Polarographic Analysis: Principle, component of polarogram, Ilkovic equation, diffusion current factor affection affusion current and half wave potential, quantitative analysis and applications. plarographic maxima supprissor, oxygen wave, supporting electrolyte.
Polarographic Analysis: Principle, component of polarogram, Ilkovic equation, diffusion current factor affection affusion current and half wave potential, quantitative analysis and applications. plarographic maxima supprissor, oxygen wave, supporting electrolyte.
Benamor.belgacemقضية الشعر الجاهلي في كتابات ابن سلام محمود شاكر
Mech0110
1. I
I
I
""a'18UqJ (Contents)
1
VlanafiUlA1aUlsrl"::>ltJ (General Principles)
2
1.1 fHlfl'lffl'l{ (Mechanics) 3
1.2 uUJmlJJfil9l~'W:[l'W (Fundamental Concepts) 4
1.3 'I1U1Uf11'lll9l (Unit of Measurement) 5
1.4 'l~UU'I1U1UfflfHl (The International System of Units) 7
1.5 f11'lr11UJblll'llW'IJ (Numerical Calculations) 8
1.6 lTIfm'V1%j~1i1lml~'I1 (General Procedure for Analysis) 9
l'il'Vlu'VlU'Vll'W (Problems) 11
3
l::>nlAElSUSJ (Force Vectors) 13
2.1 ,fflfHnfll,,::nfHl'ltlW'{Scalars and Vectors) 13
2.2 lTI'Vll'lI'Vlf1i1f1'1JtI'Il1fHl'ltif (Vector Operations) 14
2.3 f11'lUlfHlflll'ltlf'IJtI'I'I1i:llfJU'l'l (Vector Addition of Forces) 15
2.4 'l~UUf11JJllJn'W'lJtI.Ju'l.J~t1~lm~'W1Ulfi1nn'W (Addition of a System of Coplanar Forces) 21
2.5 I1fHl'ltlfl'W'l::1J1J'Vlnl9l111f) (Cartesian Vectors) 28
2.6 f);'lUlf)U"::f11'l"Ul1fl1l'ltlfl'W'l~u1J'Vln9l111f) (Addition and Subtraction of CartesianVe.ctors) 33
2.7 I1fl1l'l[)f'j~1J~h!l'l1U'l (Position Vectors) 39
2.8 nfHl'ltl{II'l.J~iiViI1'Vll.Jl'llJJUUJlff'W (Force Veg:or Directed Along a Line) 42
2.9__~~.~rul;.Jfflf1~1{~~ct) 47
l'il'VlU'VlU'Vll'W (Problems) 53 ... '.
2. {u~atJaJaun1A (Equilibrium of a Particle). .
1-
3.1 trfll'WtrlJl'ltl'Utl'Hl'4fllfl (Condition for the Equilibrium of a Particle) 77 '
3.2 vr,r)9itlvtr':i:: (The Free - Body Diagram) 77
3.3 'j:;UUtl'j-:Jh!':i:;'U1Ul~tl1n'W (Coplanar Force Systems) 81
3.4 .':i::uUU'j-:Jtrl'IJljiij (Three-Dimensional Force Systems) 86
l'il'VlV'Vl1J'Vll'W (Problems) 93
lJaaW8S:UUllSJ (Force System Resultants)
~..J~ v ~ ~ ~I ~
'1 4.1 J:-la~ru'Utl-:JnfH9itl'j'Vll~J:-laa'W1Jlu'WnfH9itl'j (Cross Product) 109
4.2 llJllJ'WIPi''Utl~U':i~-~t1trlJf)l':itrlf)mf (Moment of a Force-Scal.ar Formulation) 112
4.3 lmlJ'WIPi''Utl~U'j~ -~tltrlJfl1'jl1fH9itlf (Moment of a Force-Vector Formulation) 114
4.4 rnllJtrllJl'jt)1'Wfl1'jl~tl'WlillUl1ll-:J'Jtl-:JU'j-:J1W:;'Utl-:JllJllJ'WlPi' (Principle of Moments) 124
77
109
4.5 ' 1lJ1lJ'WIPi''Jtl~1l'j~'jtlUllf)'W1~ <) ~nll1'W~~'W (Moment of a Force About a Specified Axis) 127
.>'4.6 lmlJ'WIPi''Jtl~u'j~~rn1J (Moment of Couple) 133
4. 7 fl1'Jlfl~tlWrltJ'Utl.Jll':i~U'Wl91tlU;U'Hf)1.J (Movement of a Force on a Rigid Body) 140
4.8 Na«'Wli'Utl~u'J~ua:;'J:;uu'Utl~u'J~~rn1J (Resultants of a Force and Couple System) 142
4.9 fl1'Ja~Hl'J.Jlla:;':i:;1J1J'Utl.JU':i.J~rn1J (Further Reduction of a Force and Couple System) 147
4~ 10 " fl1':ia9i':itl~cl'I1Ill1Uf)~f)'j:;YllUUUf)'j::'illtJtlthffi1J)~ .--
(Reduction of a Simple Distributed Loading) 157
l'il'VlrJ'VlU'Vl1'W (Problems) 165
fU~atJaJ5~QlliJJLnSJ (I;:quilibrium of a Rigid Body) '.
5.1 iitl'Wl'U-a1l1'r1JtrlJl'ltll9itlll;U~!f)~~ (Conditions for Rigid-Body E~uilibrium) 193
_ trJJl'la1'Wtrtl.Jljiij (Equilibljum in Two Dimensions) 195
5.2 vr-:Jl9itlvtr':i:; (Free-Body Diagrams) 195
5.3 1:1"JJfl1'j1:1"lJ~1;1 (Equations of Equilibrium) 204
193
3. ./
v
5.4 'BlHhlHHl-:JI1Cl:;l.YllJl1':i':] (Two-and Three"':Force Members) 213
~~~CllUl.YllJiJ~ (Equilibrium in Three Dimensions) 215
5.5 N-:JllllfjVl.Y':i:; (Free-Body Diagrams) 215
5.6 l.YlJf11':i'UCl-:Jl.YlJ~Cl (Equations of Equilibrium) 219
5.7 ~Clihnmhl1ful~HlU~-:Jlf)~'1 (Constraints for a Rigid Body) 220
l1l'Vl£hmvnu (Problems) 231
I'
iI
n1s5lAS1::vifAsuaS1U (Structural Analysis) • 253
7
fl.1 lm-:Jt1fH!UU~ltJ (Simple Trusses) 253
6.2 l~ fml'11~9I9iCl (The Method of Joints) 256
6.3~lJ'rilUI1':i'l~l1mntJlu'ti~Cl~uriTu~1lifu!!':i'l !t1UIJ'Ut! (Zero-Force Members) 262
6.4 l~f11'Jl'11f11fl~9I (The Method of Sections) 265
0.5 lfl'J'It1nl.YllJmi (:Space Trusses) 271 .
6.6 lm'lmClu!!t'l:;!fl~Cl'l,]mflt'l (Frames and Machines) 273
l1l'Vltl'VlU'l'llU (Problems) 291
llsufl1s1u (Internal Forces) ~7
7.1 !!'J-:JfI1tJlu~!fi9l~ulu~urilulm-:Jl.Yfl-:J (Internal Force Developed in Structural Members) 317
7.2 N-:J um:l.YlJfll'J'Utl-:J U'J-:J !i1u'U!!t'l:!:l.YlJf11'J'UCl-:JllJ !lJUI'l~9I
(Shear and Moment Equations ana Diagrams) 326
7.'3 fnllJfflJl1Ulf'J:;i1i"1'llhtnl'fl m:;'I'h UUU m:;'illtJ f11':iL{fCl'U UClUlJ ;~TJIlt~
"
(Relations Between Distributed Load, Shear, and Moment) 331
7.4 LflLDt'l (Cables) 338
l'il'Vltl'V1U'VlJ'U (Problems) 350
4. nUla8~n1U (Friction)
8.1 af)1Jru::;'Jil'lmllJl~V9l'vn'Ul!UUUr1'l (Characteristics of Dry Fric'tion) 371
8.2 iftJ'l1lI~WH)UmlJJl~V91'V1l'Ul!UUlIr1'l (Problems Involving Dry Friction) 376
8.3 i;1lJ (Wedges) 387
8.4 lI'J'HffVI9l'Vll'UU'Ufff)~ (Frictional Forces on Screws) 389
8.5 1I'J'llffVI9l'Vll'UU'UfflV'V'I1'UlIUUlI1Jl,ll1VU (Frictional Forces on Flat Belts) 393
8.6 11'J~Iff(J91'Vll'UU'UlIU~~IIUUU ftil f), lIU~'lllUU.uill'i il~lJf) ftlJ IW::;llH'Uf) ftlJ
(Frictional Forces on Collar Bearings, Pivot Bearings and Disks) 396
8.7 IL'Wfftll9l'Vll'UU'U!!u1'l!!UU Journal (Frictional Forces on Journal Bearings) 400
S.S mllJI'i'1'U'V1l'U1'Uf)lm~'l (Rolling Resistance) 402
1'U'VlcJ'VlU'Vll'U (Problems) 405
~lJuarbJlla::munsa8~ (Center of GraviW and Centroid) •
~I ~ Q v
9.1 ~l9ltrlWmlII(j:;~l9ltrlWn(jlll.rm1:fl/1'Jm:;UU'lJillil'4mfl
(Center of Gravity and Center of Mass for a System of Particles) 431
9.2 'U19lf1''Ucfril~, 1l19lf1''Ucfnftll'lJlft l!ft::;I9f'Umiltl~'IJillil'lfiq <tI q qJ. q
(Center of Gravity, Center of Mass, and Centroid for a Body) 433
~J 119lt:jU1::;f)ilU (Composite Bodies) 443
9.4 'Vltl1J~'JillllUuifffllft::;Qft~hrff (Theorems of Pappus and Guldinus) 447
371
431
9.5 Nfta'V'lli'Jil~1::;'lJ'lJ1I'J~l!U~m::;~mJJ*T "Resultant of a General Distributed Force System) 451
9.6 m~~u;Ut1~l11-(j (Fluid Pressure) 452
11l'VlcJ'VlU'Vll'U (Problems) 460
5. "l
·10 :1.
•
rUlUU~lJa)A~)I1fQa8 (Moments of Inertia) • 487
10.1 iltlllJ'Ue'lllJtlJ'WI'1'Uti-~mllJt~etl'Ue'l~'Wyj (Definition of Moments of Inertia for Areas) 487
11
" ,
10.2 'V1f)lJ~tm'W~'U'Wl'W'Uil,:rw'W'Vi (Parallel-Axis Theorem for an Area) 488
10.3 ff1iil'iJt'Ji''W'Ue'l~'Wyj (Radius of Gyration of an Area) 489
lOA IlJ tlJ'Wl'l'Ue'lmllJ t~e tI1:hl1fll'i~/'Wyjl~tlfl1'JtJ'WVi tf1'JI'I
(Moments of In~rtia for an Area by Integration) 489
10.51lJtlJ'WI'1'Ue'lmllJt~etl'Ue'l~'Wyjlh::;f11llJ (Moments of Inertia for Composite Areas) 495
10.6 ~'Hli!lru'Ue'lmlm~mJ'Uil'l~'Wyj (Product of Inertia for an Area) 498
10,7 IlJ tlJ'WI'l'Ue~mllJ t~etl'Ue.:J~'Wyj'JelJ Ufl'Wt~tI'l
(Moments of Inertia for an Area About Inclined Axes) 502
10.8 J'I~ft'jJl'jJ"j'Ue.:Jl'jJt'jJ'WI'l'Ue.:JmllJt~etl (Mohr's Circle for Moments of Inertia) 506
10.9 IlJtlJ'WI'1'Ue-lmll.H~iltl'Uil.:Jm" (Mass Moment of Inertia) 509
1'1'YltJ'YlU'VlJ'W (Problems) 516
J1ULauau (Virtual Work)
11.1 iltlllJ'Ue'l.:Jl'WU"::;-ll'Wl'ffiie'W (Definition of Work and Virtual Work) 539
'V ~ 'U ww
11.2 l1ftflf)1'J'Ue'l.:Jl'Wnl'jJ1l'W'Ue.:Je~mflUft::;JI'I tlu'U-ltf1'J'l
(Principle of Virtual Work for a Particle and a Rigid Body) 541
<V "'" 'U ~ c;$.c:::t~ ,IV
11.3 l11lflf)1'J'Ue.:J-ll'W'fflJe'W'Ue'l'J::;uu'UenI'lQu'U.:J tf1'J.:J'Yl!'liillJl'lil fl'W
(Principle of Virtual Work for a System of Connected Rigid Bodies) 542
11.4 U'j'.:J1l~ffnr (Conservative Forces) 550_
11.5 vnl.:J':l1w1fltJ (Potential Energy) 551...
11.6 mtucvl'Ue.:JVli:i'.:J.:Jl'Wrrf1V'Ue.:J'ff~ft (Potential - Energy Criterion for Equilibrium) 553
11.7 n"fitl'j'mVl'Ue.:J'ff~9,!ft (Stability of Equilibrium) 554
l'iJ'YltJ'YlU'VlJ'W (Problems) 563
1/
539
17. 2
'"liJfll61C>SllSO
(Force Vectors)
1J'I'l~ 2 iff) cirJ n'l'11 t1f) f)11"'UU'l U1"'l UCI::l1i f)11"1"1lJ 111"'lIf)11"unifqJ'111 U1"'l i'l f) ci111UU<tTU9IeJU
U(l:;f)1'Hl~fl U1"-:J l<U1'11111f)U l~fl-:J'il1f) U1-:J lU'U1flm tim f) l~tl{ ~'l~llUUl'lfl-:Jtl1rr(Jf)tl'lJtl-:J nfll~tl{
'I'l1-:JVl'llflrul'l ff1'11jUf111'W1l11U.n 1~(J'il::tlli'lJl(Jfl111J'111J1(J'lJtl'lmmrumf1l:n{lw:;;mlJ1runf)II'lBf
.1'J1J.vf'lf)11"~~U1f)~~U~lU'lJU'I n f1 II'lU{'I'll'lYl'l1flruI'l lvimh::~fll'llil1i1'111 f1 (l f'll 'iY~flf'l1 m1"1.1if
2.1 ama1sua:l::>nlflEJS (Scalars and Vectors)
..nm~tJ1"
~!fl j!Jt_C~clll;!d _ 1111J1.t:Y'iYlFHllflUUhlvt'l tili11C1'U~ijrhIlium fl'11~Bau oM U~lJlru
j;'flf) Cl1{~1'lfnU1J1f1lUj;'f() I'l f'll j;'f~fll'luri lJ1 mr:l1, m1J11'l1" UCl::fl11lJ (J11
!1fl!Iil€li (Vector) Afl mlJ1ru~ih~'U'U1~llCl:;;Vif'l'l'll'l · 1"11J.vf-:Jl'ltJ-:Jlll'U1u~11Jf1~~m~(J1J
1'l1'U'U'UlU mmrunm~flfl'Uj;'ffi~f'llj;'fOl{lYf1'il:;;iiriwl"HCI'lJlll'U1nf) Muri i-11U'11U'l, U1"'l Ul:dlJllJU~
mlJ1runmOlBfllj;'f~-:J~1(J{lflf'l1" ~'l'il:;;1Jtlf1.vf-:J'UU1~IIC1dlf'l'l'll'l 'UUWI (Magnitude) 'Utl-:J nf1
lOltl{U'I'l'U~' tJ fl111.1 tJ11'UtJ-:J{l f1 f'I'l Vif'I'I'll'l (Direction) U'I'l'UI'll tJlllJ1:;;'11'h:J Uf)'Ue1-:JV,Hw:;;nr'UuU1
rmm::'l'll'IJfl'lClf1f'11" 1'11flVl'll'li'U mmrunmOltlf A 1'U1U~ 2-1 ij'U'Ul~ 4 '11'Li1t'J Vf'I'I'll-:J AtJ 1.11.1
20' 1~m'Ul~~U1Wf)1'il1mlf1wS'1'lV'lIW:;;iiVif'l'l'l1'l;j'U1u~1'l'IJ11 ~~ 0 lUU'ff1U'111'1 (Tail) 'IJ~'1
nmOltJ{ UCl::~~ P lU'U'fflu(JtJ9I'11~mr, (Tip or Head) 'UtJ'InmOlu{
1I1l1ll'f"'j'lJtNf11'lm~''h (Line of Action) ~ _
~ri1'il11l~ (Tail) ~
o
ffl'11j1Jf)11"l~(J'UmlJ1runml'luf 'il:;;HI'l1t1f)fl1"UCI:;ij{lf)f'l1"U~mi'iBI'lTtlmj1"i'lf1ci11 l'1iU A
'ff1U'IJ'U1~'il::ll'1'lU~1(J rAI '11~tltl1'ilU'I'l'U1'11tJ I'll'W1.1vl'u f) ~ Ji. ~'11'lflJ1f11'U f)11"U'I'lUj;'flJ f)11"j;'f1f) mf
l:hUI'l1'WlJvl'nrU'11U1 A 1iu'I'lumlJ1run fllOl Uflu j;'flJ mm flll'luf !vifl~'il:;Mlwflfl111.1 UOl f) I'il'l'l'l1-:J
fl ru Ol m j;'fI'lfj:;;'11')1'1mlJ1tiln fll1'1 €I{UCI:;j;'flf1a1fl~fl V1-:J lf1 I'l'cl-:J
13
,", I
.~
1
18. r
.2 58n1JlnAilAlJall::>m~aS(Vector Operations)
fl1'jflrull":::flTnlTinflll1lel~~'l£Jalmn~ (Multiplication and Division of a Vector
" .. " " . . . .. ~ '" d::'1 .. "" .. ""
by a Scalar) u"ilJ1UHlflllPltl"i A tll:l::'f1'If)i;'11"i a 'il::IPlll aA 'li'1lu'Wu"imUHlflllPltl"i!!l:l::lJ'IJ'WlPl
ril'IJtl'1 a iirillU'Wlnfllrltl a IU'Wrillnfllll:l::ril'IJtl'1 aA iiril!U'WmJ Irltl a !U'Wrill:lU ~'1J'W
mlJlrunflllPlvf~!U'W1;!1Jhi''illflm"it!ru mmrunfllIPlVf9l1tJ'f1'!fll:llf (-1) ~'1~1J~ 2 - 2 -ffll1i"'U
flT.il1l"i1J'1mrun fl !lPltlf~, tJ 'f1'lfl l:llf'il ::utJllJ1Pl tJ1'lifl~ fll"it!ru~ltJil 1l:l'IJ~!UUlft'IHh'W ilil til'1
li'W Ala =(l/a)A, a -oF 0 ~'1~1J~ 2-3
//
nl~ 2- 2
"
nTJtlrull~~fllJmJ.rlU(j'lfl(l1f
'j1J~ 2 - 3
"
fl1'jmflllflll1lel~ c{--oVectQ~ition).. mmru!lf1llPlvf Am!:: B ~'1~1J~ 2-4 (fl) 19l
1lfl!lPlilfrl''Y'ffl R = A + B. 1PltJ1'1ffl~#Il1~tJlJ.®J.1J.1l1Q'W (Parallelogram Law) A Ul:l:: B 'il::
~m~mJfl'U~ff1'U1J'l1m,f'1~ ~1J~ 2-~ ('lJ) U'f1'~'1!5'U1J"J::'IJ'Wl'W'il1flri"Jlnr1'IJtJ'1ul'itl::nf1llPltJfm11
miPlf)'W ru 'ilPl~nll1'WPl U"i'1rl'~,r R fitJ !5'W'Vl!W'1lJlJ'lJil'1"i1J#!11~tJlJ~1'W'lJ'Wl'W~l:llfl'illflffTu1JmtJ• q •
'lJil'1 A Utl:: B 11Jv'1'ilPl~Pl ~nll1'WPl'lJtl-.l15'W1J"i::q
~R=A+B
fll'J1J1fl!1n1I'l!l f
'j1l~ 2-4
"
R=B+A
fll"ilnf1l1f11lPltlf B Ul:l:: A 'f1'llJl'lfl1'lifll'l'f1'fl'1~1J'f1'llJ!l1~tJlJ (Triangle Construction)
1PltJthnf1l~ltJf B mUlflflUllf1llPltlf A 1'1fl1rl'flffl'W'l11I'itlflUril'W111'1 (Head-to-Tail Fashion)
~'1~1J~ 2-4 (ft) Ilf1l'l'afrl'~,r R fiil 15'W~mfl'illflril'Wl1l'1'IJil'1 A 11JV'1ri1'W'l11'lJV'1 B 1'W'vll'Wil'1
1~,tJlfl'U'f1'lmmV1l1~i'W~1J~ 2- 4 ('1) ~'1J'W 'f1'llJl"JfI'f1'till~il R = A + B = B + A
1'WmruVllft'1l t1mf1l~ltlf Am!:: B iirl'flllru::1U'WU'Wl1«'WI'l'l'11~tJlfl'W (Collinear) UMI'1
1'W~1J~ 2-5 R
A B
R=A+B
)'l1~ 2- 5
"
19. '.
