resolucao-cap03-estatica-e-mecanica-dos-materiais-beerjohston

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Resolução-Cap.03-Estática e Mecânica dos
Materiais-Beer&Johston
Mecânica Dos Sólidos (Universidade Federal de Alagoas)
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Resolução-Cap.03-Estática e Mecânica dos
Materiais-Beer&Johston
Mecânica Dos Sólidos (Universidade Federal de Alagoas)
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3.1 A 20-lb forqe is applied to the control rcd AB as shown. Knowing tbat the
length ofthe rod is 9 in. and tlnt a=25 ",determine the moment ofthe force about
point B by resolving the force into horizontal and vertical components.
?.5"
Problem 3.{
l l l l l
3.2 A 20-lb force is applied to the control rod AB as shown. Knowing that the
length of the rod is 9 in. and that the moment of the force about B is 120 lb'in.
clockwise, determine the value of a.
25' Resolv,r'g To-lb fzrct-irtl-o
brr lPonJi ls ?onl 
g 
"
20 lb Mn'= A (4 ; " , )1 , 8
B,t l G; (Zo tb) rns6
-Thvsi 
f4 o = (zo rc) us o (f ;6
lZo lb, in, =( loo tb, in,) h$4
0 = +8,2o
c ( - 48 ,2o-L fD
r t = 23'2o {
firr € ryw.ll.sf valve o* P, {vrq-
y1l ' .be plrp*nct iculai. fo l ine
1: oin,-ng fr o^ol B- Thrs, we rhzsl
hale d = ft ?
8ul- e"0 = ffi
q,= 
F = 22,6o
l4B = F (fr!)
N ' f = P(o,26on)
P = +ro'N
P=tyoo N3 d,=Zz,60<
3.3 For the brake pedal shown, determine the magnitude and direction of the
smallest force P that has a 104-N'm clockwise moment about,B.
' lc{
klso :
240mm,
I00 mm
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450 fr/
l00nrrr'
3.4 A force P is applied to the brake pedal at A. Knowing that P = 450 N and a:
30 oo determine the moment of P about B.
100 mm,
moment about D.
3.5 A 450-N force is applied at A as shown. Determine 
(a) the moment of the
450-N force abooi D, Aithe smallest force applied at 
B which creates the same
+50 N @) We r€.$u lv e 4ke 4 50- N fi,rrce
in fe f hc htv,t pone r'tf 3
F = - (t+50)s in3o"= ''&t't N
'nt 
5=-(+ro)coeJo' . : 
-381, ' l N
4o = 88.8. f'l"rn ) <
(b) 
-Ihe 
srnallest force Y applieJ ot, [i tht'h 
c reafcs
{he Sarne wronrent hust h* FrP€ndicutar fo 8D.
Si^.e Bo = t6,rzsioq*ll = o,|'ts,,,'
w€ have 
Ms = [' ( B'u']
6 E.$ N.,vrl : P ( o,g?5 rn)
P. = ?31' N T753,1" <:
f (
x -
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3.6 A 450-N force is applied at A as shown. Determine (a) the moment of the
450-N force about D,Qtthe magnitude and sense of the horizontal force applied at
C which creates the same moment about D, (c) the smallest force applied at C
which creates the same moment about D.
D
T-
0'!25'r'
L \l I
n= -(4ro)coeJoo: 
'381,1 N
d - t
I o= 
gg,8l 'J 'rn ) <
(b) Hade on{a I furc. f clppit4+(
ak C rvhic h Cr(a*tsfhosaune
f.'llorv1 en+ 
"
/ '19 = ?(o ' lz f r t r )
I |*7*a A/,tn = p (o*z?s,r.)
f = sqs rrl '--
(o) Tttc syncrlles| fiirce ?
appl lod at C u/hich crrafes
t ha Sam'e yfio,nent m usL
be perp(hd i c uloo ta C. J>
Sin.e C D = (0,2?s n)vZ = o,J tSnr
We hin V€ i
UD = 7(cD)
8g,Bd,n, = P(o8tg-)
P = 271 N yz ttS'4'
?
I| l
JD
T-
0,??Srn
L
-r
0,? Z.srn
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3.7 Compute the moment of the 100-lb force about A,(a) by using the definition of
the moment of a force, (b) by resolving the force into horizontal and vertical
components, (c) by resolving the force into components along AB and in the
direction perpendicular to lB.
too lb
lZin.
d -
A6= tff i=/3in.
