'liUmaO'!'lJ6~~illfl~~1If1~~u~i11
nnl'lllml (Proof) 1~6fl111J0111~n 1 dL ~a~1illfl~"~nJ~ 9~18 ",1JU1UlJUnUli'Ju
1:0: 21tr ~0h1lii~1~U"'lU~ii~u~i11 dA ~ 21tr dL ~1J~r:;lJ~"'lJMfl~l~o~lmljlJlillfl~
J-ll1lJ~'H}1Hln'U vl-ltTU A :: 21LJL
f dL mjH~hn~llJ 9l1Ul1U~ r 'U{)~lqf'UYl'HHJyfehl1i'unYu~iJ
fll1lJU1')vf.:ll111~ L ffllJl'H1l11'i)l nnlJnl'~ 9-7 ,xUftll r= rJ dL/L ~,J,fu ~u~r:h.,.f,Jl1lJ01 A
~ 21t rL 1.,ovi%J[i'1t«1J11iii01mljumh~lIlJPruU1JU .:1.il1
liio A
8
L
A = 8 iL
~1J~~l'IJa~01mljlJ
1I1J'II0~011m~1J1.,1IJltl~0~111~ OU, 8"; 21t
1:O:.r~~1n.1nunU'IIa~011"'1I1JlIJrJ~I'lilJmoO.!'IJa~lillfl~~1If1~ifu
fl11lJ011'11a~lillfl~~lIiHifu
~,j~ 9-]8 ~tI~ 9-19
LlSU1~S (Volume)
(9-9)
,HlJ1.,1~ 0~l<I q~,ii.,.lnn1mljULyh nlJHnf,l tU'11 o~~'IJ~~lIfl~il'1Jua:1:o:iiIfl ~oul~v
1'lilJmov.!'IIa~~lJiiiillfl~mlJlm
011l'llll1 (Proof) l~a~lJiil~n 1 dA "~ltlii 9~19 ii"'ljU10lJUnULi'Jm:v: 21tr 11:
ria1lTIR"1~U'11IJ~iim1J,.n dV = 21tr dA mIJ1911li'~"'1J"":ti'i~1~0011"'ljU A 16UUnIJ
0i~,J'¥'J V = 21ftr riA f) l'lflII Vl Ill)'")cf , ~,nH)ril'il..)1'i10lu'),1J1l11~ 9-6 tr dA - rA ,ijl) r
Ii'JU.i1Il'1li~''liUl~160'! C 'IJ6~~uii~lIiH~'1J A ua:mlJ1m V = 21t i A .:'1,)11
V = 8rA (9-10:
47. v
8
A
mlJ10l'~ll~nlm1l1-l
lllJ~ll~ nl'l1ljrr"I1-!.~tJ'IJflmL~ U1-!, 8 ~ 2rr
l:0: iN10~lOLL01-!~ll~0lll1lltiltJO~L'l11-!YL'"o.r~ll~~1-!iiiii'f{1~~1-!
K .,j'" ... .r
'W'tfV1Vli'llH'UU
ltJm~~tJ,:oounu (Composite Shapes) ClllJ1,~tJl:qo0l1'll'YLqllliyf~i'fll~nULff1-!111ll
:u~.;j~tJ,:nlllJ;'}10"1"1f11-!tJl:nflU 11-1m OJd~1-1ii~1111mnlJ1myf~l1lJ"iii'f{H ifWtl1-!1f1wYllJ
L~lJ~ll~~1-!~~111~ll1J1IJ1"lliii'fh~1uLL";":~1-I1f1L~th:nlllJnU L,jll~'lnLL";":1flujj~lJnl"'ll1
L~lnlJ 8 miillunu LL~:':O:.l0 LLnU~ll~nlll1llultJo~L'l1umll0.r~fl~LL";~:1flulh:0 fl1JLvil nlJ
r i.nYu
A = 8LrL (9-11)
V = 8LrA (9-12)
449
A:Jaa11ln 9-10 - -__________________________....
•~LLi'fL'l~·l1ifuiiihJll~m~O~lJ fia A = 4rrR' LL""mlJ1"" fio V = jrrR'
,tJ~ 9-20
•
,..lli'tll
nunfb (Surface Area) ~uiiih~fl~V1HO"lJ"~lUii 9-20 (0) LnL>l.lnnl'l1ll1-!~ll~LffU
t ~,,""jm"."."""" , ,~'".",..","",.;;1',""',"""' , ~ ",. """"" ,
48. '(
i
JI,
liil
•
450
fiJathlln 9-11
a
'Uv~unul1lJu lriv~~)m9!UmVU.rLll~VU~)UlJlJ e= 2n miiuu li'ivlffHm,nnlJl.w(,;lflJnn
~ 9-9 ~::'l.ll)
A ~ eiL; A ~ 2nC:)nR~ 41[R'
mlJ)m (Volume) illlJ)~~'Uv'm'nnlJ Qn 1fh~1~Vn11l1IJu"{uYi~1,,~ n"lJ~,1ultlYi
9- 20 ('1) 1VUl!OU x1ilflJn)~Yi 9-10 ~.Mi)
v ~ erA;
~~11)lJ1 <I'll v,fiVU n' .,Yi.lV~ n)~LvialIr), mU.illri,,r~1u.tlYi 9-2 1 m))Jl1U)UUU'lJV'
fiVUO'iL'l Po = 2.1 Mg/ m' uvn~)mfu ~m)~uYi~l'IJV~mU.llU
I--Lm----1
-1!iJ~~(':W1~-
,,4.5
il
,t.,f l44m
",1 4 In 4. L5 m I
_I 1~_
03 III
H
0.15 til
('IJ) (il)
~tl~ 9-21
Ina (Mass) )Jln'Uv~mulri'l11L~vh'lrfri)l1u..m)lJl1"U1uUU p, Lriv';v~n)n
mlJ)L'l~'IJv,fiVt!O~" ';v~m)u~uYll1lr)~LiL'IJv'muoil~tl~:nVU.llultliiL115UlJ ~uor)lw,.tlf
LlI1-n1'icHJ;;l~'ilfl c U~~ C ~-3'lI<nl~ 9-2 1 ('J) mlJlPl'Hn~Pillflu~a:;Vru~ijfhrvhnUfHHl• 1 l: .. ..