, 01'dt;l'l.ll1fl1V1t)'i (Vector Subtraction) I'ICI«'rn3U9lflI'iW:i:;'Yrh:Jm'IJlrunfH9ltlf A 1m:: B
ffl'IJ1~m~tJ'..!tl~lU~U'IJtl.:J
R' = A - B=A+(-B) /I
1'11:1~llJI1m9ltlfuff9l.:Jlu~u~ 2 - 6 f11'J1:1U 11JUmruVi lffll'IJ'tl.:J fll':i1Jl mvi'IJ ~.:J,ru tnlJ1~fllci'fl~
fl1~1Jlfll1fH>,rClfu~:;~flIll1'1i'luflmlUl1fHllltll1~ /
/
I L~7 M;'
B -B
';Iuft 2-6
"
"f11';1Uil'ilty'M1'Utl.:.ll1fl1V1tli (Resolution of a Vector) I1fHllltlfffl'IJ1'JflUm9l'19l£Juu'Itltl f1
IU'WI.Yl'l.nJ'J:;f1tlufftl'ltflu CJ1'11'i£Jf1':h uUlf11~m:;vh (Lines of Action)'hwlci'f1~~tl~I'I1~£JlJ
~l'U'IJ'Ul'U (Parallelogram Law) I'lltl~1'11'li'U t:11 R l'U~u~ 2-7 (f1) ff1lJl'Jfll1f119i19lmlU'Itltlfl
IlJ'Uril'UU'J::fltlUfftl'lth'Um:;Vll1J'UU'Ul a U1:1:: b 19lm~'IJ'IIlf1ril'Ut11'IJtl.:J R lvi'lJ1'd''Utl~:;'lJU1'Ulu
nUl'd'wl'Ul a UCI:; b lm:;ril'UU'J:;fltlU A UC!:; B 'II:;CI1f1'ii1flffl'U'M1.:J'lJtl'l R ltliJ'I'II9lI'l9l~li191~'Uq •
lllfH~'Utl'J::nuu'W1 a UCI:;b ~'1f1ril1,r'U t:11'Jll../U~'1 A IIC!:: B l'ih~l£Jn'Umi'ln'll::I~l1f1lIllUf'IJtl'l
~C!cr~1i fitl I1f1lllltlf R ,r'tHtl.:J
a a
r------=
L
~----='
b)/ _b'JU~ 2-7
B .
'lJ
2.3 01SU3nl3nlVltlSUtlUVla1t1USU (Vector Addition of Forces)
t:1lil U'J'Il../l fl f1il ffU'I11'J'I'lJl1Jl fl fl'U ~U'IHfl ~~u~I1-1 ~£J'IJ ~1'U'lJU1'U11nuty'f11 U~l'1il~'UIII tl'U
ilUU1'11'1i'U t:11iJ1I~'I F, F U1:1:; F . m:;vh~ll91 0 ~'1'JU~ 2-8 ~m«~,r'IJu'IU'J'IffU'Iu'J'Il91 "1 F.. 1 2 3 q ' <u 1
+ F2 uci'11JlfHvi'IJnUI1~'1~ffl'IJ'II:;I~p.jC!«~,r'IJU'I11~'1vf'l1-1'IJ9I,rufiu FR = (F, + F2) +
- 13 f11~lci'f1mu~1'I1~.~'IJ~1'U'IJ'U1Ul'Uf11~1JlfHI~'1'IJlf1 f1-)lfftl'l U~'1~tl'll'li1~fll'Ulru'l'11'1 F2
.,
1'l'Ulfl ill1ll1m:;9I'ilflruB ~ilJl1h:;~fl ~lvi'IJI&i'IJluf11'J1-11 fil'IJ tl.:J U':i.:Jvf.:J'lJUl ~u1:1:;ilff'VlWU tl.:J p.j1:1«~1i
ifty'M1trffll../1~flunUty'Ml19ltJl'li1~~:;uuVifl9lmfl (Rectangular-Component Method)
~'1'il:;tl~U1£Jl'U9Iuul'iul,j
15
F3
...3U~ . 2-8
20. --
L6
- P13afi1on 2-1
(n)
'"..,"
I!
liillT'Hbl'lr1.l?!fl~J~li _(p'rocedure for Analysis) ifqpnvhnfJ'J'11tJlnlJfiW.f'JiJ
!!J.:J~eJ.:J !!J.:Jl!Cl::iitleJ.:J;1ulbhjymlJfhtlllJl'HlunifqJ'I1l19ilf!I:JI'1i1~llml::MI9i'I~tJ1U-Q
flJJ~tl~l'I1gtlll~TU'llU1U (Parallelogram Law) tl~llfllWrfjl'J '1 Utl19Flnl1'lJdfinml'ltJ{
If! fJHfit]~U~I'I1~I:JlJ 19l1'U'UUl'U fllliJ'Ulull9lrn1'l11~lJfl1l:Jl'U~u~m~I:JlJ 19l1:tJ-'Il'Ul'U'il1 fi ~ul'lll
l1''lJlflrul'l'IJtJlI'il'l'HJifuJ'111 ..rl-Q vHn'JlJ'lJtJllJlJ..rl'l1lJ~'il:::#ltJlM 360
0
rillJlJ>'il1'l #lUlutlf!ll'11'
'If~l'il'U..rl ril~fllt1:::1~f 'UtJ fi'il~~'d~u-l;~~~iJ~-~~~~'~19l1~~~~;£;~~~~~~Inl1'lJd fi ri'J'U
111>'itJ ri1'U'I111 UlJlJ~UtlllJ l'I1~I:JlJ'UtJ.:JeJI rlU1'::: fiUlJlr'U '1
ll~lfiruilCl (Trigonometry) 1i'l1clfinl1'I'l~lfiruD~'I11;'Jmh~hhmlJrilh1'illfi'11mJil~il
l'U~utlllJ l'I1~tllJ fll~'lJl'Utll'lJI'll ~I:JlJhjil~'lJl~il ril 90
0
fitll'lJl1'ftUl fit]'UtJl9I1I:JUllil:::lfl9l1 tlU
lJlU1':::~fiI'l1'lf1I9lI9i'I~U~ 2-9
".itl~2-9
"
('II)
. .~ .
fiJJ'lH)l9Iltlii (Sine Law):
A B C
sin a sin b sin c
C = ~A2+B2 _2AB cos c-
~U F Uil::: F 'ill'l1l'U'Ul~llil:::
~tl~ 2-10
l'
1 2
360°- 2(650
)
----=115°
. 2
. ~~L-~,-~___
(f'i)
, ,
21. ..
~d 0'
~]jm
fl~4U~I'I1~tJ:IJ~TU'IJU1U (Parallelogram Law) n~~U#m5t1lJ~lU'llU1UU'il~.:!1u~tl~
2-10 ('II) 1'l1uU)'11-i'l'1':l1Ufh.J-:J'iltl-:J fitl'UU1~'Utl-:J FR lU't~l.JlJ e'illn~tl~ 2-10 ('II) ~U'illlJm~tI'lJ
mllPltl1'ill'lJl),()'iltl.:!1u~u~ 2-10 (fl)
n~l'IjlfltuiJ~ (Trigonometry) FR ml~I~(JH'f)t1'11tl-:J1f1'lil(Ju
FR = ~(100 N)2+(150 N)2-:- 2(100 N)(150 N) cosIlY
= ~~0000+22500-30000(-0.4226) ~ 212.6 N
= 213 N
1.J'lJ e ml~~tltl)'~~f)Pl1'lif)t1'Utl-:J'liltiu uCl~1'lifh FR ~ri'1'Ulru'lJ1M'Ii'1-:J~U
150 N 212.6 N
sine", sin11's°
sine 150 N (0.9063)
212.6 N,
e == 39.8°
v
l'l,nT'U Y1i'1'Vll.:.! <j> 'Utl.:.! FR l~'illf)U'Ul),l1J 'il~1~",i1
<I> = 39.8° +15.0° = 54.8° d <l>
pbm.hun 2-2
,-, .
'il-:Jm,h'Utl)'~f)()1J~()(J'U().:.!11'j'':'! 200 Ib ~m~'I'h~tlMl.J~~.:.!U'il~.:.!1'U~tl 2-11 (n) t-u (n)
Y1i'1'Vll-:J x l1Cl~ y UCl~ 1('11) V1i'1'Vll-:J x' UCl~ 'y'''
y
:t<;
; . -1 _ __ _ ---= 200 1b
I£-_ _-l----'~I-- x '
~r
('IJ)
(n)
'jU~ 2-11
" ('iJ)
17
~"'.
/ .
22. ,'J "
18
"
" ."
.' '
,/
/
"'...1Ji111
, 1'Uullia~mru'll~1'1fnOlU~!l1~tJlJI9.l1'U'U'Ul'ULYiVl11 F 1'UlU'UVILL~IavtJ'ffVI L;L~Iu"'1lh'lu'fff1.l
~U'ffllJ!l1~tJlJL1 fHl'lVfLYifll11 ~a~1La'Ul~tJ1'1fl'ljlmulJ~
(n) mnnnL1fHl'lflf F = F + F LL~l1'U:iU~ 2-11 ('U) l~tJmVmflal'l~'l 'WU':i1fl11lJ
x Y, CU I 1
tJ11'UVILL:i'laVtJ ln1~1'l11JLLn'U X ua~ y li'ff~ll Lff'UU:i~~'U'Ul'Uf1uun'Ul'lllJnOluiY!l1atJlJ ~1'U'U'Ul'U
'IIlnlU'ffllJ!l1~tJlJL1m~vfillU~ 2-11 (f1)
F = 200 cos 40· = 153 Ib IlHlUx
F = 200 sin 40· = 129 Ib I'II'I'Uy
('U) m:iU1fHlfHl'lflf F = F + F !L'ff9l'l1mu~ 2-11 ('l) U:i~tJ,f)~f)fl'Utl.:J9fll'Jlr!La:x' y cu "6J
1'1f'lJVl.Jtl~flQ1'U~U'ffllJLl1~tJlJL1fHI'lV{~I~U~ 2-11 ('II) 'II~'Ml1
Fx' 200 Ib
sin 50° sin 60°
F, = 200 1b(sin ,500) = 177, Ib ~HlUx sin 60°, , _
Fy 200 Ib
sin 70° sin 60°
F = 200 Ib(sin 70°)=217 Ib 1'1flU
Y sin 60°
U:i~ F m~l'11IlivlmlmvU~llU~ .2 - 12 (n) liii'U'UWI 500 N LL(I~'ffllJl:imLUlVvmlJ'U
'fftl'l!L:i.:JatltJl'lllJ'lJVI'iV AB !La: AC 1l'll11lJ,lJ e ~1~h1!L'U1'jlU LYiflYi'll:MLm~fltJ F
" AC
Vift''I11'1'illf)' A 'lucY'l C !Lt'l:ii'U'Ul~ 400 N rt ~<
~B~_ __ ~,
/ ('I)
'F=500'N _ .J;. 1
(0)
-.,/_-
23. .. ,
~500N
. () l:W 600 .
~c=400N
(.:.t)
'jtJ~ 2-12
"
"'.. '
ltiTll
lflIJHfl~ltl~tl1~IJlJ ~lU'IJU1U fll'l"Ul fl n ~I~ ()f'1e~IIj.:J ti{) 1J~.:J~{).:.t'il::;I~~1;1 l:l'Vnfi.:Jltl~
2-12 ('1) «.:Jlfl~111~h~{)llj;]i:l'Vnfflflllr1ltll1Julmti{)V~{);]lIj.:.t F 111;1::; F ~-lflm::;1J~'"WIIUlI qJ AB AC 'U q
nrll'U{);]f11jm:::l'll ltl~llJtl1~VlJnfll~{){~{)flflael'HI~fl;]1ultl~ 2-12 (f1) ~lJ e ~llJTl"f1
l11i~1~V1'J5f1~'IJ{);]'b'1VU
400 N500 N
. sin' .<!> sin· 60°
"' . (400 N) ,sin 'l' = - - sm
500 N
<!> = 43.9°
60° = 0,6928
e = 180' - 60' - 43,9' = 76.1')' ~9
1-hf1~ 8 .QMlJltl'l::;~fl~1'J5n1Jfl~'1{)'lIf1'b'lVU'il~M F AB ii'IJU1~IVllfllJ ~~_~ .N 'Il;]U
ftllJl'lflflTl.nullmil F 1I1;1:::tlTU1u!'VI1'V1f1'V11;]~'lfi{) l,jlJ 8 ~1~'illf11l'Ul'nlJi'lII~~;]1ultl~ 2-12
(-l) l~vtY'lfl-lihmti{)1J F {)fj 1um&n 8= 16.1' 111;1:: FAB = 161 NAC ~
,I.
.
/
" 19
:-:
, I<
24. 20
"
.1-
--
I (
1.:J1t'l11'Ultff~.:Jl'U~1l~ 2-13 (n) lnm::;'I'll~ltJll'j.:Jfft).:]It'j.:J F 1tC1::; • F t11~tNm'jIt'j.:]1 ~ 2
~'V'nj~ii'n"h~ i kN lw::;iiVifl''V1l.:]yj':]C1'll'll'IJU'Ul~':] 'il'l'l1l (n) 'U'Ul~,Hl.:J F 1tC1::; F lntJ
.... q 1 2
e= 30· 1tC1::; ('U) 'U'U1~'UtJ'l F lW::; F t11 F iifilUtH.lVi'l'~
1 2 2
-:i1l~ 2-13
"
"'...ltifll
(n) 11Wlfll'l'lfl111 '1 l~tJlnUfll'j~lflllflll'ltJfl'll'IJfltl~tl~l'I1~tJ'IJ~l'U'U'Ul'U~'l~ll~ 2-13 ('U)
'ill n~llffl'IJl'I1~tJ'IJn flll'lt)f~ffrl'lI'U~ll~ 2-13 (fl) 'U'Ul~'UtJ'Hil F1 lW::; F2 ~.:]rNlli'Vl'Jlu,r'U
1"1'IJl'jfI'1111~VItJ1'Iifl tl'U tJ.:J9l'1tJU
Fl 1000 N
sin 30° sin lJO°
F[ =653 N
Fz lDOO N
sin 20° sin 130°
F2 =446N IlHlU
' , :4: .
25. f ,
..
('IJ) tll'hiM1:::'l.lfh e 'il'UlU~llJt'I15rJlJ'lJfl-:J!!1-:JlulUl1fHl'ltl{rl-:JlU~ 2-1 3' (-:J) 'f>I1y,h F2
, 'i):::(1'llJl'Jtl1J'JflnlJ F h1"'HnrJl1il'vlfllY11~!!'ml'f>1'fi 1000 N l~tJlll'f>ll:::mh.:,jt.:,j mllJrJl"l,rVrJ~~1'l1 -.-- .--.•-------- .q
l1~fl'IJ'Ul~'lJfl-:J F J'U11:::ln~9J'UlriVU'I,r.JI~'U1!1.:,jm:::y'li.:,jlnf)nu F ri"l'U1'U'ViffVm~'U <') 1'Ji'U OA. _ • • 0- - 2 ._- -_. - . -- ' ~' - " ... - .. . - -" - .. - ---- _. .-. . 1
-~~V OB l1:::vhhr F iifhlJlf)f)",h ~.:,jJ'U I~tl e= 90· ~ 20· = 70· 111:11 F 'il:::iifh,rvrJ~1:I'1'l2 2 •
1l1f)l1l1:l'1lJt'I15rJlJ1'Ulllii 2-1 3 (11) 'f>IU'l1
1000 sin 70° N = 940 N ..