= ax F- ( 1 -
Na = +'Fsrno
wi ' lh 4,=f r7= 17;n.
F= /a0 /6
$: Qt90+60"st72,f
lltp =($(toopin /7Ld)
= (tSYloo) s;n 7,.+o
AA = 167,0 lb- in. .)
= f0z)rso) -(:Xas, A1+
= (eao- n3) & 
:
Arr= 167,0 lb';nJ {'
4 B = lS in '
0= 30 ' -22 ,6o
= 7 ,#u
l F ! ? ' . \
f fo= t3r-x ( t I t?L, l' '=/3f , '
' 
l t r t d r z l ^ 6 ' . r r r \ l : u= ps,inlftz.Bs tb),k
t r t l l . l l l t
f4; = l(;7.0 lb-;n.J {_ H
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3.E Determine the moment ofthe 100-lb force about C.
l l l l l l l l l r r l
U sin.e h o riz on tu [ a n4 ve rtica I'\}
Ct ]m Poncnts I
I
f i i =h x F
' - C - r .
=Qt! -til) x (- 86,0 i, tsdd)f
z!oo&-H33i
- 96,6 i
-'r
J tE .
Problem 3.8
l l l l l
A
tl-j n. -*l
I
l n' l v
too )b
3.9 and 3.10 It is known that the connecting rod lB exerts on the crank BC a
2.5-ld.l force directed down to the left along the centerline AB' Determine the
moment of that force about C.
42 rnnr
& = c B 3 a (p,ott4^)j + (o,osg rlj,
N =!yF'c 
= (o,ct+L: +qos6j ) u 
(tooj ,luoofi
= 100,8 & + 3q,2& = 140,0 4,
N = lttl,}ltf'nr )-C
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3.9 rnd 3.10 It is known that the connecting rod l.B exerts on the crank BC a
2.5-kl,l force directed down to the Ieft along the centerlne AB. Determine the
moment ofthat force about C.
Problem 3.10
- ' ' f= ' -Fs in |J
F: -(too u)
Ag=,|ff iy= rson,tr
a)n A'- l tZ / , .e A - l l i l t t+sinft=#
! = fr = - (o.o+z'll - (u, oi6 n)-i
N'=LxF- C -
= (*0,0,+L!- 0,0r6j)y (-700 i - L+00i)
= loo,o&- iq,? fr -- 6 r,6 &
f; = 6l '6 $n)
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j.// Rod AB isheld in place by the cordlC. Ifuowing that the telsion in the cord
is 300 lb and that c = lb in., ditermine the moment about B of the force exerted by
the cord at point Aby resolving that force into horizontal and vertical components
applied (a) at point A, (b) at Pint C.
l l l l l
c
T
IBin',
l2in.
_*
F= ftwaj t [:s;'n.cX'-= dio oio 6i :ft0t,/4I'Bi
r = [,io tb) ! . (r+o 
tb) r
t = m = -(22,5in) i - ( tz i ' . ) j
E:;;*z{s ! - I? il xftBo! + xL+ oi )rtr =-:;{r; 
il t''tio .k':'-(tr-io i',ri4=-(170tb'w+
Aa = 1,70 lb'f t )
r T
19irr.
J =- 43rqotbi,)ft= =fi: i: fpi 
"
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3.12 RodlB is held in place by the cordlC. Knowing that c :42 rn. and that the
moment about.B of the force exerted by the cord at pointr4 is 700 lb ' ft, determine
the tension in the cord.
C
T
tZin.
I
I
-.l
l 2 , f i n ,
F ='- F"o,EN ! 
* Fs;n ilX1
F--affi rt+#'t
| =' BA = -(22,'3in)!-(tzin'\i
f l : bxf = F22,5i-Ee)I-B- -'- -\ 
ert!+;uj)fr,
,F L . at t , r r , t \ f - - i -?\E nA[ 
-
ii+tz$.k+2t u/,)E =F W"rX
ftept,l I ot g'n t _C :
,t =':fr, - (+tin,) i
Azin,
-I
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3..13 Determine the moment about the origin of coordinates O of the force F = 4i -
3j + 2kwhich acts at apointl. Assume that theposition ofl is (a) r= i + 5j + 5k,
(D) r: 6i +1 + 3h (c) r= 5i - 4j + 3k.