'JU-lyfU~l1Ul ~'~ Llu::n u :a~u Vl 1~1~ tJ 191 UllHl tI~ lufllHyf1 ,l!t1lrl'~ 19fUVl"itluilfl ~fJ11 ~l Ul
e= n(90' /180' ) = 1.57 miivu ~'" C LlJU1: V:'UV' r = 4 .5 - 0.1 = 4.4 m "nllUtlnn• 1 1 OJ
'Uv'n1nll~u (~.. 0) 1u'UOIoYl C , Ll'iu,: v:'IJa, " = • + 0.1 5 = 4.15 m ~10 Q 1i1flJOl
9-12 ~:1.lmlJ1I>mf'111J'"
49. v ~ 8LiA
I.S7 rad[(4.4 m)(1 m)(0.2 m) + (4. 1S m)(O. IS m)(0.3 m)]
1.676 m'
111ft'J~~ftflUn~~;):M11
m ~ P,V ~ (2.1 Mgllll')( 1.676 m') ~ 3.S2 Mg
fllilib (Surface Area) ~uYi111'1 a~~tlhl~11l1~om1L1u~lli'u1aultl'lfl~~uYil11rWr~
li'JULltl~~',.lii1UciDU'JD~Ili'U,r~ltlYi 9-21 (ft) L1"'tl1"~n<>ralJm1Yi 9-11 ,r~Ju
A ~ 9L iL
~ I.S7 rad[4 m(O.IS m) + 2(4.1S m)(0.3 Ill)
+(4.3 m)(1 m - O.IS m) + 2(4.4 m)(0.2 m) + (4.S m)(l m)1
~ 20.4 01'
451
9.5 waawslJaJs~uullsJlluunS~01811:J1Ll (Resultant of a General Distributed Force System)
hn11-«O~ 4.10 lvlnal1{i-3in~'i'~viv(1~:t1'U~N LL'J-l"HWltmHrt''ijtJuL~(J>l LL'J-3L~ (J1'U(),nY111'Uf)
nmhYlm:;)loaJ,,alJ8~llJllnu'lD~~1~tl~'11;OlJtiuf1'1 1ul'hullll"nft110.'iiinm1nu;1u~1
I Itllm1~ '1 L1ft:Qnm:nl.l1Um1nmllmhl11l'nllUU>;" '1 ;,j,,"1'll'llnm1u,hLTaYl 9.6 ,uunn
l11Jll1unn'j:::vhclvnr~l'ill'i[)~1J~1'lJV>31P1t.l~~lJhl'Uf)-31'11(1
nl1fl1:mUfll1lJvlllUllvlllib (Pressure Distribution Over a Surface) V1;)111l11
ll~uUlmU'luu,r'ltlYi 9-22 (n) YiQnnmh.l1UVI~n...ruJ111tinnmi1 p ~ p(x, y) l~~ 1
lhffftla Pa = 1 N/I1l~ L~tl'V1'l1Jvr'lI{;1Jif ffllJl'10111'U'Ul~'H»lLlHl~n1 dF ~n'l:::yjl~()~l~~
lan~ dA m' 'J8'll~IIU1~ 8IiYl~~V1n~ (x, y) 'JU1~llHil'<iioulultlmjl~~lu dF = [p(x, y) NI
m' ](dA m') = [pCx, y) dA] N J111tinm.nlvfmlJ"utlllfiuUl~m'11u1:UU'JD~1l1J~'Itl1U
~ii~llntl1lJ~1 nVl Lla::LL~ i;'l~ Ll).:jn·n:vhu'W;.fl.A~lgn'1 dA 'j::;'U1JCUU-lu'n ticu'W l'Uifu::; L~tJ'Wl~~lu
lUltl~D~1l1~""i,~", FR m:yh,.ilu~~mrm (X, y) UIHl~UUHvl~ltlYi 9-22 ('I)
,~u6n
lilt1J~ i,
I'
(n) ('I)
'nJ~ 9-22
•
50. 452
'Ul!lOl'Ua~!!,~.rVli (Magnitude of Resultant Force) l';;m11'lJmOl'lJV~ FR ~l!UU.ru.
nlJ!l~a"II'.lan 1 dF ~m"~luuv1u~ihvr.'I1lJOl'lJu.II~U1m A ~anlJ;ln11J1'0l.u8U1u~1.1
mUOlfflM" luuiiuvitmOl , "M11
- (9-13)
VlU11 p(x, y) dA ~ dV ;uri1ummmlan1 ~um1unOl.lulU~ 9 - 22 (n) ".Ju
'lJUl"'"a.m.""Ii,"1>i1nummmvr.'I1lJ~m8MrT~,.h'l1Un~m",lV
olll!'I1U~'UO~!!H"VI~ (Location of Resultant Force) oll11m,i. (x, y) 'UU. FR
11110lvri1'l1uOlllJllJu.1"o. FR 1>i1nullJllJu.1'UO.ll'i.vr.111J0l dF 'iOUllnU y Ui'"x A1lJa1"U 'In
nJ~ 9-22 (n) Uti" ('II) HnlJn1'i~ 9- 13 ,"M11
_ J, xp(x, y) dA J, x dV
x= . = - --
J, p(x,y) dA J, dV
_ J,yp(x,y)dA
y=
J, p(",y) dA
J, ydV
J, dV
(9-14)
'V'ltJ 11U'U1 nl'l nl::vll 'U e ~ ll'J-l t1'Wli~ ll-ll'l:; ~ ~ ~ Yl'UUf) (;ll-'l~tJ I'll~ 1l 'U 1fl ru?l111V lctfUYll tl Ufl'U()~
mIJlA'im8M,r.J1'11,rnm:~1!!1JUm",lV
9_6 A:J1ul7lumJlJ1VlCl (Fluid Pressure)
'lnnQ'IIO.U1nft11l (Pascal's Law) 'IIo.1'11a~oiinu~':rioll1liiOlft-lllJ"U p ~~0l1~V1nu
1tJ"lfl1';]f'f'Vll~ 'U'U1Vl'UiJ~ P 1~lliJuU1-lYiDl1-ct-ll1UJU'UD-lrr'U~ {l-Hl~nuJnnrrl'illml:; y 111ilfllllJ
l1t11111-hl P 'U"~'IIU~111(HI":m1lJlln z 'Ua"~"'ln~'uih"m.·l'l1tl m1lJffm;uin1lJl,m'ii8ulu
.uvn.mUOlfflnOl{ M11
(9-15)
l~a g !lJum1lJI1.'Uo.m11J1lT1J~1. (Acceleration of Gravity) n1Jn1'~ 9- 15 ,"1'll'M
Ill"1:'U"~h1"~ ~ nnlJ)J ~115OJ,h'hi'l" (1,jjj nmu~8UlIUMu1m m II~IU§vulU~1.1uOJ1lJ
m'tju:~m'i1J) miJaulunnu'lla.'lJo'll1tl1vr,1u ii19HlIu~fl.11101~5~il (mlJlmiimJlu5uu
lltlo.fl11JlYilJ{U'I11fla~ o.111.r) ltO"I~u.illn~111J11l!1uuuiin1'il1.1fiUWIUtI. ~1lJ~11lJiUII"'
Qru'l1~ii n)Jm'i~ 9- 15 il"1,jl'f1lJl1~1'll'1.r
Y;"'!lJ1II"UUH~'1Ji.ltO~"1u,U~ 9-23 rill1u~~~nllJ~~UWl"UU1" I~U"'ln~" A 110'
B jjm1lJftfl Z il1flr:h~o"'I1t11 m1lJ"U~,~"'n~11jj"lJlOJ p ~ "z lul'hue"~81nu ilA C2 • 2 j ' 2 •
1l~~~111Jfin z, ",.Ju p, ~ yz, 1UnJruvr~'I1lJ"m1lJiunm;wf.U1nUn~nuv1u~ih dA ~