1000 sin 20· N = 342 N
"
VlflU
VlflU '
024308
21
2.4 s::uun1sS3unUlJt))lLsJIla~1us::u1ul~f.J3nu (Addition of a System of Coplanar Forces)
1'Uri1'Ud'il:::Ufll'IJuWl1lI11.:,jUI'll:'l:::U1.:,j1I'Wf)1111I'lfllu'UlmvvrJ F 111:1::: F ~.:,jtlv1'Uuu"luf)'U xv x y <u
IW::: y 1'l1lJi:i'1I'1U l'1.:,jlll~ 2-14 (n) 111:1::: ('IJ) 1I'lrJf)OlllaI11~rJlJ~1'U'lJU1'U 11:::Ml1
F = F +Fx y
111:1::: F = F" + F"x y
y ~y
lZJ_
F
- < /
(f)
lU~ 2-14'IJ
F'
('II)
'illmtl~ 2-14 Yif('V1l'l'Uv;m~l:'l:::li1'lljmJll~I'l'lil1rJ111Mff'J 1'Uf)1'Jllml::...hl:::l91fl'liiff'wtrfllJru'U • 'lI U
• Yif('V11~'lJV.:,jriTUU1:::f)VUvvti~i'lln f) nu'Uv'I UI'l l:'l::11 fl Il'ltlfl'U'J:::U1U I~rJ1 nU~lrJ 1191 rJ m:::vhM
'~
1:I'tyl:lfllJo1mmni (Scalar Notation) 111'1VVrJ F 1'Ulll~ 2- 14 (f) IU'UI'il1:l'lfl(;1l{
1J'Jflvr'l F IW::: F 1'l1lJYiff'Vll'l~IU'U1J'Jfl'IJfl-:Jllfl'U x Ul:'l::: y I'lllJr.11~U -a111i'UI11.:,jVVrJ F' 1'U'J1l~x v y • IV 'U
2-14 ('IJ) uti'f>lUl1 F' iiYiff'Vll~1UIlf)'U y vitU'UI:1U l'1'1u'U 11:::11911!1'lVVrJ F' iirillUUI:1U
y y
atyil'ollWldml'le,iltf:i:::1J1JYlOVllllfl (Cartesian Vector Notation) f)1~U1:I'I91'1111'1civrJ1'U
11l'IJV'I11 flIl'l v111i1'lm.brJ1'U1:::1JuVlnl'll1f) (Cartesian Urrlt Yestors') ..,~:;if1l1:::1t)'IIYlJlfl1'Uf)1':icu . . ,- _ .• ____•
unU11J11l1'il'Vlt11:1'1lJii~ ri"l'U1'Umruh'VlcJ1:I'v,dJ~ I1fl1I'lV{11~'l11)l!~1'U,':i:::1JuVl,rll91lnn i l!l:'l::: j 11:::
1i'u1:I'1'l'l11ff'Vll'l1UU'U1I!f)'U x 111:1::: y '9lllJ r.111'11J ~.:,jlll~ 2::.1~-(fl}nJlI~V{~'If)ciJ111:::ii'U'U1191 Iflfl
~ ,,~ ~ d '" ~ ~ u ~ -l ~i
fll'f>l t!1:1:::t'JlJ I'l ImV'Il1lJl rJU1flILl:'l:::I:1U IU'Ulll1'l1lJ'Vlff'Vl~'I'IJ()'I111~f) ff':iIUIlf)U.x ,11i:'l:::,, y 'I'I1)J'UlJ"lf)
26. I ' [
23
lUfl'iru.t1hJu'j~cimJhH!f)'W X ua:: y 'Jfl..:Ju'i~awfi'lJfl'lU'i'llu'i::tilul~tllnulOl'1 ffllJl'Hl '
'~fJtiflylu~tl
(2-1)
)' )'
----------~~----------x
(0) ('U) ,
)'
FJ FR '- . '
x
FRx
, (i'l)
'itl~ 2-16
~ "
lrim.h::tlo~1'li'fflJfll'iii'il::9lfllfhU'l()..:Jli'l~fl'l'f1lJltl'Ufl'lU'j'lrifltl~l'l '1 ' ~llJ!!UlUf)U X 1m:: y• q
ill'iliifhdJu1nf)'f1~mlU fflUU'i'lawi~lnl9l'il::9lCl'lI;UtluluVifl"'I'11'l~!'f1lJl::fflJ~llJUUlUf)U x ua::
y ~I~U~ 2 - 16 (fl) ;'UU1Ol'UCl'l FR 'f1119l'illf)'I'1flH~'Wlil1m~tlu (Pythagorean Theorem) JUfiCl
e =tan-l IFRy 1
FRx '
l _~
27. 24
"
y y
'------- x x
F'b' = 200 sin 600
N
FJ = lOON
I (n) ('I)
'jtJ~ 2-17
"
= ....
• li'Yll
UtlJ~flllrua!mn1 (Scalar Notation) l~e'l'Jlfl F m:;'I'111'mJHflU y Yid'Ju(j1J U(j:;'JlJl~u 1
'IJe'l F fie 100 N UJ'ltitltJ1:l'llJl'Hll~tJu'l'U'H.h:1'lmnl1~il1 v
d ..... OJ d ,
'YI,)'1Hlfl'UtJ'YIWnl
lHIU
1~tJfl~~tl~!'YI~tJ'IJ~lU'lJlJl'U F2 'J:;u~fll~d'J'UuJ..:JvmJ x U(j:; y ~..:J~tlrl2-'-17 ('J) 'IJ'Ul~
'IJtI'III9iCl::UJ..:JVtltJ'J::'VI1Mi~tJ~11muii~ 1~t1'1'Jlf1 F m:;'Vh'lUYlf1''V11'l -x 1m:; F m~'I'h'lu
a ~
F2x
= -200 sin 60' N = -173 N =, 173 N ~
F2y = 200 cos 60' N = 100 N = ' 100 Ni
lHI'U
liIelU
iltlJinllru!1fl!~HI~1'U'j:::'U'Ui'ln~Hllfl (Cartesian Vector Notation) fllml'IJ'Ul~'IJtI..:J
ll,)'1VtltJ'lJtI'I F2 ~'I~tlrl 2 - 17 ('II) allJ1Jml1:l'~'I'l'U~tlllfll~t1fl'UJ~uuVlfi~mf1il '
FI = Oi+ lOON (-j) "
= {-100j} N
F2 = 200 sin 60' N(-i) + 200 cos 60' N (j)
= H73i + 100j} N
liIiI'U
liIiI'U
28. I .
I
I
I '
J-
25
~~aB10n2-~ --------------------------______________________________~
.. - 'iI'lvnu'j'lcitl(J x un.:; y 'Utl'l~~'j'l¥ ~'lH""~N1'W~u~ 2-18 (n)
y y
F= 260N
(n) ('11)
'jui! 2- 18
"
='" 0
llim
. U'j'l'il:;~L~mlJ'WLL'jl~tltJ x un:; y ~I~U~ 2 - 18 ; ('U) fl11lJ'lI'W'Utl'lU'W1Lff'WU'i-:Jf)'J::;'Vll
""llJl'jmL""~'l'h)'illn""llJL'I1~(JlJfl"lTlJ'lI'W 1~(J evnM'illfl e = tan-' C~) un:;'I11'U'Ul~'Utl'lu'j.:Jcitl(J
1~1'Uvil'Wtl.:J~~hnn'Uf11J F 1'W~1tlcil-:J~ 2-5 tlcil'lhn~llJlli~~l(JW,il ~tl Hff~ri1'W'Utl'llJlJ_ . 2 "Ii
'illn~U""llJm~[JlJfl"'1[J~1'i1'V16nl'l1'U~hi' ~I.J'W
(
' Fx )_(12)
260 N 13
F = 260 N(12) = 240 N 'x 13
Fy = 260 NC
5
3) = 100 N
1"jUil'U'Wl~'UtllH'j-:Jcitl[J1'Wu'W1'jlU F 'il:;11ii1~(Jm'j~tl!'U'Ul~H'j'l~1(JV~'iTri1'U'Utll'U1U'U1'il1J
'Utll ffUJm~[JlJ mllJ'lI'U~'I11'i~1[J ~l'U[Jl~~-q~'Utll~U""Ailm ~(JlJlJlJll.l n 1'W'U tl!:;~'U'Wl~'Utl-:J U'jl
citl(J1wL'W1~'1 F 'il:;1~1~(Jm'jfltl!'U;:n~'UtllLL'il~1[JV~'ilri1'U'lJtll'U1U'W1~lvn'i~1[J~1'U[Jll~ff~y <u i " "I
'Utll~U""llJm~(JlJlJlJll.l n 1'U~i1'il::;1'lifft1Jc)n'lJruffLfI nif
Fx = -240 N = 240 N (.,-
Fy =-100N = 100Nl
ill F H""~lLtJ'Wnm~tli1'U'J::;1J1JVin~ll.ln 'il:;Mil
F = {-240i-100j}N
,.,
mJ'lJ
17HJ1J
29. 6
• ~baEhJn 2-7
y y
FI =600N F2 =400N
,..--..
--------~~--~-------x
I.....-¥~-"---I-f....., --x
(Il)
~d 0 d
11i111I1UU'fI 1
y
582.8 N ;1--~FR
....".-:...,...,-----x
236.8 N
(ft)
('lJ)
tlUH:lfl1Hihumn1 (Scalar Notation) ifqj'VIldllm~(JHf)~lui'f!'l1~(Jmhu'1'l..!li!
mh.:JhfililllJ ll::;Hlilfllll'ii;l::;IlJ.:Jllhm'J.:Juu(J x lli;l::; y ~.:JlD~ 2- 19 ('1) mrnllJll'J.:JUU(J~.:Jf)ci11
'I'lJ--t~'liflWI'I fi'Tfl'u~YlI'I"Yll.:JUlf)'lJtl.:Jtl,:]rlU'J::;f)tlUll'J':] x lli;l::; y I'IllJ1:1':lJfll'JY1 2-1 ll::;Mil
~ FR =LFx ; FRx =600 cos 30° N - 400 sin 45° N
x "
= 236.8 N ~-7
. '
FRy =600 sin 30° N + 400 cos 45° N
= 582.8 N i
IlJ'll:1~TI~'lU1:1'~.:]l'UlUY1 2-19 (,fl) ii'1'Ul~ .
FR = ~(236.8 N)2 +(582.8 Nf
=629 N
e== tan-1
(582.8 N) =67.90
236.8 N
{;HlU .
I?HJ1J
30. . .,;
l~,hIl1J'lJfI 2
aruaflllrunfll~W~1u':i~'lJ'lJ-Wfl~Ulfl (Cartesian Vector Notation) 'Ulm'!J~ 2-1.9 e'IJ)v ~
LL~l:l::U'i.:luff~.:Ilu~'!Jn f1,LI9l{l'flu'i::uu'i'l n~mf1
FJ
= 6QOcos 30·i + 600 sin 30·j
F2 = -400 sin 4Yi + 400 cos 4Yj
FR = FJ
+ F2 = (600 cos 30· N- 400 sin 4Y N) i /
+ (600 sin 30· N+ 400 cos 45· N) j
= {236.8i + 582.8j} N /
'lJU1~ml::Vlfl'V11.:l'IJCl'l F 'I111~1'U'vhuCl'lI&itnnU'll1'lI'1U- R '
/
ImtiU lli tluffCl'llfid 'V'IU":h f11'l1i'-ff()j ~mHlr1.'Hf1~11ihh::ffVllif11'V0J1f1 f1il d'iCl'l'Ul f1 IImiCl tI
fflf1l:ll{ffll!l'Hllnl~~tI 19l'i.:l lli,}lIUU1'1Cl'l IIff~'1'l'!J'lJCl'l uvi l:l::u'l.:ll'I1IUun f1 II9lCl{uuulm::u1J'Vl n~~ .
111f1 nClUf11'l1JJ f1 11'l'lVCl {} Cl vl.:Jl'ln19l1l! f11'l1 Irrn::rfnf1 II9lClllm::uu'i'l n~lllf1li'!J'l::lt1'lfUClVl.:1l!lf1
l'Uf11'iIInif()j'l11ffll!iJ&i
~3af.hJn 2-8
'!Jcntll'1i~.:J 0 ll.n'!J~ 2- 20 ef1) f)f1m:::'IhI'1JtllI'i'l1'U'i:::'U1UI&i<r:Jtlwm:::~,)lJ'i)~I&itlJn'U[;fllJ!t'i'l<u 'J q 4
~ ~ ~
'i).:I'I11'IJ'U1 ~ 11l:l::V1fl'V11'l'UCl'lII'i.:li:l'V'l1l
y y
_r;, = 250 N
--~~~~-----------x
F;=400N
----~~~F=~--------x
(fl)
~ld
':iufl 2 -20
"
27
31. 28
2.5
;
~-
. ..
..
"'''' .'lfi't1l
t!~~::t!'.l'l!l~fHih.j!l'.l'lvmJ x U~:: y UIl'~'1i'U~u~ 2- 20 (~) '.l1'JJU'.l.:JV'tH./ Xl::l~il
-:!:-+FRJ< =LF,; FRx =-400 N + 250 sin 45° N - 200(~)N
= - 383.2 N = 383.2 N ~
lfl~tl'l'VilJ1tJ~lJli-JO:il F m::'I'hl,j'l'll.:J~ltJiitl ~'Ufitl i'UVlfl'l'll-J x ~Li'J'U~lJIl'l'lJl'.lt1Uff~-l. RJ<
I~Wlilf)ff'.lI~n '1 f11nlJ lritl'.l1'JJt!'.l.:JVtltJ y I::1~il
+ i FRy =LFy; FRy =250 cos 45° N+200(~)N
= 296.8 N i
FR = ~(-383.2 N? +(296.8 N)2
= 485 N
I1nf11'.l-U1nl1fH~tlll'U~,j~ 2- 20 (fl) l,!'JJVlfl'l'11.:J e fitl
e= tan-J (296.8 N)= 37.8~
383.2 N
Vl'l'.l::anill1'.l'l FR l1ff~'li'U~,j~ 2- 20 (fl) I::ritl1'111n~N~!'Ii'Ul~t.nnlJu'.l.:Jvf.:JIl'1'JJ i'Uiu~
2 - 20 (n)
13m(;lE)s1us:uullin~01n (Cartesian Vectors)
'.i::U1J1lfl'UiiCl'IJ?l SRight-~a,n~ed ~oordinates v Syst~~) 1'11.Q1t11Uliiitl~111lll'~'l
Vifi'l'l1.:JlJ'l f)'UV.:J Uf)'U z, ibiiv'U'llv1.:JffVi l'VIihl'lY'UflT:iV1J lIf)'Uiillff~'lVifi'l'l1'llJ1 n~tl.:Jl1n'U x u~::iitl
~~'Utltlf11u fitl 1If)'U Y cg'lilVifi'l'11'llJ1f)tltlf)I1f)rr.:Jiitll9l.:J~u~ 2 - 21 .
- l::U1J'wnilll1fl'IJCI'Il1fl1~CI~ (Rectangular Component of a Vector) 11f11~tl{ A
tl11ii"r.:JvtltJ~~.:Jmf)n'U ('Vi~.:JII'.l'l, Il'tl'lU'.l.:J 'Vi1tlffl'JJll'.l.:J) ~l'lJll'U1IIf)'U x, y lIel:: z l~tJiiVlfl'l'll'l
'IJtlmfH~tl{ffmr'UinlJl1f)'UI9l.:Jf)ci11 I1f)~U~ 2-22 1I')vH'f)~~1J.~!~~v'JJ~i'U.'J'U~'U.U~~t1l"11
I1fH~tl{I::1~il A = A' + A 11~:: A' = A + A lritl'.l1'lJff'lJf11'.lI9l.:Jf1ci11 N~'.l1'lJ11f11~t){ A
v
z x · ...~y ~.--
~tl.:JU'.l.:JVt)tJv1'1tl"l3J fit) , ~ .
~- .----- ~
A==A +A + A, y z (2-2)
32. ..
z z
I
I
x
'jtl~ 2- 21 'jtl~ 2-22
" "
nfllYHI~'I1'U,:j'l1'1htl (Unit Vector) l~£J~llllnfH;H)'f'Hi1.:j'l'n.l';HJ f)t1 . nfH;ltI{~ii'IJlJl~
L'l'hrru 1 61 A llhmfHPltl{~ii'UlJl~ A"* 0 ~<V.:jJ'U nmPlu{'Hi11'H'I11£J'iI:&Vifl''I'111l~£JlfilJ A 1~£J
(2-3)
(2-4)
A (i:1'lf1f11{~ii~_lU'Jfl) liJ'U'IJ'Ul~'IJtlmfllPltl{ A lln::: u
A
(nf1lPltli'1fij~) !iJlJYlff'l'11'1'IJtl'l
nf1lPltl{ A illll~ 2-23
" ,
29
-
-'
33. x
'jtJ~ 2-26
"
" ,
~~UU 'UU191'U~N nfHl'lfl{ A ilfill'vhn1Jfi11Jl fl'Ufl.:j'll flYi fffl.:j'Utl.:j vHl'll:IJ ll'l~V()fHJ mllt1.:jfftl.:j
nH'YI1~'lJB~nfllm)~1u'l::;1J1Jwil~Hl1fl (Direction of a CartesianVector) Yifl''Vl1~'Ufl.:j
l1fll~11l{ A ftfl lPJllff91~Vlfl''Vl1~~l91,]lm::;1J1JVln91 (Coordinate Direction Angles) (X, f3 llCl::; Y
191trr91riTulll~'UflmfHl'lfl{ A ltJv.:jllfl'U x, y llCl:: Z ~l'Umfl~I~tJ~ 2-27 ll~Cl::lJ:lJiJri1'l::1I--.h~
o1i~ 180 fll'lllllJlJ (X, f3 llCl:: Y Vl'il1'lrul,]lflm~ll1tJ'Ufllnflll'lfl{ A ~m::1'i11J'Ullfl'U x, y llCl::;
.Id ~vv "" .!{ d .1 d ~ , . 1 ~'"
Z ~UVl 2-28 l91fl11(l~~'UVlll'll~l~uffllJmCltJlJl'UllI'ICl::lu ']::;19111
(2-7)
~llCl'U~I fl rill llJu1fl'll'1 tJUllff91~Yifl''Vl1~ (Direction Cosines) 'Ufl~ A fi1lJ:IJ llff91IVIfl''Vl11Yi
l91llm::uu-WO91 (X, 13 !HI:: Y1I1M'illflri1flUl1fl{t1'1fl'll'1(JU (Inverse Cosines)
,~hllfl1l'1fl{1I~~mll(J'Ufll1f1'll'ltJUllff91~Vlfl''Vl11'Ufl~ A lu~tJnflll'lfl{l'U'l::1J1JVln91ll1fl A = A i
+ A j + A k (fflJfll7~ 2 - 5) 'IJ::hi'11y z
(2-8)
x
, '
.~ ..--?-
31
34. 32
x
tnt) A== ~(Ax)2 +(AS +(AJ (ffllfll'rYi 2-6) 'Illflfl1~lm(Julii(JunumJfl1~~ 2-7
i'lUl1tl':li>i'lh:::fltlU i, j Utl::: k 'IJ()~ U ffllJ1':iml'V1'W~-::w~11fl'lfl[J'I1uff~,jVii'1'V11,j'lJ(),j A tT'WLtl,j
A
UA
== cos ai + cos ~j + cos 'yk (2-9)
x ('II) x
.'al~ 2-28
'II
v •
IW::: U lJ'IJ'Wll'ltyilnu 1 ~~,r'W 'IllflffllflWVi 2-9 ~,jlJmlllfflJi~''Wi1:::'111'llfl9ill'J'I1Uffl'l~Vii'1'V11~A
A = AUA
= A cos a i + A cos ~ j + A cos Yk
== A) + Ayj + Azk
/ .