Problem 3.13
So = Ztp+ 224 
- 23-
, (9,) \= Ylo: t l ; - 22 b
r- i!
.5 -[l 3
+ -3 Z
? a t 
hMo= *+ 1+ r
3.14 Deterrrine the moment about the origin of coordinates O of the force 
F = -i +
3j + 5k which acts at a point l. Assume that the position of ,4 is 
(a) r = 2i - 4i + tq
(D) r:4i + 6i + lOlq (c) r = -3i + 9i + l5k.
, r r L r r r t t t l l l l l l l l l
i lts
2 -q I
- l 3 5
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Problem 3.{5
tB tn.-1,
,t"'Jrn,
3.15 The line of action of the force P of magnitu de 4201b passes through the two
points I and B as shown. Compute the moment of P about.O using the position
vector (a) of point A, (b) of point,B.
kl/r Arsl. .Jrfu* min2 tAe com.pone'tj 0f tltc
firce ? :- -rD
7 = ? boo = r #=(+zorQ 9ili !! i; ')t 
-0a;') frt _ P | 1 p -
! , " r t o = t 
. : -
- n 9 l + 1 .- 
l r L ) 2 l in .
T =0sorb)i +(rzatb) i- (seotb)&
(a) f l = t , .xT =_ D - F I
P.
f{r,e onSwPr i5
(zte, tl,,in,)! tUlzotb,i^)X' +fteo tb,in)& <
Problem 3.16
t t t t t
3.16 A force P of magnitude 200 N acts along the diagonal BC of the bent plate
shown. Determine the moment of P about point E
325 urrn
[Me +irsf def*rnine lhe donltooeats af clr. $rce p;
P= Ple.= P #=(zmN)-%W
p = - (a, N)j +(tzo N)&
ry=--e?x f=l;, i.r:l = s6!+l,rit-szr&-F 
l ; : ' - i ; ; , i r l 
r- i c
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-l
I
U
zlr
-_]-+Problem 3.il
I I- - l - - f 
t t t t r t l
3.
fo
(a)
17 Iftrowing that the t
rce exerted on the plat
-ffi---l--l-t
l{< de fe
excrfed al
ension in cable AB 1800 lb, determir
e at A about (a) the origin of coordinr
1t t t t t l l t l
rmlne fhe hm (2on t rn I s o
? h bv vvr,'fi^i
I I 
'r 
I i t" r l-l--T
EFyFiF
F^-ffi 
-aB+^--AB
i t t l
E = E - r. .-l,tuoll-',4 
! , ,-7, ' ? , ,
)b , Fr=Blo lb, F* :
v t 0 ' . 
o
re the moment of the
ttes O, (b) corner D.
-fr-_]--r-+-
| +1. fot.. t
T
i /vl
{ F-o
+-l
I ' do l )
2f
' ?ot
. Abo
i
'?^
F,
J
I 0c/bF,
0nen l
I.
Lf
r
r' I
(a)
D
tl
14 oment abovl -
enofi ,rX b? A r, t
D l 
' '
,d, AZ the con|onentg of a!.= ' 'h.t,
Itr. i +. I,
l i=| 4 0 o l :
[ l-aocr 8oo -r t+lro li
4- ft (" rO IN.+r) I {- 3 z(6 L H, A
t
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3
fi
.18 Knr
)rce ex(
lwing that the tension in cable BC is 900Ib, determine the moment of the
:rted on the plate at C about (a) the origin of coordinates 4 (D) corner D.
- l t t t l l l l l l l r ' | '
We drfernirtt fhe nrnBone^fs of fhe
rycrfecl dt , 
6T w ri Fi'ng 
, ,
t
Tuft,€ f
F"E,F,F
f
4au- Udqe- er*u-E
,r -L ='-n-=-E-= Wv " -+ 4 : ? 6
( '*) n
Fn
'l'o^",
t ;
= - 6oo lb,
,? abouf
i
a
de
Fb
g
?t
FL
: 3oo /b
l.t
ao Ir
f i: [4on
De nol-
t"1 =-D
e nf abovl D I
n$ b1 M,A l A z fhe cornfor{hf r of A!= DC,
| i , r l , le I
=l; b :eI
[ -o* 6or t r ,o l 
i
M ( l+ D T L + ( L 0 .[h, I t
r
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the
l l
Determine3.19 A 200-N force is applied as shown to the bracket ABC.
moment of the force aboutl.