DQli~r;)ril1nIVlvl~llh~ D-2 3 lq{(1'}Jfl1J~ 9-15 ltfl'!:r'W'i11f1r11'l.'J"()~ 9-5 tlJ,n.,jl~Y~'i1~t11H'-l.r~:
'lJ (l-J fl l1n 1::'11 tlfl11lJ ilu'U O-lttl (l Llft ::'J::~~1U11ti~U'U~I'll D-3llHlllJ 1>J~lJlJ ~U'YI,).:j'U fH LLHtiUl~ll::
y;j)1,on1.r(n1J"'''''",if
51. 460
f"minun:lU
9- 1 i1~111~~!!U6ri1~ X '1W~Uvi~~Dlu~uri1Ul1;j~'1D~f1~~1~
• .,. ~ o J , .d •
fl ~1J LlYl~Utnl1U fl ~Dl1U~l1U1umlUU11 0.5 Ib/ ft UDfli11fl
Ju i1~111u~~uBmUl1uLlU1>lU~!lU~D~i'mlu'1uU B 110:
D~flU~:flDUUDU X 110: y 'IJD~LI~~uBmV1~m-l~ A
~---~--'
8
~tJ~ 9 - 1
9- 2 'il~111Yi1Ul1t!~ ~il tr.tH)(i1>3'U tl~lrvi~~ii~U~l1trHi'~~~~
tilf111Ul1U1UliU'IJD~Uvi~"u1fi1l'nu p~ kx' ,riD k flD filf1~Yi
~uYi 9-2
•
9-3 'Il':'111 ~lUl1'1i'W:tfUV1'HIU~ X 'U€hUtYi;J).:JflOli lVltHl(f~~
ftlWlOU1l-1:u'UO~7f1'ii r ufl::~)Jfi'5-l((Tu'Utl.Jd"l-llrr~~tJ1"'f1all ex
,.
a
C
,
a
I
9- 4 i1~111';1 Ul1li~'IJD~ i1 ~ffu6fl ~1~U1 ~'1D~Llvi~ ~ll.1unJ
ri1Ul1;j~'ID~ri'lUll1'~~U1~~~U' ,
,.
--'lL------la- x
/<... 30~
9-5 ;)-3111'J::!'l::'11~.t ~~ilt!'UUril.:j'Ufl~Uyj-3~i.h11-3'W1':ill1JU1
tillL'vi-3ih.i'lnrnY1cn1i1-311'l.hVfl11lJUll 0.5 Ib/ ft 'il-3111U'H
u1imUlYi~lmD~i'uuuuV~"li1.!fluYi 0
9-6 i1~l11':U:Yl1~ y Yi~~!!U6ri1~'IJD~"yj~~DiUYlH
'Yl"i11U~1
r0.5 ft
I-,---.~. ;;;;;;:;;;;;;;;;;~::::::.------x
°-____ 1 ft - -- -
y
I d~4-lx2
'-,-<;m6,---",~,,·z-.10-+-'~'·~;'-8m_I-"-6,'c-! x
52. ,
y
, I
y
~1~?~_:~1_;- I
Ih,
"M!';' ~lIj<> J
''':!'' y:ax' j r
-.J-<'''------ +"----x
1--- b--1
nlli 9-9
•
• • .., %..:
9-10 'il-llilV11 'l'Htt-3~qf.lVl ·H)U VI'}t).J),I'UYI U'H~l
r
I----a---~
nl~ 9 - 10
•
y
. nx
y=aSln Z
I-- - L---I
" . . . ) { ..:.
9- 12 'il-3111VllLm'U~l9$lJl1Jfltr~'lJtl-3'Vit1l'lU'H-31
y
-.
Y==~X'"
a :4: Ih
I
I-----a- - - --I
nl~ 9-12u
" . ..,r ~
!) -1 3 ,F1l11Wl Lll1U.:)l91UlT'HIUt,;l'Ua-.1lt1lUVlU'H.J 1
)'
r y =h-4,x"a
461
53. t~ ,,.
.,
u
1
I
i:1
j .
462
o . . . . : : . . ; . . ; . . . v
9- 14 ~~111"lUl1lJ~!'l1lJmDO"~D~111J'1Uj!n1~mjjDUmV
wln1uHlI1H:llllJm, y = a
y
~tHi 9-14
•
9-15 "~111oj1Ul1U~!'l1mllDo.r~D.ll'IJ~l1d~h!iili11J~D~2tJ
'Nl
>tJ~ 9-15
•
y ,
y= 2k(x - ~) _
2a '
-Ika
-+-~~.. ~. '--...a~~L--x
•
y
Jt1~ 9-17
" "ic..!"l.
x
~tJ~ 9-18
54. y
I '.~.
4m I -
I y=4-
i6x2
~+---~~--~-- x
8 m~
1t1~ 9-20
v
9- 21 ~~11,jllL11U~I'lIlJmDU.r X ~fl~~'J'villll~lIU1",l{i1~u
Ul:lihwll1ilJYin,,1YlUHn()'IJfl~<ii1J«lJ (Simpson's Rule)
9-22 ~~111oill1l1U~I'l1lJmDU" y 'IJfl~lilJ~mm 1lti1~l'l6i",U
tlmjhwlliilJVin,,1.lUHn!J'lJfl~<ii1J«lJ (Simpson's Rule)
y
--/-------------------:----- x
,,-------2 fl-------j
nl~ 9-2119-22v
, <:: . . . '
9- 23 UfftHl1(lOl1'Ul 0. 3 m llft::lJfl11lJl1Ulll'll'U 7850 kg/
m
3
'i),nnv11Ul1U~'U()~~~fJuafHnnn(j "l11.nf-ll11U'l-ltlBmUl
~!1'U,)tI"~1JM1J~Llft::aa19tilt
y
/ =2x_
1." 2 m
,
2m
463
9- 24 ~''Hloill1l1U~I'l1JYllllU.r X 'IJfl~~J~lIml uti1~l'l6i~u
tll:ljjJOlliiJYin~1~u1'll'n!J'lJfl.<ii1J«1J (Simpson's Rule)
9- 25 ~~11oilI111U~I'lIJmDU.r y 'IJD~i1lJ~Uml 11ti1~l'lal~a
tll:ljjlJfmiiJYin,,1",u1'll'n!J'lJfl~9ilJ«1J (Simpson's Rule)
y
--t=~~2~fl~==j-----x
1t1~ 9-24/9-25
v
9- 26 'H"t11v11Ul1t.i-lLC)f'U''1'j()U~ x 'lJV-J~'U~U'H-31 unhlVlul~tJ
tll:dhlfmiiJiin"l",uHn!J'lJD~9ilJ«lJ (Simpson's Rule)
9-27 1)'ll11v11U'HU-319f'WVl'HIOvl Y'lIn,r~''U~Ll'H-ll uti11lVltJl~w
lh:IJJlJOl,ijlJVin1l1~uHn!J'IJD~<iilJ«1l (Simpson's Rule)
y
I ( =0;';
-'1'
.~
~~"
-f-~-"-----+----X
Im------l
~t1li 9-26/9-27
9- 28 ~~11oilU111i. r 'IJfl~I'111l11'DU.r C ffl'HflJ1~lDlJ'lJfl~
itllll1Jili~", r' = 2a' cos 2 e, ( -45' 0; e0; 45' )
./
~u~ 9-28
55. 464
9-29 lI~I1Whlll1i~ r 'U~~I'litJmou,{ C 'Uo~ ,hlJulJ'Uo~
~t1mf~uou r = a (I-cos OJ
lU~ 9-29
. , . . . co
9- 30 'l-l111¥l1 Ul1'U.J t9fWYl'Hl lI<il'U tJ.;j fl11'n.flll~'UU1~'J
x y
z
,.