(2-10)
(2-li)
,,-.
35. 33
2.6 n1SU~mla~n1SaUl~nt~as1us~uUWnfla1n (Addition and Subtraction of Cartesian Vectors)
x
'i..J~ 2-29'IJ
~lati1l1fHl'ltlftl~lUtl'lrf..J':i~fltlulu':i~uuVin9lmfl ~hmh'll'Jiu Vi'ill':iill1l1fH~ltlf(1'tl'lnfHl'ltlf A mu:
B 11'1£Jlf'lfilJViI'f'Vll'l1J1fllumtlu x, Y Ui:l~ z ~'l':iU~ 2-29 til A = A i + A j + A k UCl~ B = B iq,t qJ . x y z x
+ B j + B k ~'l,ru nfHl'ltlf"~TI R 'jJ~lJtl'lrfU':i~fltlUlUWm':i1lHHfli:l1f'Utl'ltl'lrfU':i~fltlU i, j Ui:l~y z •
k 'Utl'l A UCl~ B ,yu~tl
R = A + B = (Ax+ B)i + (Ay+ B)j + CAz+ B)k
fll':iClUn flll'ltlf'jJ~r;'jJl':iill1IUUmUir;li'fll'Utl'l(Q':i1J1 fl n flll'ltlf l'UlUtl ~l'l~la'il:;Cl1Jfil fflfl Cllf
1'I1lJ1i'1~U'1Jtl'ltl'lrfU':i~flVU i, j Ui:l~ k 'Utll A 'I1~V B ~'l~1tl~1'll'Jiu
R' = A - B = (Ax- B)i + CAy - By)j + (Az- B)k
.d. 0, d Q.J OJ .c:I .c::o.
'i~'lJ'lJI!'il'flfl'i~'fll'i1:tJ'i.!V1IVltn~_'!I (Concurrent Force Systems) 'I1i:l fl'VI11V1'1fflillrl'Utl.:!
. n fl 11'1tlf.ull ~'UffllJl':imhlJll.h~tIfl rl1i'M1V1 alurmu.QY-li:l"VITIU':i.:!lJ,ilrvilnu Y-Il:"l'i1lf !luun fll~ltlfq
'1Jtl'lll':i.:!vfl'l1lJVI~m:;l'i'witl':i:;uu ff1lJl'im~f.J'UlUUfflJfll'iMi1
!.
(2-12)
Intl LF, LF UCl:; LF lUUY-lCl'j1lJvhflrurlrlllJJ:i'1~u x ,Y 1Ii:l~ z 'I1~tltl'lrfU':i~fltl1J i, j,x y
Ui:l~ k 'lJtl'lU~i:l~u'i'llu'i:;u1J
i
.~
36. I
/
I
r !
r" ~;,,,jL·.: ; .:- .... :. ~- .. __:.r.-~----~ ~---r -. -~" . . ., ' I
. , I 1l.:J'I11'lJUl9) ua:;'lJ'lJ ua'MVifl'V11.:J'Vii'9)1110 'J:;1J1J'Wn9)'lJeJ.:JII'J.:Jrl''I'liii0 'J:;vl11JU1.:JlI'M1uhmJii" . _ 'lI
2-30 (0)
FR = (50i - 40j + 180k)lb Z
F2= 150i - 100j + IOOkjlb Fj =(60j + 80kjlb
J---;r--H"'".---'-- - - Y f"'ft-'-----~- y
x x
(n) ('I)
'j't1~ 2-30
"
""'" 0
1fifll
I'cteJ.:Jil1flll'J.:J U~a:;u'J.:J UVlU~1£j'JU nflll'lc)'flu'J:;1J1J'Wn9)'n01l'J.:J rl''I'll F ua'9)'11u'Juv1 2-30 ('lJ)~ R ~
.. ....
FR = LF = F j
+ F2 = , (60j + 80k)lb~; (5Oi - 100j + 100k)lb
;/" .
= . {50i - 40j + 180k} Jb
- -r-"--~;~ -- ,
FR = ~(50)2 +(--40); +(180r
=191.0 Ib
=..::tQJ ~ <V "1 9/ . I <I oJ • d 0
l.jlJVlfl'VI1'1V119)illm:;'U'U'I'l09) u, j3 ua:; Y 11119)ill.D.HJ~ £jeJ£j'IJeJ'Inflll'leJ'Jl1U'll1UJtlVlf!~:;V11 ;.
1UYifl'VI1'1'lHN FR
' I
cos a
cos j3
cos Y
= 0.2611;- 0.2094j + 0.9422k v
= 0.2617, a:::; 74K
=-0.2094, j3 = 102
0
= 0.9422 , _ Y= 19.6
0
?ltl1J
?lel'U
l.jlJll1'h;fua'9).:J1u~t1v1 2-30 ('lJ) l9)£JIU'l'll:;eJ~l.:Jti'l'iJ:;'I'l'Uil j3 > 900 I'WeJ'I'iJlf)U'J'I~eJ£j j 'lJeJ'I
U:J,illlhHl1J .
FR
37. A:Jaehon 2-10
.'h'l!ff~m!':i-3 FJ1u~uYi 2-31 1u~unm9lvflu':i::uuYln~ll.1fl
-~--- y
x
':iu~ 2-31
"
"'''' ,lfi'Yll
l~fl-3111fll.JlJIlffWl'lVifl''I'mYir~111m::uuYl n~':i:;lllr!'vi(J~ffV-3fiJri1Ul.JlJYiffllJ ex 11 ::'Vil1~111 fl
. .
fflJfll'rn 2-10 uufiv
'~OS2 ex + COS
2
~ + COS
2
Y = 1
cos2 ex + cos2
60°.+ cos2 45° =1
cos ex ~~1- (0.107? - (O:~? = ±D.5 --- ',
a =cos'! (0.5) = 60' 'I1~V a = cos'! (-0.5) = 120·
1I1muYi2-31 'IAIU1l ex ~ 60· !dV-3111fl F flv1uVlfl"I'm+x..., . x qJ
HfflJfl11Yi 2-11 IW:; F ~ 200 N 'il,::M11
F = F c~i + F cos ~j +f cos -yk~ - . f
= 200 cos 60' Ni + 200 cos 60· Nj + 200 cos 45" Nk
== {100.~ f+Too:Oj + 141.4k} N .
, -- -...•- " ' .. ,/ .
u':i::~fl9l1'li'fflJflI':iVi 2-6 ' 'lAluil'U'W1~'UV-3 F ~'er '
F=~F; ~ F; + F;
= ~(100.0)2 +(100.0)2 + (141.4)2 =200 N
35
J'
38. 36
roathurl '2-11 - - -________________________....
z
x
l
F Il
F=4kN
y - - y
Fx
' ~ /x
(n) ('U)
'J.J~ 2-32
"
""'" 0
11i111
l'Ufl'H:u.Q'Vlfl''Vll.:J'UeJ.:J F fleJ lIlJ 60
0
u{l:: 300 ~hi1'lflllJ'Vlff'Vll.:J(;lllJYlnWl~1~'illm::1J1J'VhrWl
'illflfl1':ilh::~fl~1'l1f)lJ~t1~!'I1~tJlJ~l'w'U'U1'U !!':i.:J F ffllJl':ifl!l~f)eJeJfl11J'U!!':i.:JtieJtJl'UHmllf)'U x, y
H'a:: z i.:JHff9l'l1'U~t1~ 2 - 32 ('U) 'illfl~t1ffllJll1~mJ~Hml'Vi1J
'y
( >
F' = 4 cos 30
0
leN = 3.46leN
F- ':--4siri-300-leN = 2.00 leNz .
F 3.46 cos 60° leN = l.73 leNx
F 3.46 sin 60
0
leN = 3,00 kNy
F = {1.7~i + 3,OOj + 2.00k}leN .....
1I
'f
~
.J
• I
39. A.d. 0
1111'f11
~,,------ y
x (0)
x
z
I
.".;n:-:,------y
cr )'
(fl)
'j''lJ~ 2-33
"
fI"llJn1JIII'nHh~~ 2-11 lJlJ'IJ{)~ 60· !!ll~ 45' !1JUi1f1'VlWJil~ F hi1'lilJlJi1fl''Vll.:J~l~'iJlf)q •
.t'y1~1J1JVln~ !~il1j';j~~f)9l1'lff)~1U#!'!1~lJmrTU'IJUltHyj{)U9lf) F !uuu';j.:jehw x, y Uf1~ z 1II.:j
U'ff~.:jlU1U~' 2-33 ('IJ) 1~tfi1i9l11muil~ 'lJU1~'lJil.:jumhw ~{)
l"~"f'l l .:;2
100 sin 6Qo Ib = 86,6 IbFz
F' 100 cos 60' Ib = 50.lb ./.
Fx 50 cos 45' Ib = 3SA Ib
Fy SO sin 4S' Ib 3SA Ib
~.:j';j~1~.:j F iiilfl''Vl1.:j!UU -j 'iJ~Mil
y
F
F
Fj + Fj +Fk
{35Ai - 35Aj + 86,6~} Ib
37
40. 38
~.
x
..
"1'd':]'l1'.h{j~m::'Yhhj'Vif1'Vll,:]'Utl,:] F J'U~tl
u =!. = Fxi + Fy j + Fz k
F F F F
= 35.4 i- 35.4 j + 86.6 k .
100 100 100
= 0.354i - 0.354j +0.866k
ex = cos-1(0.354) = 69.30
~ = cos-1(-0.354)= 1110
y = cos-1(0.866) = 30.00
r'Hwl.:]ncill'nlltr~.:]l'Ultl~ 2-33 (1'1) ~
ll'j.:]trtl':]II'l.:]m::vhllitl~::'Utll'Ultl~ 2-34 (n) 'il.:]mlJlJ!!tr~,:]Vlf1'Yll..:]Vil9Hl1m::1J~Yln~'1Jtl.:]
F lvlv~ll'j.:]ft'l'nj F m::Vh~l'l.J!!n'U y ~llJ'U1J1nllC1::ii'U'Ul~ 800 N2 _. R
.rr-.--;---------- y
Fj=300N
x
(n) ('11)
'itJ~ 2-34"II
"' . . 0
l1im
.
(
lvlvllnuwm'if ll'j.:]ftYnjHt1::H'j.:]ritl{j~.:]trV,:] F HC1:: F trl'l.Jl'jml~~.:]l'UltlnfH~tlfl'U'j::1J'lJ ._u 1 2
Yln~lnn ~.:],r'U 'illnltl~ 2-34 ('U) ')lllJ'U~ F = F + FR 1 2
41. ih::tlf)~1'JffflJf)1':i~ 2- 11 ' ~t
• FI = FI UFI = FI cos a) + FI cos ~Ij + FI C~fAJ;fr~·,:~i ·
= 300 cos 45" Ni + 300 cos 69° Nj + 300 cos 1200 Nk' '
= {212.li + 150j - 150k} N 'r. ,' ..~
F2 = F2 UF2 =F2) + ~J.:r F:k-
l~fl~Jlf)lll'l'lvifunll 1t'J~~VHj Fii'U'WWI 800 NU R
FR = (800N)(+j) = {800j} N
II'WitW i, j Iti:'l:: k Vifffl9lfl~tFl'l'l1~~1'W1ti:'l::'Ul1iiflt'vhn'W 1l1f)fI1':ilViUUL'vhll':i~cimJ x, y
, v
II":: z 'U(N F t'vhn1JI!':i~ciuu x, y U1:1:: Z Viffu~mltl~'Utl.:J CF + F) i~u'W, R 1 . 2
o = 212.1 + Fzx
800 = 150 + FZy
F2x = -212.1 N
F2y
= 650 N
o =-150 + Fzz ; Fzz = 150 N
l~tl~Jlf)'U'Wl91'Utl'l. F IW::It':i~citlU~'I'l':il1J ffTlJ1mli'fflJf)l':i~ 2-11 !vitlln a, ~ 1!1:1:: y M__ _ 2
_1(- 212.1) 80
-212.1 = 700 cos <X2; <X2 = cos '700 = 10
650 = 700 cos ~2; ~2 = cos-
1
( ~~~) = 21.8
0
liHl'U
_1(150) 0
150 = 700 cos Yz; Yz = cos 700 = 77.6 ~H)'IJ
2.7 bmUlElSS::4Pl1UVlUU (Position Vectors)
• ...,y I ~ QJ rI 0 I
ffl'W'WJ::f) 1:111 f1~lH'l f) f)l':i'U tl'l I'd f)!~ tln::1J lIl 1I'I1'U1q
:!~,!:,:!:!,~_ttVl~lfl x, y IIg:: z (x,y,z Coordinates) 1l1f)~tl~ 2- 35 ~911'Wmrul1J5mu~i1
~llt'l1ti~a-'l1l~''Wln1J~91!~lJ~'W'Utl~I!f)'W 0 191Ul9119l1lJ,hi1Jllf)'W x, y 111:1:: z iltlcil.:Jl'W~tl 2-35
Vlnwuu.:Jll91 A r~1l1flll~ o-hw x =+4 m 1lI1lJUfl'W x, y = +2 m 19l1lJllf)'W y !W:: z = - 6 m- q q A A A
1lI1lJllf)'W z i'l,r'W ffl1Jl':im~u'Wl11vcil'W':itl'Utl~Vln91i~il A(4, 2, -6) l'W'Vh'Wu'H~Uln'W f)l':il91". ~ ~
~llJllf)'W x, y IW:: z 1l1f)~91 0 i).:J B ll::iMh)~'Utl.:J B ftU BCo, 2, 0) 111:1:: q6, - 1, 4)
,39
..-'
42. ~o
~..
'/'''''.',- .-
, .
I
'jtl~ 2-35
"
~_ln1~a{'j~'4vlllm'll-:j (PositioD_Y.e.etOr.s) nfll9lvf:i~1JlPlllm'li~ r fiv nfll9lVf~'l~vi'Cl-:J
m~'Vh~~~1~~~'1d-:J'1~CliilPlllm'll':)~lI'lj'UCl'Ulumrulu~nru~t1iJ'vr'UTInu~~~'U ~lClril1I'1i'U r (llfl
'llfl~~I~lJvi''U'JCl-:JVln~ 0 fi-:J~~ pex, y, z) ~-:Jltl~ 2-36 (fl) ~-:J,r'U r U'llJl:ifll~£J'U1'U
ltlnfllPlClfl'U:i~uuVln~mfl
i, = xi + yj + zk
i:1mJru~m:iU1fl n fllPlvflluuril'U'I11I'iVril'U'I1ll'JVlCllrftl:i~flClU~-:JU'llJ'l~lvi'nfllPlClfr ~.:)lU~
2-36 e'J) hWI~lJ~~fli~lJvi''U 0 1fl£Ji~lJ x: 1l1'Vifl''Vlll +i, Y1'UVlfl''V111 +j ua~ z 1'UVlfl''Vll-:J +k
ltliJ-:J~~ pex, y, z)
z
z k
P(x, y, i )
o yj
--y
x i
(n)
'jtl~ 2-36
('11)
"
'l'Uflnihfl1tl nfllPlClf:i~Ulllllm'li-:JlJ'V1f('Vlll'llfl'l~ A lt1'l~ B 1'Umrulu~nru1~ '1 ~I:itl~" , , " 'U
2-37 Jfl). nfllPlClfiliilYqJi:1mltu r U1-:Jt'lf-:J'l~iifl1:i61-:J~lnfllPlCl{1~£JHilmj:iU'Cll~lMCl£JCl~
43. ..
U'ff~.:J111~1J1Jln~~1~'bJ§u'l'~~~~1~ i.:Juu r 'ffl1J1'Jt'll<1I'JU1~U~1l'lJ{j.:J r AB ffl'l1fU r A lW:: r B
l~nJ~ 2-3 7(fl) l1'ff~'H)mj'mri'l1rl~~lt~tJ~'illni~rialn'illfl'il~t~lJ~U'lJtJ~~«~
v d QJ <I ~ < V . 'I. ' / '"'-1
'illfllU'Vl 2-37 (n) l'IClflfll'JUlflnfl1I91{j'J'ffTUl'Ill91tJ'fflU _H ; (~ade-to-'ftil Vector
__ Addition) 'il::M-:h >-
(2-l3)
i'lJu {j,3flu'J::fltlU i, j, k 'lJrJmml9lrJf'J::1J~hul'l11'1 r rJQ1UlU~«9I:ltJ'I,hum'l'lJrJ,mml9lrJf
,. <I' v "'" rv.d .. " v ' Q J '"_ ''':" ~5 OJ • IIV I
A (x , y ,z ) aU~ll'J'Wfl~'Vl'ffrJ~f)1:H)'1'H)'1'fflUl'Il B (x , y , Z ) 1Jlfll'laflfll'JUlfl'fflUl'Ill91rJ
A A A B B B
rilUm'l'lJrJ'IrJlflU'J::flrJui'l'ffl1JM r ,rU~rJ l~1J1Jlfl A lu B ilU'ff~'11U';Ju~ 2-37 ('lJ) BU!!'Jn
"
!~1J'illfl (x - x ) luVlfl''Vll'l +i, (y - y ) 'lUYiPl''Vll'l +j !!a:: (z - Z ) Vlfl''Vl}~ +kB A B A B A
z
ClIL- - - - - - - Y
x (n)
'JtJ~ 2-37'IJ
~bafi1Jn 2-14 -,
'il'lm'IJU1~!m::Vlp('Vll'l'lJrJ~nml9lrJf'J::1J9l1WI1:UI~iJVlf('illfl A hJv'l B lUlU~ 2- 3 (n)
x
x
('lI)
41
44. 42
=d.