3.19
5=-pooN)hs3f 
=- lT3,z N
n : (ZOc, N)cos 60" = f l0O N
- (tzs,z N)t + 0 oo N ),&
3cn= (0,0a n) i f (o,oun) i
D4,*= *eu* F i
i i & l,
0,06 o,ot t 0 l 
= 7,s t - 6 i - 10,314
o -l7J,z 100 tr
E^ =(ZT N,n)1-G r *)l'-ftl,J'/ u,nl/e ,
3.20 A smalt boat hangs from trvo davits, one of which is shown in the fig,re. The
tension in line ABAD is 82 lb. Determine the moment about C of the resultant
force R7 ex€rted on the davit at A.
f*, = (+s tb) ! -
l l l l l
- (14 /6)l
6i-zrsj-s4€
IAus , 8o= 2 4rr fn o= ffB 
tb)j - (zwtt)i - Gq/l) t
Hfso' tah=0E ft)i 1. (z+1 4
dr''t g, = Sozo x Rd
t! & lt i
N"=[o7,iy i t ' ,-- 
l+B -22( -2+ l , r . , . ,_^
It. = (T{t /A'fr)4 +(t't4lt+Y}i 
-(srf /['#U 4
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3.21 InProb. 3.15, determine the perpendicular distance from the line of action of P
to the origin O.
3.15 The line of action of the force P of magnitude 420 lb passes through the two
points I and B as shown. Compute the moment of P about O using the position
vector (a) of point A,(b) of point B.
-l We firsl Je{ar^ine the c$ttpoi,teafs o{ tie
fwru ? ;
tZ in.-1,
,""1ro,
- 1 - 
l u l t t L I t
-a'
P - ? )' = r E--(+zo4 0;n)i t?';n )a 
-Qa;") fr
' i : - 'Ag f t6 
/ ? l i n .
-a,
'D 1) \ p Ag
f - f A - ^ = fi g f t5 , * 
/ ? l i n .
f = (l no,b) 
j + (tzo Ib) i - (.3s0 tb) &
i i l e
J--
l, b6th *&i,
I tas Eo -leot 
-
[4 - : - (zte, I A,i n,) t t (r lto tb,,i,^)g-o
r i l l l l l l l l l l l l
Hr= rftzt60)'+ (rl 3ao)'+ (3e olL :
Bvt No= Pc{ , and a=I&=t,tl iW'
' 
+G6o !b. ir)&
t l l l l l
ut8+ j lb,i n,
d = l l , , I3 in ,
Problem 3.21
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3.22InProb. 3.16, determine the perpendicular distance from the line of action of P
to point E.
3.16 A force P of magnitude 200 N acts along the diagonal ̂ BC of the hent plate
shown. Determine the moment of P about point E
llil) nrrn
D
l =
lfil, f;rst, deta^in" lhe co'rrrporrerrll d$ fl'e fz,r'ce
p )^ --= p = (zaarv):fo,l+)f= \(,:UIM'Bc , 
BC 
- \ - ' - - . ' / 
0 ,375n
P;
7
925 rnrn f =-(toN) i r]zoN) &
tr- = fE , f =l :, i,r:^[ = s6,i +2u p tzz +,-E 
lo - t6o r ro[
tL = (le ru,')j + (utt N,,nU + G2 N',,) t
l'+. E
By!
l l l l l r r r
iiy + (2+)t dtz),' = ,3,9 N 
'rn
= pr{ ,and d=#=W ='0,26l m
d = L69 nt/rn
3.23 In Prob. 3.20, determine the perpendicular distance from the point C to the
portion AD of line ABAD.
tr* r*iboat hangs from two davits, one ofwhich is shown in the figure. The
tension in line ABAD is 82 lb. Deterrrine the moment. about C of the resultant
force Ra exerted on the davit at A.
r r r l l l l i l i l l , l , l l
We hmpvte tfu moff irnf- abovf- C aF
th" -force fr ererfe'ol bf fhe 
' 
line an g
Problem 3.23
G7z)'+ ( tqt+)' 
- 3q 8,1 lb,tt
F d. . ot = N, = p1!:,!'fr) - ' - f 
- 
SZ|L
.Dd 
4 =Lf ,?6fy .d
tb)*Jrltv
r)t
6f+)!x [- ul.qj + (6? h)i
37L tb.+t)! - ( t++ tu t
I
+(
+(
[4 - t wF-c --D/c n tDa -
l'4 ='c
13,'t
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3.z4lnSample Prob. 3.4, detemrine the perpendicular distance from pointl to wire
CD.