~t1~ 9-31
,
'itl~ 9 - 32
•
57. 466
x
j,j~ 9-37
•
9-38 'il~'Hl~lU'HU~'J()>3~YlfJ'LHrfh,J X 'Ufl-l2Ufl~';]Y1HnftlJ
1'11111'111.11mJu'JD,3"1ffYl utIHll1~>311.1'U~norl"tHrvfl ~ (ill viu 0
'hhuIJ Po ~~IJr:;, 1~u1-15{mi1IJ1,jnm~n1J1;ihlri~1,j~i~
YJ;~n Clll li1lli'lJn l 1UuViImOi Im"HNCl <i'>11f~"~1~1'l6,j'u~ 9- 33
x
~,j~ 9 -3 8
v
9- 39 ""'111vilUl1U>3l9fUl1'H)U~ Z 'U(),;]~tJf)nUAVl'Ua>3 n';i1tn,;]
n mJ';lU~11jj"
x
~tJ~ 9-39
9-40 1,j~1~m~n~lljji'fljj r vh~ln{un~U1Jl~ 1 m~ '
~ii~11llmI1lIUUl,,j"hmllfl11lJ~~ p = kz ,flu k IUU,j-
tNl1 'l"'l11lJ1" It'':::1:::tJ:::1'l1-.l1I 0.:1 ~1Ul1U-l~ ~fJt,£fnenrlJ')Cl G
1,j~ 9- <0v
58. 9 - 41 01D,!nlt1J-3~1nUVlU~1'ilut),!nlfll~tJl 10 kg fl'j~l'h
~,.vn'llJ 2 m 'tJl'm.rlu'l71ujja'llo~~"!~lJ.rU il~111~1!!mj~
j ll~ "ln~ m 'IIa~aljfl11l Pp p
y
1m xP I~ 2 m ---+-3 m - -+'-'"-I p
-)--Q 0 I~'o-'-~--xSkg 2kg #flip
1tl~ 9-41
•
9- 42 il~111~11!11j<'II0~~"';1U[rn~1<lJli1 (.t, y, n 'Ila~
at~nlf1,.r-3 i'l'llJ
1m
·n.ll1 9-4.2
•
467
"-.0' <,)u'i1ujhnh.r".r<~tl il<111~11111j< y 'Ilo<L9!UmaU')
C 1"uhjii"mllJl1U1'lfl<1i'f~ILa"0111ri'«51ln 1 ~~1J
y
I
c lU150mm
JlLl,
~ I I I !
40111111 / 40mm
20mm
9 - 44 uNum~nLL"~DaiJl'i1UlJlh~flaunu~')OffO'j,JU~ ~m.iu
lli1:li"~,,nlJ~!T< 1l~~~L1~UlJHjjfl111Jn-h<Il~~ l~iiffVl1~ Z
200 mm lm::fll1lJl1'Ul 20 mm nlfll1lJl1'U1LLuuY1 A
llil" B iitil P, = 7.85 Mg/ m' llil",i111flJ C iilil P" =
2.71 Mg/ m' il<111~1!111";<'Ilfl<~"';1u6n01~lJl~ 1"u1l.ili"
'UU1'il'J(l-3~fl ~
y
lOOmm
I 2°omm l
A I
~~~~·'·!·~~=====.=t---X
I) I
300mm----l
1t.1~ 9 - 44
v
59. I ,
l;
.,
f.1
~""
468
9-15 lm~omh~lmr1illJffllJ Ilvi~"illJffllJiifl111W11 4
m 1l~"lJ1~ 7 kg/ m Mluii01lJ1o~a~Il~1J1l1~mm~1VU~a
1l~"ftl1lJlIIJ1'lW~<ilIJ(i11J ~m 1na"m~ d Vilftlii~ll'lllU
11'vau"~,, 101alm~0'n~"1ulllllJl~v~naniflJ
)'
r---x
A 4m
~u~ 9 -45
9-46 ilUffllJVf~ lllJ'Ila~lm'mV11 1"au.! odllJffllJihh
.., . .J . 0 ,
1IIJn,"vlllJ~1IIJ1aft11lJU11 4 Ib/ fl ~.l11,"l Ull lJ. ex, y)
'II v.~I'l~IJU01. 1I'lU1uIi<I'Il1J1I'l'lla.lIlj,"~'v"oj a UO",I1111
1I1J1'1la.illJffllJ IJvOnmrlJ ~.l11UmJ5mal~~lmv,f11u1111
d . .... .,;
"I'llllJlJnUVl A
y
E
I3ft
!-,-
I
60lb
I
-'-'-""- --:-------,----x
I-3 ft -l--Ht ~
nl~ 9-46
•
r6 in.
I
- - a - -+-- a --I
1t1~ 9-48
•
9-49 ~.l11Jll1Un'lla,villlmj. (X, y) 'Ila.~<lilIJU01. G
'tltJ~NU.:3rrU~U r.,nr.:JJJfll1lJU11 10 ft uCl:f1v'l.m1I9liifll1l
til.HillVn ::: y = 150 Ibjft3
"
nJ~ 9- 49
•
60. 469
9-50 ~-1lfl~lLmU-1l9l''U'Vl'HHJvr y 'Itl-3ftltjflu'Un~~Viih11rl 9-53 11-1111vi"Ul1,j-l (X, y) "lHNL9fUYl'H)u~ c ffl'l1f1J~U
;mJ~lm~01"~.t1 vhnh.r"~lJl1n
9-51 'l-1111 l"~ U :: Vn-1
m'l1~Il" t1
,t1Ti 9-50•
9-52 :U-111 1 ~lU111j~ Y 'lH)"]LC)5'UVl 'H)£J vf'tJu~l1 th~Vlfll 'W
1.h::n vtJ i11n 1'l1'U~19fLLa::fl1u~ijil n nil,,]
0.65 in.