111m
1'1111ffllf11'j~ r7"",13 wn~'IJ'fllri1'Ui111'A(1, 0, -3) !H't::;fJflCtU~1UWn~'IJ~lri1'U,r1 B.(-2,
2, 3) 'il::;1~11 - , ,'~~
( T , ': (-2m y m)I + (2m - O)j + [3m - (-3m)]k
~""') = ;~-3I j2j + 6k} m
'illfl~1J~ 2-38 ('IJ) U'jlri'flU~lffl11'IJ'fll r ffllJl'H)i111~~UOlJ'I lOltltfl~'fl'U'illfl A l1JrJ'l B
1'll1lUfl'U x={-3i}m; Oll11Ufl'U y={2j}m U(I::;1'11lJUfl'U z={6klm
'IJ'U1I'l'IJ'fl~~ ~::;ii-h
r = 1(-3)2 +(2)2+(6)2 =7m
fflJf11mfllOlf.}'{l1l1~Wii1uhj'Vifl''Vll'l'IJf.}'1 r 'il::;l~-:il
"C""" ~
r - 3. 2. 6 k
U= - =-l+-J+ -
r 7 7 7
11'j'lri'flU'IJ'fl'lI1f)IOl'flfl1l1'lmi1tJ'n'il::;MllllUff~.'lV]fl''Vll'l~l~'illf)'j:::uui'in~
VltllJ
a =cos-t~3) =115° VlfllJ
13 = cos-
1
(*)= 73.4
0
VlfllJ
y = cos-1
( %)= 31.0° I'ltllJ
lllll'l'lfl~l'.l'd'l9l'illflUfl'WUlfl'IJf.}'1'j::;U1Ji'inl'l~ri1'wi11'l'IJ'fl'l r 1'l'lllffl'l'll'W'j1J~ 2-38 (1'1), ~
2.8 l:lntVlEJSllSOnufinn10Vl1UllU:llaU (Force Vector Directed Along a Line) /
. l'Uutyi11"'(lOlm"'Ol{"'lllij~ V] ff'Vl1'l'IJiN U'j'l 'il:::J'::;1JlOl(J~l'lll'l"1 "'tl'l~I'l~i'lr-.il'WU'W1f11'jf)'j::;i'h(
:I9ll21J~ 2-39 lrimm F iHifl''Vll11'11llU'W1lff'U AB Cj}I"'llJ1'j(H~tJ'U F l'U21J'IJ'flll1flll'1'fl1l'U'j::;uu
i'inl'lmfl 1~Uffl'llfifl''Vlll~1tJl1fllOlf.}{'j::;1JllllUl1UI r 1I'ltJiiv]fl''illfl~~ A l1JrJI~~ B U'UU'W1Iff'W4~
l91U-vflLt.lVlff'Vll'l'il:;J':;1J1U'WI1fl!I'If.}{l1l1'l'l1Ul(J (Unit Vector) U = rlr l'l,nr'W ' ,
UJ''l F !tffl'l'll'W~1J 2-39 tU'Wl1UltJ'lJtllUJ''1 cB'IUl'lfll1i1'l'illfl r l11ili'in91 x, y !!(I:; z
Cj}liil1U1tJllJ'Wfl1111tJ11 I9llJ'W UJ''l F ~'lhjffllll'jm~(J'Wl'U21J'IJ'fl'lffIflCtU'UUf)'Ui'inI'l1~
45. F
.' • J ~
) - - - - - - - y
x
'Jtl~ 2-39'IJ
53n1Sa1V1Su51AS1::vI (Procedure for Analysis)
v
. IlJ{) F iJYifl'Y11':]ill~UmHiHnml1fl~9i A hJ~9i B 9i.:]'.!'U F ~l~l':itlJoi(J'.ll'.l~tlnmil{)f
i'.l'j::1J1J~n9imflM~.:]if
nflllltJ1'J:;1JVhmnl~' (Position Vector) rmhu'I1u'lnmil{)f r L9itJl'I':i'l'Jlfl A ltl B
. ,
mI1rll'.l1tU'11'U'.l19i'U{)':] r
<,,; • .,. d • d ~I •
nfll~~'J'I1'U.:.I'I1'U1£l (Unit Vec~or) '11nmil{)'j''I1'U':]'I1'U1CJ u = r/r 'li'llu'.lfl1'j'1J~1J{)fl
YiffYll.:]'U{),nl'l rIm:: F
' nflllltJ~I!H (Force Vector) '11 F 'JlflflTJ'j'J'IJ'U'Ul9i F u,,::Yifl'YIl'l u u'UiiitJ F = Fu
Pl::>a~.i1l1' 2-15
'liltJi'l'.l'l1i1'l'vju~i'.l~tl~ 2 - 40 (fl) ~'Hff'UI9i{)fllKltJU'j'.:] 70 Ib 'J'lU~9i'lU':i'l~'lm::'I'hn1JlIl'U
':iU':]1lJ A l'U1tlnmilt)'fl'.l':i::1J1J~n9imfl uCl::'11l?l1Url'll-:!'UU'lu'j''llKltJ
z
z'
r
. (
• B (12,-8, 6)
('I)
43
~ "
. "'
46. 44
"' . . 0
1lim
U)~ F tHjl'.mJ~ 2-40 ('lJ) 'i1fl''I'11~'lJtl~I1fl1~tl{d u ''nl~'illfl11fl1~tl{)::'lJ~1!m'll~ r ~~alfl" " q
'illfl A lU It 'i.::J~u~ 2-40 ('lJ) t1'h:Jt1'l.Ifll) F lU~UI1fl1~tlfh.j)::'lJ'lJ~fl~ll1fl 1~l'Ji~flTJi.::J
~tllud
I1fl1lnt&i:::'l!Ylurtnl-:J (Position Vector) ~fl~'lJfl-:J~~Uft1tJ'lJfl,:mrU!GJiflfl fitl A(o, 0, 30
ft) ua:: B(12 ft, -8 ft, 6 ft) t1"hmfl!~tl{'J::1J~lUl1'1l.:J1~1'J"'lJ~fl~~t1'tl~fl1i'tl.::J x, y u,,:; Z '1Jtl.:J A
flU B 'il::l~il
r = (12 ft - O)i + (-8 ft - O)j + (6 ft - 30 ft)k
= {12i - 8j - 24k} ft
lu~u~ 2-40 (fl) !;"'~'lfll'j!~tJU r 1~tJ~'j'l'illflfll'j!fl~flU~'illfl A{ 12i}ft, {-8j }ft !!t'!::
{-24k}ft lUtJ'l B
'1JU1~'1Jtl.:J r ~.:JU'I'1Ufl11l.1I'J11'1Jtl,:mrU!GJitlfl AB fitl
--I l1f)mel~11rtnnhtl (Unit Vector) "'flmfl!~elfl1d':I11'1l1tJ~iil'Jll.1'i1fl''Vl1'l'IJtl'l"f'l r !It'!: F
'il:Mil
u =E. =~i _ ~ j _ 24 k
r 28 28 28
l!flIlnel~II'J-:J (Force Vector) tiltl'l'illfl F ii'1JU1~ 70 Ib ua::iiVifl''I'11.:J'j:;1J1~£1 u i'l,ru
F=Fu=70 Ib(~i - ~j - 24k)
28 28 28
= {3Oi - 20j - 60k} Ib
U"'~'llu~u~ 2-40 ('J) lllJ!lt1'~'l'i1fl''Vl1'lVil~'illm:U'lJ~fl~t1'll.11)fll~'J:'1,dl'l r (l1~tl F)
"",-<vd..",. "":'v"'; , ~&.
u,,:ufnnnfl'IJtl,n:uu~fl~'VllJ~9l!'Jll9lU'Vl A'illflUl'ltJfltJ'lJtl'lllfl!9ltlW!U.::Jl1UltJ
a =COS-1G~) =64.6° .
~ =cos-
1
( ;!)= 107°
(
-24) .
Y= cos-
1
28 =149°
47. f1~afi1Jfi 2-16
ll~·h'!1J"*~l1d~fli:1l1lU~l1~ 2-41 efl) tJfl<Hl~r1JlJl-lriJ'U1~tJ!fI!Di:1 AB th!!~-l'1Jil-l!fI!DfI~fl'j":
yj~~tJYl:'Jf)~ A ~il F ; 500 N 'il~~~fI)l~X 1'U~l1nfliYlilll'U':i:1J1J~n~mfl
A (0, 0, 2)
~
I2m
ly
y
l cos 45° m
x
(0) ('I)
'JU~ 2- 41
" .
~'" .111m
'illO~U~ 2-41 ('J) F lHiffYll.:J!~(nn'Un1Jnfl!Yldm...jl'llU'YIU.:J r ~~i:11fl'illfl A 111 B
110IVl61'J~!vllllml-:l (Position Vecto~) ~n~'Uil-l~91U(I1tJ'Uf).:J!fI!Di:1 ~6 A (0, 0, 2 m)
. UfI: B (1.707 m, 0.707 m, 0) ~.:J,r'U
r = (1.707 m - O)i + (0.707 m - O)j + (0 - 2 m)k
= {1.707i + 0.707j - 2k} m
fl'llJl'Hl'YI1U~.:J~il(J.nl~tJfllfl A{ - 2k j' m" 1'11lJ!!fl'W Z, {1. 707i} m IilllJUfl'W x Ui:1:
{o.707j} m flllJUfl'U y l11cY~ B
'U'.ll91'Jf).:J r ~il
r = ~(1.707)2 + (0.707? +(_2)2 = 2.72m .'.
d d t
I1flIVl6'J'YI'U-:l'YI'U1£J (Unit Vector)
u=E. = 1.707 i+ 0.707 j - ~k
. r 2.72 2.72 2.72
= 0.627i + 0.260j - 0.735k "
I1fllvm11m (Force Vector) l~il.:J'illfl F = 500 N 1m: F iiVlff'V1l-l U 'il:ll9i'il
. -..)
F = Fu = 500 N (0.627i + 0.260j - 0.735k)
= {314i + 130j - 368k} N : ,i
'illfl!!~.:J~iltJ.n 'l'/lril'U<ul91'Uil~ F ~il 500 N 11'1(J~
F=~(314?+(130)2+(-368)2 =500 N
,
Vltl'U
VleJ'U
45
48. 46
fl'::>aEi1un 2-17
u'i~~tl~l'.Hfl!1J(l F.;.B = 10? N um: FAC
= 120 N 1Jm~!!'111'U~ A ~~~U~ 2-42 (n)
. 'ilJj1':!'J'U1~'Utl~u'i~~YnfVim::vlTVi~~ A
z
y )-----r--y
B t4, 0, 0) ./"-___--Y
x
(0) ('1/)
'j,j~ 2-42
'IJ
"'''' 0
Tlim
!!'i~~'I"nj F !!iY~I'i,jmlyJyjnl'U'i,j~ 2-42 ('U) iYltJ1'ibl!!iYfII!!';ildl'U';iUnfllfltlfl'U';i::1Ju'Vlnfl
R <u <u ,1" <u
mn lf1f.Ji~lJ!!miY~l1iYlJfl1'i F !!(l::: F !1I'Unfllfltlfl'U'i:::1J1J'Vln~mn!!allJ1fl1!';ilr.i'flfJ'iliYtllAB AC
Ylft'1'll1'Uill F U(l:: F flm:::1Jlf1ufl1';iiY111nn!fltlf"1~llnbu U U(l:: U flllJ!fl!1JCl
AB AC ~ • AB AC
nfllfltl111~lm,bud'il::M'illflllfl!fltl'h:::1J9hUl1U~~iYil~flaeN r tw::: r 51~t)~'iU~ 2-42 ('U)q AB AC <u
ffll1i'U F 'il::Mil
AB
rAB (4 m-0)i+(0-0)j+(0-4 m)k
{4i--:4k} m
fAB ~(4)2 +(---4)2 =5,66 mv
100 N (rAB) = 100 N (-.Li--.Lk)
fAB 5,66 5.66
{70.7i -70.7k} N
49. ' rlTHfu F AC ~~1~il
rAC = (4 m - O)i+(2 m-6)j+(O-4 m)k
= {4i+2j-4k} m
rAC ~(4)2 +(2)2+(-4)2 = 6 m
FAC = 120 N (rAC) =120 N (~i+~j-~k)'
rAC 6 6 6
{80i+40j-80k} N
FR = FAB + FAC
= (70.7i - 70.7k) N + (8Oi + 40j - 80k) N
= {150.7i + 40j - 150.7k} N
FR ~(150.7)2 +(40)2 +(-150.7)2
217 N
2;9 waJ;]rulBualna1S (Dot Product)
hnJl-:J flr-:Jl-Wffi'i ll f'flffll{1J::ii fll'l'l1llJlJ'J::'VIil-:J !ff'UffV-:J ny'U !'If'U n'l-:JrivtJ~'IJ'Ul'U'VI~V~-:Jll1fl t1U
n'Ul!ff'U rll'V1i'uifty'l1l ffV-:Ji] ~ffllJ1'lfl1ill11fl ruii ~ !deJ'l1Jlml~~ll!'li-:J!'l'Ulflrull1~~ltJ nvirll'V1fu
'h,,HJifty111ffllJii~1J~vll1~ellfl vll1t11'ltl-:J1i1~fll'l'l'11'l!1 mlleJ1'lfltJUf) l~fJthmJ'lJeJ'l fH'lf,] ru fltl 1ii
lll'Wl::-ffl'V1i'Ufll'lfl run fl !llV{ffVIn fl !llv{ 1irll11i'u nnifw'111'1l1-:J1'l'U~ u
J:-H'lflru'Uvmmllv{ A !I~:: B !~tJ'UM':h A· B 91'liiDtJllJil~H'flOl'IJtl-:J'IJ'Ul~ A !Ii:'!:: B !tTI~~ , ~
lfl'lflfJ"r'IJV'llJlJ e 'l::whlril'U'I1l'lJ'lfi~m.J~ 2 - 43 1~tJ!~fJ'U1'U'l1l'IJVlfflJfll'l 1J::M:i1
q " 'lI "
~' .~.'IV~'7'II't:~~~"",~~~",
".', rcA." !,.~B>= 'ARcos "!!.~_" ..,.....___,,;i
(2-14)
.d v d d , ..J, d'
!lJtl 00
~ e ~ 1800
f-l1:'lf,]OllJ.fl1J::!'ltJflVflVtJl'll1'U'lll f-l"f,]OlUUUff!fli:'!l'l (Scalar Pro-
duct) 'Uv-:Jnml'ltl{ !dVl1JlflJ:-.l~c1'Wi~-:Jflcill!1J'Uff!fla1fhn'lfnm~ltl{
nnms16nu (Laws of Operation)
1. fltl'ffc11J~ (Commutative Law)
A · B = B·A
.. B
'J1J~ 2-43
"
47
i,
!,
50. 48
2. I1T:i~tu~1t1tilmn{ (Multiplication by a Scalar)
a(A . B).= (aA) . B = A . (aB) = (A· B)a
3. 110m'jm::1l1tl (Distributive Law)
A . (B + D) = (A . B) + (A . D)
':itl~T:lJfll':il1fl1l'1tl11m::;'lJ1J'Vlflflillfl (Cartesian Vector Formulation) ti'IJm'j~ 2-14'
1il11~1:'l~tu'IJtlluii1:'l::l1fH(;ltl5'1'l'ct'l'H'li':HJlW'l::1J1TVnr~mf) ii'1tl~l1l'li'W i· i = (1)(1) cos 0' = 1
lm:: i· j = (1)(1) cos 90' = 0 1'W'Vh'WUlr~tnn'W
i . i = 1
i·.i = 0
j.j = 1
i· k = 0
k· k = 1
k· j = 0
'W1l1'ltu1 ~(1~ru'IJf)'Hlmflf){l~"l A U(1:: B 1'W~U'lJf)ll1mfltl{l'W'l::1J1J'Wnflml1 1l::1~11
A· B = (A) + Ayj + Azk) , ~B) + Byj + Bzk)
= AxBx(i . i) + AxByCi. j) + AxBJi . k)
+ AyBXG . i) + AyByG . j) + AyBzG . k)
+ AzBx(k . i) + AzBy(k . j) + AzBz(k . k)
(2-15)
ii'lJ'W m'll11 ~1:'l~tu'IJtll11f)1(;lf)ll'W'j::lJuYl«~'inf)titl'l11f)Ifltl5~run'Wfl1J.j fflljU,'l::!1~JJ ~-L...y
11(1:: z ~(1a'l'l1~M1l::liJ'W1(1'IJ-W'lfflru(1) ' Idil.:l1l1f)~(1a'l'l1IiJ'Wtilf)(11{ ~.:I~f).:I'J::;,r~'J::;l.:lhj"Hj ·
;~~I(;ltlf'H~'1'H'lbtl1'W~(1a'l'l1J'W
m1'ih::;f,!flvl1.jj(ApplicatioI!~2 ~(1~ru1'Wl'lflf)(1fYl'ffflfiJm'ju'l::~f)fl1i~ri1rltytlQtif)~11l-
, 'iI , r! <! ~ 'V dOJ tV
. n fll"Ji:'f11'-U;!:ln::;'I111'll1fl1l'1tl':iiHl'll1flll'ltl':i'l1':iellnnlll'U't1I'1f1fl'U (The angle formed
between two vectors or intersecting line.) l.l'IJ eoR'ItlQ'l::w.i1'1ffl'Wl11.:!'IJtl.:!l1fH(;ltl5 A U1:'l::
B ii'':!~U~ 2-43 tillJ1'ltl1111~1l1f)ti'IJm'l~ 2-14 hw
e=cos-l(~:) 0°:::; e :::; 180°
oR.:! A· B 1111~1l1I1ti'IJm'l~ 2-15 t11 ~ . B = 0 U(1:: e= COS-I 0 = 90' Uti~.:!11 A
v
(;l~I'in flfllJ B
®tl'lf'itl1::;fltl'lJ'Utl-3l1flll'ltl1'U'Ul'U!!(l::;Ill-3'inflfl1J!I'U1I1:l'U (The components of a vector
parallel and perpendicular to a .line.) tllrlu'j::l1tlU'lJtl,mmfltl5 A ~'IJ'Wl'U'I1jtllill'Wu'Wl
l~f.nnlJlff'W aa'ii'I~U~ 2-44 iltll'IJ11 All 'lJru::~ All =, A cos 8 1'W1J11flr-3tl'lrlU'l::l1tllJi1'utifllM
1~um'l'lmu'IJtll A lJ'Wlff'Wm-3 th¥iffl'l11'IJtllll'Wllffm::u1~ul1f)!(;ltl{'I1~.:!;"nb£J u ii'lJ'W I~f)
U = 1 tillJ1'Jfll11 AII1~Uil'j.:!1l1flt-l(1~ru (ti'IJm':i~ 2-14)' J'W~f
51. , !