Fr,nn Ia*ple Pnb' 3'4:
F= 200 N) AA = - (7,58 ̂l 'n') i r (28,8N'm )i' 1(28'8 il '^) &
N t+ 
=[ Ve r)'+ (za"8f + (z\a)1 = 4 t, YS N,14
B,tf N.= Fd d= f l* - 4ttr ' t ts N'm - 0,207 n'd ' 
.Ft '= f f i
d = !,07 tmrnt
3.25 Given the vectors P = 2i + j + 2k, Q = 3i * 4j - 5h and S : 4i+ j - 2h
pompute the scalarproducts P . Q, P . S, Q . S.
- + - - - - l - - ] - J - l r r r l l l
f-9 = (Lr *A + 24.),(oL + 4_i -
=(eXE)+(rX+) + (zX- s)
l"t = (L: * A rzk).(- 
Lr: + i - 2&)
= (zX-,+) +{,X')t (eX-z)
g€=(3j r,{ i -s' t) ,(- 'r !+ io-24 )
= (EX-r) r (qX,) + (-sx-t)
8,L3 -// <
Problem 3.25
l l t t r l
q=qcco,!+\sinQd (t)
I
P=Pa,soi+fs inF- i ( t ) -- L - ' L L - ' ' L 2 9
3.26 Form the scalar product Pl . Pz and use the result obtained to prove the
identity cos (dr - e)= cos dr cos 4 + sin &sn 8.
B1 ileFinil','o^;
!,,?, = q ?, cos (% 
- gr) G)
For^ ( l) n',,t (.) : ?,, fr=(q 
"r 
0, j * P,pi,nq 1), 
(f,^ru. t +Prs,uao/,
?, -To= ?r(r(cos o, co s 0r* s ,"n sf t'u 
dr) A1)
E1'ohn3 
X( r:T;i=:];,, :,7:.:: ::] #:'*.
t____t- L r - | , l I
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3.27 Knowing that the tension in cable 8C is 1400 N, determine (a) the angle
between cable.BC and the boom AB, (b) the projection on AB af fhe force exerted
3 m
t
)
8
by cable BC atpoint.B.
l l l l l l l r r r r l
BA=-?. I l - / ,g l Ef t :3,eu
0,s lq 3
4 f tBe = FJ.o-f" {;
(S),'ProJu etian of furrc- onr ftB
(P); = P+oou),tussq.di'= (r uoo4bsr+t
{a) Si,rre {A,d;= @AYBD)cc,s i ABD, nle have
3.28 Knowing that the tension in cable BD is 900 N, determine (a) the angle
between cable BD andthe boom AB, (b) the projection on r{B of the force exerted
by cable BD atpoint B.
t t l l l l l l l l l l l t
BA = 3a
, E D ; 3 , / t l
/b ) ? ru5e. hbrr o( fi.rvt e on fq B .
( P)^, = (troo fr) cos ?0, g= (ra o N)(0,) i j 1)
B-T
t
I
1 . 8 m
Problem 3.28
l l l l l
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3.29 Thee cables are used to support a container as shown. Determine the angle
formed by cables AB and AD.
-'3) +r
5 ince f t6, hf
coS* 8 ,49 =
ftB= 53 in,
hD=-fs in.
nE= ZBt+fsi
g.
nS =-26!+\i i+ts&'
= (ts)(,+n) qs + I
f tE-B - GEi+45.
M)-
- (t,d(- l.tl+(qflkr) +o = o,rr+Iq+
- f 
, 
' 
+BhD=63'60{
J.30 Three cables are used to support a container as shown. Determine the angle
formed by cables AC and AD.