JtI~ 9 -5 2
,.
- !- -
--- 2 in. I ]: in.
r--- 6in. ----i~,.
9- 54 1I-1111vllLLl1li-1
fill!
~tlfi 9-53
l00mm
)
, ,
300101
I
30mm
61. 'i,,
470
9-55 i1~lmhll1'nj~ y ~aWIiUmflU.J i i 'IJflJ~U~11lh 9-57 )~'mhu11liJ (X, y) 'lJDJI9iUl'1'iflU,] C 'lJflJ~U~
"~'lJaJmU I>lUhj~>l'IJlIlwJM~lJloiimJ~ A U~" B lum~
,h,nru Y
~6 in, -----j- 3 in.
I 1---150 mm -------i
15 mr-'-;n ", ~~.. I
y
150 mm :~ 15 mm
I C
x - - . I-=~--'-- x
1tl~ 9-55
•
9-56 i1.111';ll1ll1i. (x, y) 'lJa.I9iUmaU.J C 'lJB~v1U~
11,r,,'mhl1r1J;mhul~nl'fl'H~1I1'ml"ll1anitJ"19i~ii'IJU1~
lvilnulm.;u1.,uloiiallli01Oj.,nu".itJ lI11lJo;ilVlnllllli'lultJ
lil11~Ullil~rllll~"hjiiVl'IJtn"'Ila.~DUI~Dll
"i1J~ 9- 56
•
;
6 in.
I~
~u~ 9-57
y
15 mm -j f- -l :-25mrn
~] n: i~" :-,C . lOOmm
~ - ~ ~
I I ' I I !r-- :-150 mm---l r----1
x
50 mm SO mm
~tJ~ 9-58
9-59 i1nmhlllni. (X, y) 'IJD~I9iUmDU']'lJD~v1U~11ljl
""
;..,-=~-~-~~ --~t--X
300mm
,}-""'----"-' -+-
1300 mm
I 1
I
1 4~m[ll
I I
~- 600mm~
300 mm 300 mm
'JU~ 9-59
v
62. ,I
l
1t.1~ 9 - 60
•
9-61 rl'-3LW ::~ltl~vlnlffihnCl 15 kg Ha::~~fJuUn(ll-3
... :::.. - o'd ..
lJ1f1l1 GT l1lJl1-3lJu!'lvj lJlJl(1 70 kg ua:-:~Vl~t!tJf1al-3lJ1C'l
~ OM ~~11111)J~,iiu~ 8 "oooo,yjovh11101'~lHi')Jft-:h
nl~ 9-61
•
471
9-62 M"1un1,"n1um"~lU~iim~u)J''l'I1~jjJ111,rn 151b
UWi1:;'Ull1Ufl 2 lb U~:;Ull ~ ft 'il-3tfli'l11~,!~'lJB-3~Ot
;;ru6011.1n~u J1lJI1011111lJVii''''1 nUt!1J11JVi~U111,iltl
1JU 111111J nIiiUOlJU11a0"1riauM"~",ill"lJ ~~1 1.."hi~"
fl11lJl1'Ul'Ufl-3UWC'l ::'Ul
~11~ 9-62
9-63 'il~111I'ilUl1U-3 X 'ItNl9f'.IVl'HJU~ C 'IU~~U~U'H-Jl ~~
Lu'.!,htrHrt~'lJf.Hl~fli;'1lJ~ii1fl'iJ r
y
,t1~ 9-63
•
63. !I
472
9-64 llN!Ul1~U~llJn1~UlJiifl11lJ'HUlfl~Yi ~~ijyh,rmJ'lJln
(i'1u~'UQf1WU.r~ltl ~~l1l';lUl1ti. y 'm.~~i!u6(i1~~ij.U~U
lJl.G
9-6 5 Ur-i'l.I1Jl.J~tlC'llJ Ll1~t1lJjjfll 1lJl1tn f1-3vt ~-lijrh,rflUtll n
d'lm.h~t:.1nw1Jvr-l~t.J 'H111~1 1l11'lh Z 'J()-3~~'U.6(h-l'lJ0-3U~U
UH G
x
,tl~ 9-64/9 - 65
u
9-66 ~.111';llmti. (X, y) ~a.~ ~lJu6cil. ff1lJci'ij ';llmti.
'U fl-l ~~fJUVri1-.1'1En nv1~::(}.J ri'D'J:: n()1.1Utl UUC1 :-:ihhH'UnUCf~-l
l"WJ tl11'f1lJclaUm1lJffllJl50,V;vunm:U1U x -y ~.11l
Uj.tl5mV1.r.U1mJf1"~uyj<1"ci'aLii~nu;fl'~'U
1. ~1I11"~ HI Ib
z. [I-3rlth::n'()Ulf)~ij~utH'l 85 Ib
3. ~ln.J 120 1b
4. "ml1!' 8 Ib
"
O.SO fl
nJ~ 9-66
u
9-67 'il-l111~~tT'l!ifnenllll"'lJfl-ln'cHn ciO-llhzoaunu ftl11/
11'U1uti'U~fl.lffl'l A U<1: B iilll PA ~ 150 Ib / ft' U~:
P, ~ 400 lb/ft' "'1J<i1.rU
"
6in.
./
.,
ll1~ 9 - 67
9 -68 ~ncwofi1'1nltlnnOfffl.il'tJ Il'luul>i<1:;ltJiii'lIjj 1.5
ft ('1 h ~ 1.2 fl 1l.l11J"U:l'l1~ Z ~ll~~~i!u6ril~~ ij. ~f1 tlij O
G
9- 69 ~n<1flofi1~1f1ltlml0fffl.;i'u lVlou.J<1:%uiii'l'Iii 1.5
ft ('1Vi'fl.f11jl,r~Yi!tJElcil.~fl.~f10flU G fl~Yi z~ 0,5 ft
'~l11m1lJ,!~ h ~a.ltlm'lV,huu'U
~tl~ 9- 68 /9-69
64. 9- 70 1~~D,~nfl1J~1uul'i~'JJ 20 in. uO~Doflnm~n .,. u
lfl'l:EJ!:'YlTJ X Vi~~fJu6fh'Hi'llhl1tTmhL'vn::'Uo.nvff~iifi1 "til.'
, 150 Ib / ft' UO~ Y , 490 Ib / ft'
"
10 in.
---
2 in. x
I in.