All = A cos e= A· u
'·;~,fu fl1~mU1:1'lfHn{'lJtl'l A 9l1lJllU111:fU'Vll'1~'il1f11!-1f1t)ru'IJtl~ A llfl:::l1f1l9ltl{l1~'1l1lbU u
~~ij!mri11vU'i'lft'V11'1'IJtl'l1l'U111:f'U 'lltl:cr'l~f1Jl t11I!-1mlVl1l1V'UU1f1 All 'iI:::lJ'i'lft'V11~ll1iitl'Ufi'u u l'U'lJru:::
~r-ifltlVlilV'UflU A ll 'iI:::lJ'i'lft'V11'1I;)'i'l'llllJnU u tl.:]ftlh:::f1tlu All 111:1'~.:]l'U~tll1f1l9ltl{~'1d
All = A cos eu = (A · u)u
,j'V~.lf19ltl.:]fttl"i:::f1tlU A ~i~mf1nUllU111:f'U aa' ~.:]~tl~ 2 - 44 1~tl~'iI1f1 A = All + A.L
~~,f~-A.L ;= A - All iilTIf11"i1:1'tl.:]iTI~'iI:::'VIl A.L iTIll"if1'V1l ell1f11!-1f1t)ru e= cos-
1
(A· ulA)
~~"ru A.L = A sin e ril'Uflf)11i11~':] fitl t11'Vl"ilU A ll ~.:]"r'U ll1f)'V1qfJQ'lJtl':]'W1Jllf1l~u'U
~ v' I 2 2
(Pythagorean Theorem) 11:::1~11 A1. = j A - All
~ _al
AII=Acos8u
'jll~ 2-44
"
/
PlJaa10n 2-18
lm'ln·H)lJ~'1ltlYi 2 - 45 (n) lnm:::'I'11~lUll"i.:]lullU1"ilU F = {30oj} N lW:::l'hYil,JlJ
'lJv~lm'lmtlu 11'1'V1l'IJ'W19l'IJtl'lll"i'lciVU~'1'IJ'Ul'Ullfl:::i'l'nnnu;'Uril'U AB
B. F= rlaOj)N
I
x (a) x
(0)
49
'-'-'-- - - y
52. 50
""". 0
111m
'U'Ul¢l'Utl-lU'.i-lV'O!'J'Utl-l F OlllJ AB ~i'ilLl'hnmH'lfJru'Utl-l F u"::nf)LOl'Of'l1~-l'l1'.i1!'J U
B
~U!'J11J'Vii1'1'l1-l'Utl-l AB ~-l~U~ 2-45 ('1) Ldtl-l~lf)
r ~ + ~ + Th . .
"B =.J!. =~ =0.2861 + 0.857J + 0.429k
rB (2)2+(6)2+(3)2
FAB = F cos e= F ' "AB = (300j)· (0.286i + 0.857j + 0.429k)
= (0)(0.286) + (300)(0.857) + (0)(0.429)
1
= 257.1 N;f -r"I" )9l' "nn l: ,
rdtl-l~lfH,md'J'Ui:1'Lf)m{U1f) F iiVii1'l'lVH~!'Jlnu U ~mJ~ 2-45 ('I)
AS B ~
UC1'~-l F l'U1.unmOltlflm::uu~n~mf1 ~::hlilAB -
FAB = FAB"B = 257.1 N(0.286i + 0.857j +0.429k)
= {73.5i + 220j + lIOk} N
U'.i-lVtl!'J~-llllf)~-l~U~ 2-45 ('I) ~-lJ'U
F.L = F - FAB = 300j - (73.5i + 220j + llOk)
= {-73.5i + 80j - 1I0k} N
'I'U1~'il:;'I111~'illf)nm9ltl{d'l11tl~1f)l'lt]'H~~1i11ma!'J'U (Pythagorean Theorem) ~-l~U~ ~-
45 ('1)
F.L = ~F2_F2AB
~(300)2 -(257.1)2
155 N
53. viil1'U~1.l~ 2-46 (n) lnm:;'I'h~ltJ1L':i'l F = 80 Ib ~l.la1tJ'l'ifl B 1l'll1llJm:;'l1il'l F flU
~ri1'Uvifl BA <nlJvf'l'IJ'Ul~'IJfl'll!~'lriilV'lJfl'l F ~'IJ'Ul'Ul!a::~'lmflfl1J BA
z
..~~~==_2ft-::::,7' S77""/_'... Y ~~r-----Y
2ft
c x
x
F= 80 Ib B '
(n)
~tJ~ 2-46
"
"'...l1i'fl1
3;!:IJ (Angle 9) nflll'lflf'J"::1Jl'i'll!'I1,j'll'lllJ BA l!~:: Be 'I111~'illfl
r~A = {-2i - 2j + 1k} ft
r
Be
= {-3j+ 1k}ft
cos e rBA · rBe _ (- 2)(0) + (- 2)(- 3) + (1)(1)
fBA f Be - 3M
0.7379
e 42.5° I'HI1J
' ( v O".J ,
1I~'lUtlU'ltl'l F Components of F) 1:l''J"1'l1:l'lJfll'Hlfl1l'1fl'J'I1'U.:J'I1'U1Vl'lllJ BA U~:;l!'J.:J F
1U~U!1fl!l'I{)flm::1J1J~n~mfl
F
rBA = - 2i - 2j+lk = - ~i-~j+.!.k
~A 3 3 3 3
= SO Ib(rBC
) = SO( - 3
j
+ 1k) = - 75.S9j + 25.30k
fse .JfO
0 + 50.60 + 8.43
= 59.0 Ib
B
('I)
51
54. I
I
!
I
i
I
·f
I.:
, .
52
. FBA = 80 cos 42.5" lb = 59.01b
'lJU1~'lJCN!I'l'ltJUtJ~~'l'il,1f11:l'llJl'ltnn1.,n~tJ~'j'l'l'11'l~~1f1U!ii&l
Fol F sin e
80 sin 42S
54.0 lb
'Y!1u1~tJ'I'lflavhn1f1l~tlu (Pythagorean Throrem)
F.L ~F2 - F~A = ~(80)' - (59.0)'
54.0 lb
.
./
C;UI'IJ -
:)
55. Pi =600N
---'-.,--- x
·n.J~ 2-1/2-2
"
' 2-3 'il.:Jm'lJ'Ul~'lJtl.:JIl'l.Hl'V'l1r F = F + F 'l1lJ'I'l.:JVifl''Y1l.:J i
R 1 1 2
1~trr~1'UVifl'm'Ut4JlJ'UlWf)1 'illfHtfl'U X yjiil'i11J1fl
_2-4 'il..:Jm'J'Ul~'Jtl..:J1!'l·Hl'Wli F = F - F 'l1lJ"r..:JViff'l'n1
, -, '. R I 2
1~trr~1'UVifl'm'Ut4JlJ'U1Wf)1 'ill fl Ilfl'U X yjiiril1J1fl
y
F, =250 Ib
''-."
F2= 37S1b
.,
40lb
'j'll~ 2-5
"
. - -~,
2-6 'il.:J'1l'IJ'Ul~'lJtl.:Jtt'l.:Ja'Wli F = F +F TllJ"r.:JVifl''I'1l.:J. R 1, 2
1~Ul~l'UVifl'l91llJ t4JlJ'UlWf)1'ill fl Ilfl'U u yjiiril1J1 f)
2-7 'il..:Jbbl91f)1!'l'l F tl'OfHlh..!'O'lfhh:;fltl1J~mjm:;v1ll91llJllf)'U
" 1
u ttel:; V 'illJ'I'l.:J'1l'IJ'Ul~'lJtl.:Jtl.:Jrllh:;fltl1Ji.:Jflrill
2-8 'il.:JUl91flll'i..:J F tl'OfHtJ'Utl..:JrlU'j:;fl'01J~Oum:;'Yh91llJ1!fl'U
" 2
u Ilel:; V 'j1lJ'I'l..:Jm'IJ'U1~'Jtl..:Jtl..:JrlU~:;fltl1Ji.:Jflri11
. ,
, .
" I
'jU~ 2-6i2- 7/2-8 :1
2-9 t.~;H."'J;''i(V-Groovoo Wheel) onH'i,u",,, ,I
t1l'il..:Jd)~tml'Ut~'Ul~.:J 200 Ib m:;vlll9imi'tl ;..:Jmo.:Jrlu'l:; / .. . I
fltl1J~tlU'lJo.:JU'j..:Jm:;vll911lJllf)'U a Ilel:; b cJi..:J~.nnfln1Jil1'U~ f
ii~tl.:J i
I
1
56. 54
..
2-10 1JlU9lflU'Il 60 Ib tH)mlhHl'lrllh~fl{)1.l{jmJm~i'h
1'l1lJllfl'U u w''l::: v ':i1lJ~-:I'H1'IJ'U1~'IJ!)-:I!).:Jfi'U),::f1!)U~,:If)ciT1
v
'JtJ~ 2-10
"
60lb
2-11 "lJi'h'h1lii~u),'la'Vnf F = 110 Ib ~-:lmf1n~lui~B
'iJ-:I UI'l fl U),':I'd'B BflIU'UffB.:J tl.:J fi'lh~f1tlU citlU ('IJU1'UU"~~'lmf1
v '"
f1UIIf1'Um::~f1~I),tl aa)
v
* 2-12 1'l::'IJB),B':l'ruu)'':Hfllijml'lfftl.:J F = 500 N u,,:: F =v 1 2
300 N t111'J"a'Vnj'lJtl.:JII),.:J~'lflci11ilm~i'hl'Uiif1*".:J1uu'U1
~.:Jm'j~ii'lJ'Ul~ F R 750 N 'iJ..J'I11l,JlJ e IW~ 0 'lJtl..J!flIU"
~..Jflcil1
2-13 !!':i..Jl'U!!Ud~..J F = 60 Ib m::yhluiiftvi.:J".:J~'iJ~ A UU ~, ,
lrmmtlUfftl..J<1lurilu 1ll'l11'lUl~'ltNtllfhh~f1oU'vflfftN'ltll
F l'Uiiffl'lllJl!fl'U'UO..J;u~h'U AB u,,~ AC nl'11'Ul'lril e = 45
2-14 1!),ll'UU'U1~.:J F = 60 Ib m::yhl'Uiif1vi..J"..J~'iJ~ A, ,
u'Ulf14'..Jf1JtluffB..J;'Uri1'U 'iJ..J'I11l,JlJ e (0 :0; e :0; 90
0
)
'llOl;Uril'U AB rritll'11tllrllh::fltlU'lltl..J F m~Y]19t1lJUfl'U
AB iiril 80 -lb ),llJ~l'll'U1~'lltl..Jtllrl1.h~f1tluu),.:Jm::y]l
v
9l1lJl!m.j'llOl~uril'U AC
2-15 I!~UU1l~f1m::y]1~lUU),..Jfftl..JU),..JYl A !w:: B ~..J~t1
t11 e = 600 ' 'il..J'I11'IJ'U1~'Jtl.:J,:,,,,a'VHj'lltl..JU),..J~..Jfftlilif ':illJ-vr..J
iiff1'11.:J~1~l'lllH~lJ'U1Wm'iJlflufl'U X iiiirilulfl
- - -- - x-----y
FB =6kN
57. 50 Ib ililflltJ'Uil~rl1.h::m)'lJu~HJm::l'hI9l11J
1m:; y'
y
'j,j~ 2-16
"
2-17 tl1-lfl'j:;'Vl1lJ'UYf'UlliJil.:J F = 20 Ib 'il~U9lf1U1.:J'ifililf1
d'JUffV,W-lfllh::f1illJm::'I'i'119l1lJU'Ullff'U aa 1m:; bb
2-18 f.J-lflU'j'::f1f.JlJ'Uf.J~It'j''1 F m::'I'i'119l1lJltUllff'U aa !,l'htllJ
30 lb 'iJ.:J11l'U'Wl ~'Uil'l F It"::il'l rllh:: f1 illJ'Uil'l1t'J'I 9l1lJ UUl
I~U bb
b
F
v
2-19 U1'1vl'l'fff.J'Iii'UU1~ 10lb It,,:: 6Ib m::'I'i'1~il1'111'l11'U
fhUU1~lJlf1~'l'~'Uil.:JIt'J'I clYnr~l'1It'l11'U'ffllJl'jf)rlJi~ ~il 14
lb Jll1l~lJ e'J::wh'lIt'J.:JIil,:mcill
*2-20 'il'l'l11~lJ e (0' :::; e:::; 90') 'J::'I111'ltl1'1vY'I'ffil'l
.d d ' <V <I 0 I .0<:1' !I.I ..J
IYHl'YI'UUl ~'U il.:J II'J'I "'I'l1l m:;VI1 9l ill~ It'I11'UlJ fll'Uil tJ'Vl 'ff9l 'JllJ./' ,
'nJ~ 2 - 19/2-20
"
55
2 - 21 rt11lf1~'1'illf)«'U~'Ul9ltJHlff'Ut;'iilf)'ffil'llff'U A tI,,:: B
Iff'UI~ilf1 A tJf1m:;'I'i'l~lu U'J~ 600 lb tI,,::iiVifl' 60
0
'il1f1
U'U1'JllJ 'ill111U'JI T 1'Ulff'Ul;'iilf1 B f111'ffll~lJ~'UfIf1~~lrlil
e = ,20' rlll1r~ll1~f11'Jru'if U'J'I~'I'l{lJurt11'il::iiVl~Yj'l~'U1'U
..:... QJ 0 <V d'.J'
U'Ul ~'1 'Uilf1 'illf1'U'U 'il'l fllUlrul1l'U'Wl9l'Uil'lU'J'I"'I'l1iU
2-22 IJ:YltJf1~'1'illf1~'U~'U19ltJ1'lflffUI~flf1J:Yfl.:JlffU Au,,:; B
IffuI;'iilf1 A f)f1m::'I'i'l~ltJ 11'J.:J 600 Ib ua:;iiVlfl' 60' 'il1flll'U1
11lJ f11Umr~{m:;'I'i'llJ'Ut'ffl!'l'htllJ 1200 lb 1'UYifl'Yjl~'U1'U
lIUl~'1 'iJ'I111U1'1 T 'IJ'Ut-ff'Ul;'iilf1 B tm::lJlJ'ffil~rHlil'l e
600Ib
'j,j~ 2-2112- 22
"
2 - 23 fil'1'i'1111li)~u'JI 20 Ib lJ'U~il'Ui,r 'iJ'IIII9lf1u'JI'ifililf1l11u
fllrllh::f1illJUiltJm:;'I'i'l (f1) 9lllJLIf1'U n Uft:: t ('U) 9l1lJUf1'U
xu,,:; y
*2-24 fil'1'i'l111liiflU'JI 20 lb lJ'Ul'iilUi:U' 'iJlU9lf1U'JI'ifililf1
tll'Uillrlll'J::f1illJUil£Jm:;'I'i'l ef1) 9l1lJUf1'U n Uft:; y ('U) 9l11J
Uf1'U xU,,:: t
y
n
20Ib
58. 56
2-25 0'1 e = 20° lla:; 0 = 35° 'il~'I11'11'Ul~'IIeJ'l F _ua:: F
, I 1 . 2
IYieJh1ll'j'lEl'Y'nji'i'll'U1~ 20 lb lIa:;iiYli1I'nlJilf)'U X Viiifi1mf1
2-26 51 F = F = 30 lb 'il'l'l11lJlJ ella:; 0 IvieJ'll1lm
1 2 , q
ElVOjiiYli1>lllJllf)'U X Viii~lmf)lIa:;ii'IJ'U1~ F = 20 lbR
'jtl~ 2-25/2-26
"
2-27 'il~'I11'IJ'U1~lIa:;Yli1'Vl1~'lJeJ~YlaElvi1j F = F +F +F
v I R 1 2 3
'lJfl~II'j~vl~ffllJ l~tJI~lJu'jf)'I11YlaElv/11 F' F +F ml1'linJ
1 2 '"
llUU F = F'+F
R 3
lIuuF = F'+F
R 1
y
F~=20N
tid '
l l1Z-Z7IZ'-Z8
2-29 'il~'I11lJlJeJflf1LI1J1J e (o~ :s; e:s; 90·) thl1rU'JtJ~tJ
AB Ivitl'l11I1';j~'lull'U1'j11J 400 lb iieJ~rllh:;f)flUrifl(J 500
lb iiYli'fI'l1'l'illf) A iurr~ C ';jllJvf~fl1'Ulru111tl~rlU'j::f)flU
v
rifltJ'lJfl~II';j~f)';j:;'I'll>lllJ'ii'Uril'U AB fhl1'U~ <p = 40·
2- 30 'il~'I11lJlJtlflf1LIUU <p (0· :::; <p :s; 90·) 'j:;l1iWUfl~fl
AB 1m:: AC Ivifl111u'j'l'l'UII'Ul'J1U 400 Ib i'ifl~rlU'J:;flflU£ifl(J
600 Ib ~f)';j:;'I'l1~~iU'Vl1~~l'W'lfltJJjtl'l'UYli1m~'il1f) B lurr~ A
fhl1'U~ e = 30·
4001b
'jtl~ 2-29/2-30'tJ
2- 31 'Vitl'W'1!~~fla1fll~(J'jf)a1Mtl~r1'U A IW~ B 'il~'I11'IJ'UWI
'Utl~ll'J'lll~a1flvllfftll Fila:; F 5111'J~El'Y'njVili'f)lfll'Jii'IJ'Ul91, A B
F = 10 kN ua::iiVii1V11~~llJllf)'U x nll1'U~ e= 15°
R
v
*2- 32 51YlaElvnf F 'lJtl~II'J~vl~fffl~f)';j:;'I'll~f)'Vif)'U'1!~ii.Ylrl'., R
1'l1lJllf)'U x Viii~lmfllla:;ij'U'W1~ 10 kN 'il'll11lJlJ e 'lJf)~. -, q
Iflllia~ti~~~f)vU B Ivif)'ll1l1'J~ F '1'Ulfllliadiifi1UtltJ~ff~t.f B 'V q
'jllJvl'l'l11'IJ'U1~'IJfl'lll'J~'1'UII~a:;lflllia-ff1l1i'u Il1lPJ fll'jruU
'jtl~ 2-31/2- 32
"
59. 57
'UEl!U)! 800 lb ' *2-36 ~!!!ff~! F, F Uft:: F h'!'Jt1nfll~H)'fb.!':r::1J1JVln~mf)1 2 3 <u
<v.,. ~ dev c:: ..,.