FC= 5t in ,
f to- 55j" '
Ll-l l l l l
lem 3.30
! l
- \ : t ,
AZ'= +sA-I|&
tr = 
-t{i ++sj 1 tE&
-63 =([+tI.,Al) crr +cflD , we 
rnauc
+C nD = 55"4o I
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3.31 The 500-mm tube r4B may slide along a horizontal rod. The ends A and B of
the tube are connected by elastic cords to the fixed point C. For the position
corresponding to r : 275 mm, determine the angle formed by the two cords, (a)
using Eq. (3.32), (b) applying the law of cosines to triangle ABC.
3.31
5oo
pinneu,ts iorns i n fnhn
s = 77fu -3ott11€'afr',
C A:725
CE=1025
qs0
I t t l
aso =(),81215 --]--l- i I 0'= 26,8 {;
l l l l l l
0 = 26,8
l t lL l l r l l l l l l l , ,
(b) Trom fart +, w< rt'nat C'A. md CB .
ls* oF /r,sihes I
(M)'=(c n)'+(c * 2 (cA)(c B)cos o'
,(i*)o = (tzr)t+(tol:)o- z (tn)(tlls)cos o
htr)J (pzii - (r?9)' = 0, Bq asLoso=ffi
c
t02 '5
Problem 3.32
t l t
C
67ts,23
t
l l l l l l l l l l r r r l
3..32 Solve Prob. 3.31 for the position colresponding to x = 100 mm.
3.31 The 500-mm tube AB may slide along a horizontal rod. The ends A and B of
the tube are connected by elastic cords to the fixed point C. For the position
corresponding to x : 275 mm, determine the angle formed by the two cords, (a)
using Eq. (3.32), (b) applyrng the law of cosines to tiangle ABC.
we ilnd Cft *^d e'8
(cs)'"- 2 (CA )(cil cosa
'qoo)'-/ (emzt)Po) cas t
."1, = 
_o,glts
0 = 3.3, '3o
(1)y',:r,:fr 6.qr') , , , , t-fi--f-t-1_l_
€fr = /ooi - 3oo; + 6oo & , cA: 67a,zs ]-l
oB = 6oaj -Tooj 4 dook , CB=Iffi 
' 
+-1
I I I 
| |
-+ --|
^ CA,CB
L'05 0 =
( 6 / a i L i ) 1 7 o c )
hse : 0,8t i f
Dirnensfou,'5 in vntra
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3.33 Giventhevectors P = 3i +2i+ lq Q:2i+1, and S = i, compute P'(Q x S), (P
x Q) .S, and (S x Q).P.
Problem 3.33
_ (!^ g),9 = 9.0*9) = f' [gx$- -1,
(t"8).f = 3. (Ixg) =- P,(g* s)= I ,
! '(9*s)= - l :
(fxe)s---r a
8*S.!: t <
3.34 Given the vectors P = 2i + 3j + 4lq Q : -i + 2i - Zl\and S = -3i - j + S,k,
determine the value of & for which the three vectors are coplanar.
l l l l l l l r l l l l l l l l - - - l - r r r r
The 3 Veclot s o.re coplanaf- ;F tieir rnl Xeot friple
F"odu" i ig ?€ros 
I
F'(Qxr)=l'-i ?+t =o
l- 3 -r s*l
+5* + 18 +t l +zLl + 35. -Q = 0
75rtr f2=0 S*=-6
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3.35 The jib crane is oriented so that the boom DA isparallel to the x a:<is. At the
instant shownthe tension in cablel8 is 13 kN. Determine the moment about each
of the coordinate axes of the force exerted onl bv the cabler^8.
We first firSsr+ninc tAe camponcnts of the f*ce
f exerl"d ol' A +
- J 1 , ,
AB= -t{.8-}.A'+&-)'3 
fr$=fznt'
-
l l l l l l l t l ..+
F = Fl*r= F 
#=(rr4l) 
.:1.',.#L:&= - rctr54
Since Er= ff" i * ft? j + Hr &, we hqve
Mn= 2,4 AM,* , My,= 
' 16,4N;m, Nr=-3ql##Ntm {
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3.36 The jib crane is oriented so ttrat the boom DA is parallel to the r alris.
Determine the maximum permissible tension in the cable r{B if the absolute values
of the moments about the coordinate axes of the force exerted on A must be as
fol lows: lM-l < 10 kl l 
.m,l@l < 6 ld{ 'm,lM,l < 16 kt{ 'm.