'D~ 9-70v
9-71 I,:J111':i::U::VIl-.l h Viln"1JiJHftH~1tmUrJnrn~ 100
mm ~"!~lU!lu'Jfl~,Dnnu' Ivlfl'ilhr~~!;!u6nolJU10'Jfl'
~tl11HH~ii Z= 115 mm -lffV,}ijrrJl)J11'U1UUU 8 Mg/ m 3
9-72 .'111'"U~l'm Z ~''ljumflu,r,tlm'~D,~nfllJ~·lU,tI
nnoYiii,n ol~iifl11lJlJ~ h ' 50 mm l~l~lu~lU'Jfl~,tI
nl1U
ltJ~ 9-71/9-72
473
x
160 mm
u . " tI ~
9-75 1V1Q'VllillfH"t1"f1~ flH'VlHnalJ PSI = 7.80 Mg / m3
U(l::D"ijL'~tllJ'jU'VlHn'j::::1J{)n p = 2.70 M 2.lm3
ll..Jlfl'ilV'l. . . . al '-' •
IJu6n olJlJ1o'Jfl~lll ~ OlmllJ il~'J <l~m~ n,~1Jfl nii~l h '
200 mm
n.l~ 9 - 74/9 - 75
v
65. 474
9-76 l"a'l<iifmiiuihm" ••mvf.,{u~U~"~.a"l9ium~U.l
~e.,{u~mm X U<11H~I]1Jii~l'fe.~v'lllhJihHm"~~~Ul'f
." . ~ ~ ..,~
,.mu~1J1m~v.~ij.U~.·Vlln<,.ln mmljUYiUmijUllnU y
-----'f--j-----'----x
~tJ~ 9-76
9-77 lYlu1fm~iiuyhm.. .m1vf.yfu~1I~<1.U"I9iUmijU.l
~a4~U~Um1 y ll<1.lin'l1Jii~lla.~ij.UlhJifl'fLl~:Q"~Ul'f
••mmlJ11>l$~a.~tluuuYin11uill~lii".lnnlmljuY1u~~llU
Ufltl x
I fl --
lufi 9-77
, ,. • • .., '" <
9-78 LOlU 9:fOl'HI'W.'YllO'JA 'il~ 'Hl'Vi t.nrlUf'l:;'jt U ~ l91UVl1f)UOl
~a.i1u~uml x Uc11Hn 'l1Ji.i~l'fij.~ij.mJlJifl'f1l""QiliiiUl'f
.m1m1J1on~ll.~ll.U~.~'iiYl.lnmm~uY1u~~aUllntl y
~U~ 9 - 78
•
9-79 ••mY1t1~~hl,uullria
')ltl11li1 (Font View)
,tl,)WIl1. (Side View) ,tl. .
9-80 HmmlJW1'~ll.1l'f'l~,)ij.n1,yj1"uullria
2 in.
~tl Yllumh (Fonl View)
l,j~ 9-79/9-80
9-81 l~lU'iflnih..rYhn'JA ~-l"t11vr1J;;iUC'l~'l:;U::Vll1 y ,~~~
19iUmaU.l~ll.Y1t1~Um1 y lI<1.Hn 'l1J~~l'fV.'Ull.lnJtlifl!
U,,: n~~Ul'f ••lllm1J1"~~a.~".u-U.~In<,,1nn1mljtlY1t1~
um11auunu x
y
y~ = lx
~ III 2f(
66. 475
9- 82 1.wl'li'n151iu,itmOl ~~lIl,ffI'~Ua:l'U:I9iUmaU" 9- 84 ~~lIlllilJlm'lla~fiflUnl,".ra~nllll'wIlVlJkU
'Ilfl~,ffu~uml i mi'11'1i'1'1~lJ~~(,fl~'Ilv~ulhJif('ua:n~;iu(' 9-85 ~~111";U~ih'llV~'IlVlJkU lYlUhjii,""lJ";uvi'lla~lhu
i~111llilJl m'll V~'Il a~ uoi~~ti'i,"~l nn151111tntu~UmllalJ UalU
unu y
"J 1Il --~
lUvi 9-82
•
I2111
I
I
9- 83 mlUQnnV~ln1-j l' m!,,,,~U <I)J)Jmlna~"'nril1
., d 9 .. , . 1 • • • ."
IUUl1U~IU<l('lU'llV~~umlU"":tJ'VW~lVU'" 26 'IlV'",)Jll'll
ifvi,tJu'lia~-jH ij~111t!i1Jlm'lla~Vlll01.,u1'li'Vl q1J~vi1fa~'Ila~
lIuuiffflm:n ~;iu('
x
; 1O(J mill
·:c 150 !tllll
I
~ - .,- ISO111m
1501ll1l!
':itl~ 9-84/9-85
•
9-86 'IlVlJ'IlV~avjj11lr,,,," A-A .iJJU ij~l1llli1Jl>l''IlV~
1erVl~Yl{)~nTH1111fUflT:i riv ff11-3f11fl~~ A - A
67. 10
TulUU6IlJ80fl'J'1UlQaU
(Moment of Inertia)
ilJUI'lifnci1111.1~01Jli1l1~tJlnllJIlJlJ.l~fl. fll1lJl~flU~fl.~'lJ~~fl.i'~ q~iilJ1"fl'~ llJIU".l
~a.mllJl;jau~a'~lJ~t1'J1l1l0JalJuOi~lhrlQJ11J1'il'imnmJ l~a'i11nilo,j1hJilfln"MI'l1DaDnUUU
{wh1l~a.lm.a11'l1i aii'1lrilll~fl.,fl~a,')mn" I'llavJflU011'Ul1~,i1illnl~i!n~nn1~Ifl~allii'llfl.