2- 37 'il!'11l'll'Ul~'IIil'l !!'J'lft~l1 !!ft::Vl ffVll!Vl1~Vl1'U!'IIlJ'U1Wfl1
y 'ill flI!f)'U X vilhi11J1 f)
y
F3 =750 N
-------------~8Z------------ x
"!tJ~ 2-33
"
F2 =625N
QJ oS' "'" dv ~ "'"
2-34 ~'l'I'Il'll'Ul ~ 'IIil'l!!'J'lft~ 11!!ft::'Vl ffVl1.:J'Vl1 ~'Vl1'U!'IIlJ'U1Wm
'illf)Ufl'U X Viii~i11J1f)
"!tJ~ 2 - 36/2-37
"
y
800 N
----~~~----~~------ x
."
"'.. ,
'illfl!!fl'U x 'VllJfl11J1 fl
y
IV.,. """ dcv ~ 4=>.
2- 35 'il'1'111'11'Ul ~'IIil'l !!'J'l ft~1l!W::Vl ffVl1.:JVl1 ~ IllllJ!'IIlJ'U1Wm
d ", ,
'illflUfl'U X Vl:IJfl11J1f)
y
50 N
-----------~~~---~-------x
t./,,'._
- -- .....----x
70N
Fj = 30kN
65 N
~tJ~2-38/2-39
60. I.
III
l
I, .
I
I
I
58
Q,.' ~ """ dV' ~ .co.
2-41 1l.:J'Y1l'IJU1~'lJV.:JU'J;jft'V'l1iUft::'Vlfl''Vl1.:J'Vl1~'Vl1'WI'lJlJtnWfll
lllfHlf)'W X Viiifilmf)
y
45°
;
:n18----'----!.... FI =200 N
F2 =IS0N
x
'JtJ~ 2-40/2-41
"
2-42 lI;junifUJ'Yll'li'v~ 2-1 1~£Jfll'J'J1lJV;jrfU'J::f)VUUV£J X.u .
11('1:: y IIUU~l'Y1~£JlJ flw·rl~v.:J LL'J.:J,Ylv1'Y1'1~II'J·:Hrl'nr
2-43 ~.:Jllniff.1J'Yll'li't1~ 2-2 1~£JflTj'J1lJV;jrfU'J::flVUUV£J X
U('1:: y UUU~l'Y1~£JlJfl'Wr:hutl,:m'J.:JLYlvl'Y1'1~II'J.:Jtfl'ffi
*2-44 ~.:Jllniff.1J'Yll'li'v~ 2-3 1~£Jfll'J'J1lJtI.:Jr1U'J::flVUUV£J X
11('1:: y IIUU~l'Y1~£JlJ fl'W~l'Uv.:JLI'J.:JIYltll'Y1'1~LI'J'ltfl'ffi
2-45 ~.:JLlniff.1J'Y1l'li'v~ 2-15 1~(Jf)wnlJv.:Jrfu'J::f)VUUtI£J X
LW:: y LlUU~l'Y1~£JlJfl'W~1'UV.:JlmIYlvl'Y11~LI'J.:Jtf~li, .
2-46 ~~Llniff.1J'Yll'li't1~ 2-27 !l9Wfll'J'J1lJV.:Jr1U'J::flVUUeW X
lm:: Y-!lU1!fil'Y1~(JlJ fl'W ~l'JV'lu'J.:J!Ylv1'Y1'1~u'J'ltf~li
2-47 ll'l'Yllil.:Jr1U'J::flilUUil£J X Uft:: y 'Jv.:Ju~a:iLl'J.:Jf)'j::'I'11
U'Wll~'Wl'Y1~mh::nu (Gusset Plate) 'UtI.:Jlm'lt1m"::l'n'W 'J1lJ
Ij . IV d'.d ~ I I V' . ;. 0' v {.
'Vl.:JU~~'l !!'J'lft'V'ffi'VllJ fI1!'V1lflU ~'W£J ~n £J
/
F2=4001b
./
x
':nJ~ 2-47
"
- -~-
*2-48 tIl e.= 60· lIa:: I; = 20 .kN ~'l'Yll'IJ'W1~'lJtI.:J1l'J.:J-tf~li
, I 1 ;
ua::V1i'l''Vl1.:J~1~o/l~lJtnWf11~lm!f1'W X Viiirilmn '. .
2-49 ~'ll11'IJ'W1'~ F ua::V1fl''Vll'l e '1Hl'lU'J.:J F !VlvVi~::
'I'111li~ft tf~li'IJil.:JU'J.:Jvr'l ~lltlf)'j::'I'11U'W~::'IJt1~iifillyh nu ~'Wtr
r--------x
40 kl'!
'JtJ~ 2-48/2-49
"
61. v v
.>2i5l~)!':i'J'I'r.:)t11:IJm:::1111J'W!ll1!1'lJ'L6'1JthI1Ufl ll':)Ml'lJ'WJ~U~::
. fitYf11~ e'lltFl F, !viil~'il:::'Vhl11!!':i.:)i:l'Vnjii~fl''Vll':)lll:IJ!!fl'W x'
~~fh1J1fl!!~::ii'U'W119l 1 kN
*2-52 61 F = 300 N !t~:: 8 = 20° ll..:J'I11'U'WlI9lU~::'
du 1 ~...... ... QJ (J'
iifl''Y11..:J'Yl119l'VI1'W!'U:IJ'WlWfl11l1fl!!fl'W X 'U il.:) U'J..:J (I'I"l1l'Uil..:J !t.'J':)
y .J Q V QJ ':( <V
'tl..:Jt11:IJ'VI m:::'VI11J'W!'I'l1!!'U'W'J1J'Wl'l1'Wfl
45°
F3 =200 N
:ilI---;--I... - - x
Fl
'Jt1~ 2-51/2-52...
J>
. x'
59
2-54 ll':)'I11'U'WlI9lU(I::lifl''I'll..:J e 'iUil.:) F !vl8~1l::.yh111'!!':i':), A
rv ,(~ """ "'" d, I .".
(I'I"l1l:IJ'VIfl''I'll.:)lll:IJUfl'W X 'I'l:IJfllU1ntm::lJ'U'Wl19l 1250 N
2- 55 61 F = 750 N !w::: 8 = 45° 1l..:J'I11'U'WlI9I!!(I::iifl''VI1..:J
• A
iii'~l'11Ut~lJl..nWfll'ill fHLflU X Vijjrll1J1f1~fl-:J U'j~ a'Y4lif1~~vll
'Jt1~ 2-54/2-55
'II
* 2-56 11'J..:J-vr..:JmlJm::'l'l1U'Wt'Vl1U'U'W1mh'l1'lrn ll'l'l11'U'W1191!!(I::
Yifl''Vll..:J 8 'Uil':) F !vlil~1l::Vll1'11!!'J'Ifi''I''lTIiiYifl'~l:IJ!tn'W x'
· ,
viijfi11J1fH!t'l::ii'U'WlI9I 800 N
2-57 61 F = 300 N U(I:: 8 = 10° M'I11'U'WlI9I!!t'l:::Yifl''Vll'l
• 1 ,
..c:t <V "" Q. d.:::t I _ <V.I 0
'VI119l'VI1tH'U:IJ'W1Wmmfl!!fl'W X' 'I'l:IJfl1U1n'Uil..:J 1I'J..:Jt'l'l"lfifl'i::'I'll
v ~ v
2-53 !!'J':)'I'l':)t11:IJ m :::vl11J'W1..:J1I'111'W ll'l'l11'lh..:J'Uil..:J fllff1'1111J 1J'W!111!!'UU1mlTI1Ufl
'U'W1191 P ~rvitl~1l::vli111'U'W1I91'Uil..:J!!'J'Ifi''I''l1l1lili)'W 2500 N
1191(J~u'J..:J p lJYifl'1t1vm~i'1'W'U11iiil -
'Jt1~ 2-53...
y
I
F2 =200 N
x'
~________________ x
62. 60
2- 58' 'il'lll'ffmll~'1U~a:::ll~'1~m:::vll1Jtm11ll'IJ'Ufuth"T1rflll1ilri ' *2- 60 1l'l111V1ff e 'lJil'ltfHU~Ua:::flTJ~~ F tviilvll111ll':i'la'Wi. 'U I 1 .
'1 _I ~ """ <V &:j .... ""'" X ~ ~ .eI
1'U~Unfl1~il'J!m:::UU'Wfl~ll1fll'll'!Htfl'U X lltl::: y 'J1lJ'Vl'l'VI1 lJ'VlJJYj-ru'UI'Ull'U1~'1Utl::lJ'IJ'UW18illLN---
'IJ'Ul~Hta:::V1ffm'l e'lJil'l F tviil~1l:::vll111u'J'Ia'WiiiV1ffl'lllJUfl'U1
X' Yiiir111J1mttl::ii'IJ'U1~ F t'vilnu 600 N
R
x'
F3=100N ..
'JU~ 2-58OJ
2-59 U'J'Im:::vll~9lt~£J1n'U"f'l'ffllJm:::vllU'Ut,nvll111til~U'J'I
a'Wi F = 0 til F = 1. F ua::: F vlllJlJ 90° 1l1fl F
R , - 2 2 1 1 q 2
~,mJ 1l'l'VI1'IJ'U19lYi~il'lfl1'J F Uff9l'll'U!'VlillJ'lJil'l F tttl:::3J3J e'" J 1 "
y
OJ
2- 61 1l'll11'IJ'U19llttl:::V1ff'Vl1'l'IJil'lU'J'Ia'Wi'IJil'lU'J'I'I'1'1ffl3Jm:::vll
U'U1'1tt'l11'U A rll'VI'U9l F = 500 N lttl::: e= 20°
1
y
400N
~'-------'----x
';iU~ 2-60/2-61
"
2-62 1J'Il11'IJ'U19l'IJil'lU'J'I F tvlilvllll1'IJ'Ul9l'IJil'ly.",a'Wi F
v d ' ,,~
'IJ il'l U'J'I'I'1'1'ffllJiir1l'Uil £J'Vl ~9l ti'iTVi I1J'UltJl~ 'J1lJ"''1l11'IJ'Ul 9lYi
.. UtWYi~9l'IJil'l F R
.,..----i~5kN
4kN F
63. "
61
F = 250 Ib 'il'lbb'i.Y~'1 2-66 ~llJ,rU Su~~~nUbfl~tl'lna'llC1l'1::flnm::'I'h~11'J!!':i.:J
60 N ~'Hn~'illnuihnhnaI'J1 D 'il'll'lllllJ:I'i.YI'l'lVifl'l'm~il'l
'illm::Uurinl'l ~ UCI::Uff~'1h.!lllnml'!tlTI'.m~uurinl'l'illfl
F=2501b
1tl~ 2-63
"
* 2-64 bl':i'l F m::'I'hu'..!l'Il.,/l'll'lllfliiel'lffll'j::fltlurimJ 40 N
nm'hl'..!':i::'..!1U x-y i'llll 'il'lU'i.YI'l'l F 1Ulllnml'wTI'..!'j::UU
rinYllllfl
Q., dlV Q,IV
2-65 'il'll'll'U'..!lI'l UCI::l.llJ U'i.Y~'1'1'1 fl''I'1l'I'1'11I'l'illm:;UUVi fl~'Utl'l
U':i'l F m ::'I'11U'..!l'Il.jl'lmlfl
z
--~--y
x
1tl~ 2- 66
"
2-67 'il'lIt'i.Y~'1uI'iCl::u'j'll'..!llll1fll(OltlTI'..!':i::uurinml1flmhl'll
v ,
U'j'lilVili F 'jllJ'Vl'll'll'U'..!lI'lUCI::lJlJIWI'l'lVifl''I'1l'lV1il'l 'illflR " IV IV q
1::UUrin~ fl fl'Vl'l11 I'l n ml'!tl1ii1'..!1::UurinI'l'illfl
x
Fj =8kN
1tl~ 2-67
"
~ 2-68 'il'll'll'U'..!lI'l UCI::l.jlJU'i.YI'l'lVi fl''I'1l'lV1il'l 'illm::uurinl'l'Utl-l
U1'1ilVili flmf'l11I'lnm9lil1d1m::uurin~'illfl
z
::::;,t-----y
x
'Jtl~ 2-68/2-69
'U
j
I
,i
I
;:
I'
i',
J
.,
.:.J1
64. 62
..
2-70 'il~Utll'l~U~~:;u'j'llu'jtlnf)ll'1tli'hn:;'IJ'lJYinl'llllf)..
.<::lo, d u ' ""'" IV
2 - 71 'il ~'I11'U'UlI'11W:;JJJJ 1Itl1'1'I'1'1 1'I'1'11::J'I'11 1'1'illm:;'IJ'lJWf)l'I'Utl'l
11'j~awi ~mf~111'111f) l~u1if1m:;uuYinI'lUlf)
z
~---.---- )'
x
'nJ~ 2- 70/2-71
"
*2-7 2 'il~'I11'U'Ull'1'UeJ.:IlJJJ I1tll'l'lill l'I'1'1l'lYill'1'ill m:;uuYi nl'l'UeJ.:I
u "
Im~W1i
z
75lb
y
55lb
'aU~ 2 - 72
"
2 - 73 m'Uflflm:;vh~11'J1I'j..:J"r..:Jtltl..:J~..:J'jtl ~..:Jtltll'l'll1l9i~:;!l~..:J
1'U~1In f) 11'1 ~fl'U'J":;'IJ'IJ'w n 1'1 m f) U~:;111~'U11'1'jlmr'1JJJJ 11'if1'1'1
" . .""'" '..... <U' """ QJ _ <S'
'1'11'1'1'11'1'1'111'1'illm:;u'IJwf) I'I'U tl'111'J"'1 ~W1i
z·
)'
. 2-74 l'iflm:;ll'1..:Jf.' f) m:;Yll ~, £J tt'j'l"r..:J 'ifllJ~'l'J"tl ~..:J1111JJJ
" " ,""'" dlU'""cv .d
1l'if1'l'1'1'1rl'l'11~'I'111'1'illm:;uUWf)1'I a, ~ ml:; 'Y 'UeJ'1 F tWtl
1 1 1 1
Yh11111~'Hlwim:;Y1lu'Ut'iflm:;11'1'1 F = {350i} N
R
2 - 75 t'iflm:;ll'1'1nf)m:;yh~ll'J1l~.:I.J.:I'ifllJ~.:I~lI 'il'1mJJJJI1'ifI'l'1
" '"
iIlrl'l'1l'1Yill'1'illf):J:;uuYinl'l a, ~ IW:; 'Y 'UU~ F trieJY1l1'l1
1 1 1 1
tt'J".:jawim:;Y1lu'Ut'iflm:;ll'1'1iifill'vi1 nUrl'Uu
"
x
Fi =200 N
'aU~ 2-74/2-75'II
*2 - 76 It~.:j.J'1'ifeJ~ F 11~:; F m :;yhvi A 11'l1'Jlf1l'1tt'J"'1awi
1 2
F = {- lOok} IbR
'il'1'I11'li'Ull'1 tW:;lJ JJ ll'if1'1'1ill1'I'1'11'lYill'1'ill f)
~:;1JuYinl'l'IJeJ.:j F
2
2-77 'il.:j'l11lJlJll'ifI'l'1i1lrl'l'11'1Yill'1'illm:;1JUWnl'l'UeJ.:jtt~.:j F, 1
11~:;tt'ifI'l.:j1'U~1I1'l1£J
B
- - -- -- y
x
65. . . .l.. "'.1 ' •
Lff.l1flm::Y1119l1tJU1'l F 'If'llHl'lfllJ1:;fltlUtltltlm:;'V11
~l1JLLflU;; k, y LL~:; z ~'l~1l 51'UU19l'lHl'l F i!iv 3 kN U~::
p,: 3D· 1l11vf.:J Y = 75· 'il'l'YI1'UUll9l'UV'lV'l1l1l'J::flVUVmJvf'l
2-79 Lff1t1flm:;vlll91ltJU'J'l F 91'liiv'lllllm1vUVVtJ F = 1.5~ x
kN LLCl:: F ,: 1.25 kN 51 ~ = 75· 'il'l'Yll'UU19l'UV'l F U~::
z 1
Fy
z
'JtJ~ 2-78/2-79
"
, ,
* 2-80 U'J'l F t1flm::vll'ViriluUU'UV~'YIVfltltJff'lYi A 51U'J'l- ~ ~
m::vl11uVlfl'Vm~.:J~1l tl.:Jlllh'::nvuvcltJvYlu'J::U1UH'Jl.:Jl y-z
.d, : """.d !IV
'IJ'IJ'Ull'l 80 Ib 'il~'YIl'IJ'Ull'l'Hl~ F !tCl::1J1H!ff9l~'Vlfl'Vl1-!l'Vll9l
.'il1m::uuVln9l a, /3 UCl:: 'Y .. '~
A
- - - y
x
801b
''':;.,
'JtJ~ 2-80
"
63
• ... F .J... 0'.1 '
2-81 fffl~1flfl'J:;'V1119l1 tlU'J'I 'b''IllMfllJ'J:;fltlUtltl Jfl'J:;
vl1fl111UflU x, y UCl:; z ~~~1l 51'U'U119l'UV'l F iifhL'yhnu
80 N ll~:: a ,: 60· 'JlllYl'l Y = 45· 'il~'YI1'UU19l'UV'IV'I1l
ll'J::fltlUriVJ~'lfl all
2-82 ffmt1flm::vl11'l1tlH'J'I F 91'1iitl~llll'J::m)1J(jvtJ F ,:~ ~ x
20 N HCl:: F
z
= 20 N 51 ~ = 120· 'il~'YIl'IJU19l'IJV~ F HCl::
F
y
'JtJ~ 2-81/2-82
"
...