i'~qil"~a.m11JllJI1JlJ.l'IJa'm1lJl~aU
487
10.1 utl1UutllJfUIUUv1UtlllA:J1UIQtlmwIJIDun (Definition of Moments of fnertia for Areas)
illUVl~ ,;1ttlJ1'["I'lll'!!ttVl,8 u.1'll a.~11iiI"ui;i11,0J11lJIlJ11.1Vllv1Ul1~.~a.~11~,auIIn11
ti'llii 8 ilJ01,ii"Ii1III OJi1: 'fllJ1,Q1'll'[,,1flUn1,UlIYitn,,, n1,ijttYiIm~llJ IlJlJ.JVlIv1Uaa''Ilil'
«u~ JUfifl Jx2
dA ci-1L~Vn-,i11w~'Uyf'ij().:!fll1lJLrtm:J'JtJ,niu~
IlJIlJlI.l'IJ 0' A11lJI11aU'IJ a'v11l~i1:,ilJI~a'hni>i'~iJn111'll1lJ!lJll.l~a.,]',mJn9i.nl:1'IlUUU
n,:i11U~lthH'il'1<l1li11 nun1l'lm,llJllJlJ.1 v11ari1'~lu 1 'Ila"hmrnm"1'I1'l1il",;1IR",.;l'IJI~a'i11n
fll1lJv1l1~O.'IJ a,'1I'l"n,01'l11JlJv1l1~l'IJil.U';lJtJ1.iiillJarill1~a~[m! 9i,l.rna111umlli1l111-uaii 9.6
m1lJv11l115aUH~a~tt~l'lrt'mi1uiio~ii'hlll1,j'fll1lJ'5n z l.r~lJfh~a,~a.'I'l" P = yz v1'
fflJm,~ 9 - 15 I~O Y(Specific Weight) 1'J1l11U'U1r11'l,ymiln'm'IJa'~fl"I'l'" v1.JlJ ~1l1",~a'
u"~iiatil"u'IJil'IH",u,,~,,ii dA ~O'U,;"tJ1'~i1lJi'l!l~ 10-1 iio dF = pdA = Y zdA
IlJ1lJll.l'llo,!t>Jif,auunlJ x 'Ila,u';lIm. iia dM = zdF = Y z'dA v1,J'" lIJI)j1l.l,R"i11nnl'
m::illUfllllJ"''''n.fJl1lJ~ M "'" Y Jz2dA nl'JHuihn'i~Uff~-31lJLllulPf'Ufl~fl11lJl~tHJ'IJ().:)vhL~'lJfl':)
U,iUUH,aulmlJ x ,iifl' illnlUIIUUnl,u"Yi1m"v1'oft11ifh11ln"fl1 a~f~0,111" (SIfuetural
Mechanics), n"fllcr",f~a'll1'" (Mechanics of Malerials), noi'11ff"flm'lIrl' (Structural
Mechanics) ua:-:nn{Hlfl UUtJlfl~[)~,rnJni:l (Machine Design) 1ft1 nJfllJ~'lHflUfi"u!'J1inl'lii
I~al'llillnl,li1"111
IL..-----~--x
1U~ 10-2o
v
68. 488
1
~
•~j
10.2
•
~
t
i:j
11111lU~'IJ~~rnlm~~D (Moment of Inertia) Y;l11!ul~U~ A ,r~L1l1"~''U~,j~ 10-2
.;1,~~ l'U1: 'Ulu X-y 1"U'Il1lll~ln"mllJllJ1lI'U.l'IJU~fl1llJt~6U'lu,~'U~1'U1:'Ulul~Ul nWBn1
" 0 . . . 0 ... ..r.,;:::;
dA ,uuunu x LI~: y ~u dI, = y'dA ll~' dI, = x'dA mlJ~lVlU lIlmUW'UVlm'f1lJ"
1lJtlJU.l'lla'm1lJl~Vum1"llnnl.fluiitmVl ,Tulia
(lO-l)
lI1lJ110mllJtlJ'U.lD'U,rUlIV~'IIa'~'U~lan '1' dA 'VU~Vl 0 'f11mlfl'U Z ,r'l1J~ 10-2 .;1,
l~uni1 IlJllJ'U.l'llU~fll1lJlriUUl;;~,j', (Polar Moment of Inertia) dJ, = r'dA l~a r lum :u:
.r~l11nIl1n~" (L1n'U z) l,ju~~'Ulil'U dA li1'f1rU~u~J~'f1lJ" 1iJLlJ'U.l'llU.fllllJl~UUl';;.,j', flU
i "= Lr 2 dA = Ix+ I)" (10-2)
f111lJ«lJw'tnfj::l1il-l J na:: I I I Lilu1t11wl'W(WiIlfl r 2 "" x
2
+ / i-l~t1Yi 10-2
o ' Y
1l1nlllJOl' 'l''.'',l WU11 I, I L10: J ll, iifl1tUU1J1n l;ja'1l11ll~ul'li'V'lU'U~~~!u'IIV'x y 0 ~
.:u:vmUnnl <1,lIV' LI~:~'U~ llfl milmr'U 'f1,hu'IJ v,llJllJ'U.l'llv~ fll1)J lriU utV;' nu fll1lJU11
tlflfl1~,rrf JUflV m mm~ l1~tl ft', in'
nf]t:J[jllnU~UU1UUa)WUn (Parallel-Axis Theorem for an Area)
01Vl'jl'UllJLlJ'UGfllJtl"fl11lJL~etJ~().,:'t.I~·HnJLLflUlLft::yjlUVl::~LC)f'UV1'H)tJ~'la.,~u~ 1)::i'I'::~1n
~11: 'f11Itlltl'1.1.1'11"'fll1lJ 1~UU'IIV.~U~11llJunU~'II'U1'1.1nuunuifl",UHVl ~1JliLIn'U~'IJ'U1U '~1)
Y;'J'Il,rillJl'11U'Vl~'lIli;j' Wlll1!ulml)Jl)JU.l'IJu.ml)Jl~UU'lJV'~'U~L1m1l'r,~tl~ 10-3 1VUtlnu x
1umruif { mil'Ul3n 1 dA V~il1:U: y' 1Vl'1 lVlIl1nLln'Ul'llUmVU.l x' 1U'IJ!u:~1:u:Yilifl~Vl
nU~" l1i" UnUfl'IJlllU x L10: x' iiuni1 dy ,iiV'1JlnllJllJu"'lJU'fl11lJl~au'lJV. dA 'VUllmi
x AD dI, = (y' + d/ dA ,r'lrU ilJtml,r'llV'm1lJl~VU~u~~u~J''f1)J''
I, = L(y' ...d/ dA
:;:: ty'2dA+2d, Ly' dA+d,. LdA
'Y1VlJlluVi'Dl"lt1n'lVl1JllJl)Ju"~V'mllJt~uu'Uv,~u~,vuunUt'llIIVI1VU.l tVlil)JflUVitn••Yi
(ij"tlJU':1UUtiiD'~lnl!nU x' ~lUVl"~l'llUmvo.f'IJu~~uil C lrUAU f y' dA = y t dA = 0
,fiv y = 0 lY1UlJauiitmYl~1I1)JUVlUifu~J''f1)J'" A ~a,!"'vi1u ~v
1!~'11O" '~Olnu l'il1Tl u I ,,,,ilvY
- ,ly = ly. +dA ~
(10-3)
(10-4)
al117U1lJLlJl.tvffl11lH~tlm;>l<tf1'ifJ1JUm.fvi;~1:nnnU';i::;'U1U x - y Ufl::~l'UVl::~~~ 0 (LLO'U z)
""~tl~ 10-3 II"M 11
69. ..l oK .,: • . ~ .......... "" cO
'J {}~ fl11lJ liW u'UtI~vn..YI 'wu unu~'UUltl Hll..YI::q 19:ftlVl'H)U~'l fl-ll1tiYItJ1 nntJ H {l fJut'J fl~WUYl un:::
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, y'
d,.-,C
d
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u
lU~ 10-3v
10.3 sAiJlmsffulJallWUn (Radius of Gyration of an Area)
i'1Iii11l11;U'v,,{U~1li':"llJ'~01 nuihni1 u,v,n11lJall U~:ItJIJUllJllU~Hcrll1i'lJn1,