2-83 U'J~Yl~ffv~F UCl:; F m::vll?itlfffl~ 51U'J~t1'V'lTI F
1 2 . R
ii'UU19l 50 Ib U~:;l.JJJ!!ffl9l~Yifl''V11~Yil9l'illm::uuVln9l a =
110· U~:: B,: 80· ~~21l 'il-l'Yll'UU19l'UV.:J F2 UCl::'l,PJUff9l.:J
VlflVl1.:JYill9l'illm::uuVln9l
z
lWH--r----y
FI =201b I
'nJ~ 2-83
"
66. ·64
b)'il_lIl~CfI_lnfll9lvf'j:;'4phm11l~ r 1~l~llfllCflVTI'U2:;'yD
VlflCflmfl mr:I'Hl'IJ'U1Cf1 LLi:l:;~lJ m,"CfI_l'Vl fI'Vl1_l'Vl1 ~'ill m:;DDVI fl~
z
v----cr------r--,;j-m- y
x
~tJ~ 2-84'II
x
2-86 'il_lUff~_lnfllPHlfDVf)Phu'YIll_l r lwpJnfll9lVTI'U'j:;DD
Wn~ll.lflU51'YI1'IJ'U1~Lm:;~llLm~_liifl'Vl1_lYil~'illm:;DDwn~
1--- - 8 ft - - --I
A
,;n.'~ 2-86'II
2-87 'il_l'Yl1ml11U11'lJV_l'1fmfl'U AB 'lJV_l1fm«f)1mH~mr~
~mflL9Ivf'j:;'4i11LL'YIll_l1'U'j:;DDwn~mf)'il1f) A 'lufk B U51
'YI1'IJ'U1~~lU
y
T
1.5 m
I~~~-L_ _ _L--x
~tJ~ 2-87
v ,
*2- 88 LrHUfW11 8 m fJf)1i~~~n1J'vh.J~'W'Vi A 0'1 X =4
m 1m:; y = 2 m 'il_l'Yl1wn~ z 'lu«_l~~1'_l~~'lJ'fl_lfl1'j1i~~~
9I11l1ff1
2-89 IrHUi:lU11 8 m fJf)'(jCfl~CfIf)uD~'U&'W'yj A 0'1 z = 5 m
'il_l'YI1i1111'Y11l_l +x Lli:l::; +y 'IJ'fl_l~Cf1 A 1~£JI~vmh~ x = y
z
67. Ul1'IJfl~~'I"Hn'li'flm~u~ AB 1~H.II~lJI11f)
. 1ImUlhn::1J1JVin~'il1f)1l1fl A ltluI B
~--x
~tl~ 2-90
"
.J d QJ ~I tI 0 ,
2-91 'VI 'i.::U::11 I)1'Yi'U-l ~-l JU 11 fllfl fl1'i::! fl1 Ul1'U~ flllJ ll'U'U
ri'UU'U1'f1l1fl 0 ltlV-l Bill):: B 'rtlv~ A fio r ={ looi+• OB
300j+4ook} mm IW:: r = {350i+225j-640k} mm 9ll'lJ
. BA ,
cililJ 1Jll1l'i::u::mnnfl 0 ltlu~Yi,r1J A
• 2-92 61 r = (o.5i+4j+O.25k) m UI):: r = {o.3i+
OA OB
2j+2k} m 1JlU(1'I'1..:J rBA 1'UJtll1fll9l0~'U1::1J1JVinl'1'illfl
x
y
~1J~ 2-92
"
65
.d. .J 0 I . d . ' •
2-93 Yl'i::u::mnl1m 9l111'l1'U..:J'UO..:Jlfl'HHiil.!ij A 1ll):m11vJij
IV ILl IV ,fv ",..,:
B jfl11'11:YlJ'V'lYlllfllJ!1:Yll'il'lll1Yl 0 1Jll1l'i::u::Yl1~ d'i:::l1'h'l A
Ill):: B ~1:::!J::mmr'U ff111flJfll'illf)'Urul1l;1f1l'i,rI'1'itll1fH9l0{
'i::!~llmU'l~iHif11Jlfl A ltlu'I B ~!ti';'Yil'U'U11'1~1~
x
~1J~ 2-93
z
0.5
x
68. 66
2-95 1J-3!!j,Y~-3 F hllllnfl!Plflf1U'J::UUWn~Ulflllal'I111PJ
l!j,Y~,'JVii'l'VIl,'Jii1~nl1m:::UUWn~']lU
':ilJ~ 2-95'II
. .' .:::.. .::lev . .o:.,1J
2- 98 1),'J'I11'lJt!119lua:::lJ1JUt'1'I9l'li1 fl'VIl-li11 ~1l1 m:::U1J''1 fl ~'lJtl,'J
u tS' o.J
U';i-la''11im:::i11i11)19l A.
x
':ilJ~ 2-97/2 - 98'II
2-9 9 'iJlIlt'1'~-lU';i-l..r-lt'1'tlllU~lll1flIIPWflu'J:::uUW n~'illfl
*2-96 'iJ,'J l!t'1'~,'J®lUllll1flt~tlflu'J~uuVln~mfll!all11l.JJJ *2-100 'iJ-l111'lJU1~l!~:::l.JJJ~l!t'1'~,'JVifl'VIl,'J~19l1)lm:::uuVln9l
Ut'1'I9l-ll1fl'VIl-lii19l'iJlm:::UUWn9l,]ltJ 'lJflll!'j-la''1lim:::l'hii~~ A
fr-5ft
--7"----~;f------;t--)'
r!--8ft
B
x
.x
';jlJ~2-99/2-100'II
"
69. ;,
r-'- - - - -
y
':itJ~ 2-10112-102
"
} =:I o.d d . '1
2-103 !!'l~ F lJ_'IJ'Ul~ 8~::;m::;VI1V1~~f)~f)'11~ "C 'lHJ~
l!'I'i~1J1~ 'iJ~li'ff~~IIJ~l'U~lll1f) l91t)'fl'U':i::;1Jul'in~'illn
z
----~~~~----~~--~--------y
x
67
*2-104 'I1,hIl11~lfHiI~~lU1CJl AB 'il~'I11fl11lJU11'IJtl~ICJiU'1::;
U'ff~lU':il 50 Ib m::;l'h~ A l'l:lJ1CJl1'U~tJnfll~Hlfhl':i::;1J1J
l'in~ll1 f) mITrlllJlJ uff~~Vi~V11~Vil~'ill m::;1J1Jl'in~
y
x
~tJ~ 2-104
2-105 IfHii'1lf)u~~~n1J~fI~ B lfi~u'j~ .350 Ib 1J'U1m~
I'I1gf) 'iJ~Uff~~UJ~~~f)ril1ill'U~1lI1fH9ltlfl'U'j::;1J1Jl'in~1l1f)
:----- --;r-- y
70. )
68
2-106 'iFHta~~tt':i~ F l'U~1.1nfH(9lill1'U'i::uuwn~'il1f1Ua1111 *2-108 lflliJaJ1rJ'UJ~a1lJtn~u':i.:)i.:)1'U~1.1 ll~Ua~~u~a::u'J':)
~lJtta~~1f!'fl1~Vif~1l1m::U1J'Wn~~1(J l'U~tlnf11(9lilTI'U'i::UUWn~'illfl
z A
- - - - y
x
':itl~ 2-106
"
2- 107 11,:) Ua~~tt~a::tr:l.:)l'U~tln f1 tl'1 eJTIm::uuwk~'il1f) U":;
t. <=>. d..... ""'" ..... ..... "
111'IJ'Wlm1lJVl-3lJlJ Ua~.:)Vl f!'f11~Vl1 ~ 111f1':i::UU'V'lf) ~'IJ il-3I1'J-3Cl'V'l1i
x
/
c
~~----~~-----y
. /
/
;--
/
/
/
/
/
" /
/
/
'jtl~ 2-107
"
2-109 1l~111'IJ'U1~Ua::~'lJUa~-:)1f!'f11~Vil~1l1m::UUWn~'IJil':)
fVd' ::i d od.
u':i.:)a'V'l1'i'IJil':)u'J~'fl-:)ail':)Vlm::Vl1Vl~~ A
)'
'jtl~ 2-108/2- 109
"
2-110 "nr1~1-:)lmil~1~liJ1CJi AB 1l-:)'VIlf111lJ(J11'IJtl~lCJiua::
l1a~'1u'J-:) 30 N m::'Vh~ A 1'11'lJ1CJi1t.j~1.1nml'1ilflm::uuwn~
mf)
;}
c,--- y .
x
..~
71. E iJ'lJ'Ul~ 28 kN 'il~Ui.1~~
. tll~H)'fbj'j::'IJ'IJ~t1~mflUa::'I11I1'l~~'YHf~11:J
z
y
*2-112 'il~Ui.1~-JI!'i-J F 1'U~tll1fH~H)'fl'U'j::'IJ'IJ~«~mf1 t11U'i-J
U I o.d.& ,
~-Jnll11m::'V11'Vl~~f1~f1Cll~ B 'UQ-Jum
z
, A
I4 m
6m
~~~4m y
. ~.
x
69
2-113 'il'llli.1~'1Ul~ F 1'U'jtlrjm9Hl{1'U'l~'IJ'IJ~«~'il1f1t11'il~
B fl~~l'llUl1'11'1 3 m ~llHl~-J'illf1tllllti C •
z
4m
I
y
'j11~ 2-113
" .
2-114 l1flflfll:JfJn£j~fl«'IJ~1~tllflliiayfli.11lJ t11U'llhwI'iCl::
lflliia m::vll'IJ'Ul1fl fl fl 1:J~'1~tl M'I11l'lllll1'111 (x,y) tYll1~'IJ
lflliiCl~ij~«'IJ~ DA lviflvllhi'll'Jlft'V'li'lfi ~~'U1'Um)fl()l:JilVif(
'Vlll~llJlln'U'U()ll1()flfll:J'Illn D ltlrJ-J 0
2-115 11()flfltJtJn£j~f-J«'IJ~1~l:JlflliiClyfli.11lJ t111l'Jll'UuI'iCl::
lfllii a m::vll'IJ'Ul1flflfl tI~'1'Jtl 'Il-J'I11'lJ'Ul ~ ua::lJlJ 1!i.1~-JVi ff'Vll'l• ~ q
Vil~'Illm::'IJ'lJVl«~ a, ~ Ull:: y 'UtJ'III'J'Ift'V'li' nll1'U~ril x =
20 m UCl:: y = 15 m
x
'j1l~ 2-114/2-~15
"
I
'I
72. 1
70
o ~ ~
· 2-116 flll1'UWll1fHl9ltl'J'V-l1:YllJ A, B !La:: D 'iJ-lH1:YWI-lil
A· (B+D) = (A·B) + (A·D)
v
2-117 'iJ-ll1111lJ e 'J::l1iWffl'U111-l'Utl-ll1fHl9ltl1vi'-l1:Ytl-l
2-118 'iJ-l111'U'U1W1'Utl-lfll'l~1l1tJ'Utl-l r IPlllJ r Ha::tl-l~th::fH)1J
tltltlfll'VHntl r IPlllJ r
2 1
z
3m
-<':~--r--:::::x<:
~4m
x
1 2
'j,j~ 2-117/2-118
..
. . ,,~
2-119 'iJ-l111l,llJ e 'J:::'YI11-l1:Y1'U111-l'Utl-ll1fHlPltl'J'V-l1:Ytl-l
*2-120 'il-l'YI1'U'U1W1'Utl-ltl-l~th::f)tl1JV{)tlfl1'V1Ultl'lJ{)-l r IPlllJ
1
r tm:::fl1'V1111tl'Utl-l r IPlllJ r
2 2 1
}'
x
'j,j~ 2-119/2- 120
"
2-121 'iJ-ll11tl-l~lh::f)tl1JVtltlvf-l1:Ytl-l!Jtl-lH'J-l F IPlllJH'Ult'ff'U
Oa Ha:: Ob tVhl'l'h111 F = F +F 'JllJvf-l'YIltl-l~lh::m)1JA B
Vtltlfl1'V1111tl'Utl-l F IPlllJ Oa Ha:: Ob lla::ff11-l{)-l~lh::f){)1J
citltl tta:::fl1'V1"ul tl1'Ut~-l mlvJy.J f) ~1 tl
b
O~----~------------a
'j,j~ 2-121
"
x
y
2-123 'iJl111'U'U1W1'Utlltll ~lh::f) tl1J titl tlfl1'V1111 tl'U tll t1 f) tlil tl1
'J:::t.l~hH'YI1j,:j r 19l1lJtif)'U Oa
z
2m
r r
6m
OV<"'-----------:::;r-~"------- }'
x
73. BA 1If1~ Be
y
x
~'ll~ 2-124/2-125
"
2-126 H'5'.:I F m~'i'h~um£J A 'Uil'l'viil.yjU'J::fH)lJ~'W 1)'1'111
'IJ'Ul~'Uil'lil'H'lu'J::flillJciil£J F IItI:: F Yim~'l'h~TlJHfl'U AB
1 2
Uti::i,l1:n fl n lJ IIfl'U~'1 fl ci1')
x
~'ll~ 2-126
"
71
2-127 t111l':i'l1'Ul!'Ul~'1m::'l'illJ'Ufffll F = {-500 ~} N 1).:1
'Hl'IJ'Ul~'Uil'lil'lrl'll'J::flillJciil£J FUel:: F ~m::'l'il9mHlm'!v 1 2
OA LLel::i'l'inflnlJLLfl'U~'::Jflcil1
* 2-128 1)'1'111l,J'lJ e 'l::'Hil'lll'ln nl'U'Utl'lfll':im::'l'il'IJtll F Lm::
LLfl'U OA
x
F = [-SOOk} N
~'ll~ 2-127/2-128
y
2-129 LflLlJClLflI9lU':i'l 400 N lJ'ULffl 'iJI'I11'U'Wll9l'Utlltllrl
lh~flf)lJcimJm'VHl1(J'lJtll F ~1'lJU'U1LtY'U'IJ()'1fll'lfl'l::'l'il'U()I F
1 2
y
'jtl~ 2-129/2-130u .
74. 72
2,..131 lI'l'l11'fmhh:;f)v1JUtHJ'UtH F ~m:;VhlPlllJLLvi'l AC 2-135 1I'l111lJlJ e iiLrUua OA VIlr11Jfll'W OC
Ua:;J~ulflr11HLvi~ 1~Wll.9l B VtJ..·~~~fla1~'Uv~Uvi'l '" 0 v
* 2-136 ll~'I11lJlJ 0 'VlLflLUa OA 'Vllfl1JflTW OD
2-13211~'I11V'Irflh:;f)v1JumJ'U1l'l F ~m:;vh9l1lJuvi'l AC
ua:;J.:J'inf)r11Juvi'l 19ltl~9l B V~~ 3 m IPIllJUvi'l1l1fltlaltl C
x
'Jtl~ 2-13112-132...
,. .2-133 lI'l'l11lJlJ e ua:; 0 ilLn9l';i:;wil'lUflU OA 'UV'IL'ffl1i'l
r11.1 AB ua:; AC 1'11lJr:i'1~1.I
2-134 Lfl LuarhtJuvr'l'ff1l'lLf)9l U';i'l ~'l~tl 'il'l'l11V'Irftl';i:;flV1J
UVtlmv/'intl'UV'IU~a:;U';i'lm:;vlllPlllJUflU OA 'UV'lI'ffl
x
y
'JV~ 2-133/2-134
"
y
x
-'JV~ 2-135/2-136
"
2-137 'il~m'UUl 9l'U 1l'l V'I rftl';i:: fl V1J UVtI fl1 v/Ul tI'U V'I LIN
100 Ib m::vlllPlllJUflU BC 'UV'IviV
v
2-138 ll'l'l11lJlJ e ';i:;w.h'l~uridUvi1l BA ua:: BC
z
- ~8ft
x 4ft
~D Y
'JV~ 2-137/2-138
"
75. ,_ ~~""lf1~'m""m',oli'fftl,mftll;'itlf) 'il~111'U'Ull'l'Utl~U'j~
rl'j:;l'il1ti(l~n:aff'U!;'itlm~v'I'hlr1'lnl'lu'j~Kl'nj 80 lb,
. aa i~iU fll'l1'U1'l e ~ 40·
O 1~"OflCllflt'Utr~il'loli'fftl~lff'Ul;'itlf) t:1111'j~Kvnj 80 Ib
*2-14 .U"
"-""""111U'U11ff'U aa i--li'll 'il--l111'U'Ull'l'Utl~II'j--l T IICl:; P1111T1T' v ,
m:;~hhi!l~Cl:;lff'Ul;'itlf) 'j,1lJvl--l11111lJ e,'UV--l P lVltl'l'h1r1''U'U11'l
'litH" P i1filUilO~,,!1'l diil T m::'I'lTVilllJ 30
0
'illf)U'U11ff'U
Yl-3flth1
':iU~ 2-139/2-140
"
II
70°
v
250 N
':iU~ 2-141
"
73
2-142 'il~111'U'UlI'lIICl:;lllJUffl'l~iiff'l'11~Vill'l'ill m:;1JlJ'Wnl'l'Uil~
F3 l~ill'il1r1'F,mKv'l1f'Uv-lI!'j-lvf'Hn:JJm::'I'h~11J1If)'U y Vii1fil
mflun::ii'U'Ull'l 600 Ib
2-143 'il~111'U'UlI'lIICl:;lJ:JJ llff9l--lii ff'l'11--lVil9l'ill m:;1JlJ'wnl'l'Utl--l
F I~V'I'h1r1'~ClK'I"nf'Utl--lu'j--lvf--lffl:JJiirill'l'iln1Jff'UV3 ..
F1
F2 =300 1b
1U~ 2-142/2-143
"
76. 74
!-145 'I'mff~N F Uft~ F l'U'jll!1nL~H)'n'U'j~uu~rl~mt1 * 2-148 1J~'I11'IJ'U1~'lJtl'H)~rlll'j~t1tlUritlUfllVHnt'J'lJtl~tI'j~1 2 "
!-1461J~'Hl'IJ'U1~'lJtJ~Uml'j!uiuft~Viflvn~~1~1'UViflYl1'U!~1l F = {60i+12j-40k} N l'UViflvn~'lJtJ~!fl!iift AB !!ft,: AC
.nWt1l'Olt1!!t1'U x Yiii,ilU1t1
y
---jl£--- - - - - - - X
F2 =351b
'nJ~ 2-145/2-146
"
v
2-147 'O.,'Jl11lJll e !!il~ 0 'j~'H';h~'lf'Ui,'h'U!ffW11~
z
O.6m~C
x
x
F
'itl~ 2-148
"
2-149 tI'j~ 23 kN !n~~'Ul~t'Jluvr~llll'U'UtJ~!flitJ.,'J!u~,
fl tJll!1'! tJflwu ill~~!flitJ~ii'Uill'1l1.,'JllUl 'O~til'! t1 tIl~i1!lJ'UtJ.,'J rl
lll:;t1tJUVtlt'J x!!!;):; y 11JJ'vr.,'JtlliU1tJ Nftm:;'VlU~tltflitl.,'J!ui!
fl tlll!I'!tlfvi!n ~'01t1!l~!;):;tl-l rll.h:;t1tlUcitl tJ~.,'Jf) rilTn