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1l1nlYlJ0l1
k = ~< ~ A
k = ~J ~A
(10-6)
~t1('(lJ nl~d~lUI"lflnl1'1ll ,i1{l-l~1fl fl cilU rltJnl'nnllJllJ'U~fl111J l~{)U'Jtl-.l-«'U~l~n '1 ~fl1JUflU
Un.llUuw'lilJ I = k 'A hl'Uf.U:~iu~,an<j dl = y'dA< , ,
489
10.4 rUluuc;ilJaJf1J1Ulaa8c!1V1suwUnr~8n1s1iufitns~ (Moment of Inertia for an Area by Integration)
l~v'UmJl,~a'11i'lJ~1J~hn:ulu,?iulnlJ 1I1I~,hlUw~n,j'ul'mft(ii~fllll~f lIlJOl'~ 10-1
;):::(JflO'UV;W')Vl iYlfllnl,IIH'IA'J ~n tl111J LfifJ Ual' fi/U'~l..~ 01iUri11HJ lH ~U~tJ111~fl 01,1fll'iil, rvhn'i{i1
~~lI'l~I~11'1 h"w,"rll" "'Icrol"mtl~ 10-2 n"iiIJYi,ml'lcrD~flf~l:lIu~Hlviuu,:djuhHlJ1J"
'V,ft lllJ I~DU I.wvt,lt1.:iin llflUYiImVlftf"~Oll~oIEUfl{uril1J~ii'Un~l1;U mllJl11Jliiuu 1
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y
y=/(x)
L-~_.L.LL_~-L__ x
--l ~ dx
>1.1~ 10-4
•
JsmSchVlSUJIAS1:1i (Procedure for Analysis)
(',iiuViIm~fll'~IiiU1 lvi111111lJIlJU.r~~,ftlllJ'~fla'llfl,~uyj~flUunu 1il,i'lU"fl~~.·l'{mhu
,ao 1 dA l'lU~j%J;Uli1tlilflQlultl~m~ulJiiu",vi;jm'lJVll1i'fl~ua.m'Un"rmrflV 1:1Uffltl
"fl~lu,"l",.,ti<lvifl"~~flU''II'''lJfl~~u~~~l~ 1 (x, y) 'iiiOl~vili'ltlbJl"'iiff",'iiivi,d·.,:1tlfflU
.,llJlIou,vifl11l1lJIlJU.r~flHnllJ1~"U
~ _ , ~.... ...J' ... .I' "" ... ,
01"'11'0 filllJ Ul.'IJ fl''!iUff.tHnlJl~O~ ~ 'IJU1U nU Uml ,,.,.,nnruulo ~'IJUllJfl'!iUfflU
~tl~m~"lJiiu"Wl"tlyj 10-4 ~.1.1',." I = Sx'dA 'IJfl<~uvi 1.1'rrlJ~15vi 10-[ tTUftfl ffl,.,i'U• • y
o~ti1il..l'1l'I"'lri":1Ulh1JjjmllJ",ulUflUlJl0 dx ,,~,rtt 'I'lmi'lu'IJu,;tt,httil:U~vil""."~tt
1lJllJU~hii[nn! flfl ih::(J::lvi1fiu x lVl 1l1fHLfl'U y
nlru~afl~ ('lfllllJUl'l'IJij';1JfflUrrllJl~o-r.,,f,mo'ln"nu rrlJol~vi [0-1 il:H'I~'I"
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lfluunuUU1llUY1:~~'t!''liUY11fla.r'llfl,{uff.'U ",,rt! 'f11llJ'lJU"'IIfl~R'lllJ'~flV~~,iuri'U~flU
1101.1 xI~U0l5HY1qllii"nuti'llUlu 011~uVilnl~'II"'Ha",ocil1;:l' ..i;'{~l'11fl' I,
71. ~ pbmhufi 10-1
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iimll~'Uv,iilHnll,j'ncill ~uffluyfmlJ<'lYi,:u: y ~ln"nu x' ~~';l,tJu"a~iiuihm"~ln
y'=-h /2 D' y'=h/2 ,i'lv~~ln dA = b dy' ,j,ti'u
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1
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12
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= -IJIJ + hit - I = - hh1 , ( " ' 1 I
12 2 j 3
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b ua: h ~ln'lia (n) ITuflfl
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, 12
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72. 492
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.,
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--IOOmm---,
(Il) (~I)
~u-ri 10-6
•
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I,jV''iI1n~tJ('htJijfl11IJmn dy ua:.,~~tfjfhri~"'''1 (x, y) Yl'tlvi lie dA ~ (IOO - X) dy
tJvn'illntTtJ '1fl1hU'1le'{U1htJe~vi,:o: y 1mlnuntJ x ."Ju nl,VlliiLn''''iI:liioUflU y'illf
Y = 0 (j~ y = 200 mm 'lI::1A'-1l
I, = fAy'dA = f, /( LOO - x) dy
= f:oo
i(IOO - y' )dY = 100f:'" y'dy - - I- f:'" l dy
400 400
= 107(10' ) mm'
""d. de:!
1um (n,run 2)
{Uff1tJlan1 '1ItntJfluuntJ y ",~uri 10-6 ('II) QllIae mil111Um,1iuiim,,, ~'ll"r
1ri',ri ~"''' 1 (x, y) 1tJn, wi' '1nff1tJ'IIe,il'mi1tJ1ua~lm:o: L~01fltJ'illnUnu x ,jnT
lin q~fiunllti'lllllU'illn11JIlJu.J'IIe'fl11lJL~V~'IIV'%Uff1WiiOIJflU"nun ffl1111J.UaLl1~OlJ NUf
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fflUn"'"'1'" ~,,.n1"n~i'u"H,,.II,.lllU«<lnu iill1iil"H.1'.111n~ Nll1lJer1f~W~1>III1U11""H
'h",U1«11era ('lJU1UflUih,n.fu) ~,iil'l~lnm llJlaUI'lYllU"'U ua:luuYlil'nci11ihn1>i lflnnf
t1Djl11~l~u1'l.i'V.JnUU'j'Hn(lVl'VI1'W 'j,)lJO-l LlHlatJ~"r11'Uhf"'fll (Screws), uu1~ ( Bearings),
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Resistance) ~:n ri110.1Ul>lilU111U'IJil.U'VI
8.1 ant.:lru~!JCl)fl:)'Ulaa~n1UIIUUm') (Characteristics of Dry Friction)
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"D~i'Vl~ ~~dil'wtnfnUl~ilUfl{ul1'd~t'lflyftlH1Yii'~q~.:JnciTltYUnlJfffffl ~ 11n~U'l.JU~:::fl'i::'l'll
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