S/k first atprcss the
inr {ornns of t'tte
fofte E *5fir,1 al n:
tension T and the
wit uectorrs""#:: 
(-:s.,,") ! + (t^\ I
A8 = s,Z"a
_ t l l l l t ' - + '
F=T,l .-='fE
= -5 -'+'Ei'tj_-='- '#,i
Problem 9.36
3.! m
z
_ t_ i _
t4_D
.Sirrce A,= l , l^ ! ,Nl t r
v/e !'tcvte
5 5.12- ' f t v
G
( t f ,88 &rl
s F.t+?* k u
Af^l =-&-T (',0 /-ttt.+*
ffl l : ftr s6 
*ru'* T
INz[ =#T6tbkv. ' tn T
The Scondif, 'ons nrc sa*is$oJ if TE 4.88 
'{4M,
Th us t No,f*,um Fn*rlcrsSib/e 
{erS,on = 4-8t'& d <f
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3.37 The primary purpose of the crank shown is to produce a moment about the x
u<is. Show that a single force acting at I and having moment M, different from
zero about thex a:ris must also have a moment different from zero about at least one
of the other coordinate axes.
l l
f = fL+Ft L*E4
bp--6i+41-sg
Io = !o* f
t\-0 = (,r F+*s Fr)l*f 3ry-6 t)i + (6 Fr-T r) &
(r)
(r)
(3)
Thus/ iF l1r + 0) ,J ancl ma cannot brth be lero ,
v
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J.Jf A single force F of unknown magnitude and direction acts at point / of the
crank shown. Determine the moment M*of F about the x axis, knowing tltat Mr=
+ 180 lb ' in. andM,: -320 lb' in.
l l
v li
A
f = F"L * Ft I+FrE
Un--6!++1-gg
fo=!o*f
&, in.z)
te
C t -
x
= (4F+*s F:)LnF 3r;-6t) i + (6Fl-T r,)&
0), t?),( l ) by 6)t+,-3) rcspccf ivel '7 a^d,add
" IuaFtons 'Fo 
e l i r r " iy ra fe ' L , Fg,ano4 Fr3
6Mn++44),-3Mt =o Ctr)
Wtr. use Eq , ( +) 
o'
l!
Subst i ;uh ' r ,g fu*= +180 lb- in , and Mr= -3zolb- in ,
we have d
6 M,r+ + (r Bo lb-in) - : (- g, o /b-, 'n) = o
6Mr, =- 1.680lb-in,
/4n? - ZBa lb'in
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lem 3.39 3.39 The rectangular platform is hinged at ,.4 and B and supported by a cable that
passes over a frictionless hook at E, Knowing that the tension in the cable is
. 1349N, determine the moment about each of the coordinate anes of the force
.;ru
exefl Bd )yul e c ablt at U.
I
I
1.5{J
I
n) --
- l :
FUG i
ei
i !
(
E
-1,
F
n L
= (
F
?C
0,q
(r
0w
J !t,
/4
)i
?Al
r,6
+(
) '=
N)
I, f,
0r l
t g
)*)
u
r(
'i-
l+
,7
7to
(r,:
l,5r
E
N)
?fn
,#
I
o ) i
- ,
- (
h,
W
l0(
C:
,l
,r5
?
L
R ' t ,
2,
k
15\J:
D - x
t
j , Eo = (t,Lf"n) G
N, = *"x Eo =(I.rr&)x (+zt !1tro4 - t06s*)
At =,4- ! r i,lTr:f:l i,{;n r,, + (i ia s il,,h)a- 0 ? 4
crefdra,' Plnr; - l5q7,S N,ilrNJ=qSB,f N:rl, Mu =Q{
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3.40 For the platform of Prob. 3.39, determine the moment about each of the
coordinate axes of the force exerted by the cable at D.
3.39 The rectangular platform is hinged at A and B and supported by a cable that
passes over a frictionless hook at E. Knowing that the tension in the cable is
1349N, determine the moment about each of the coordinate axes of the force
exerted by the cable at C.
fr3 -(lfi0nt)!+(rs0 *)4 -Q.,zfirrlp, DE :3' 55n
=, lttw 1i1 
-230i'+!'Jtoii -tL-s 'g
= - (arrn); t6 Tr,NU*(assN) &
+r = G,to n) i'r (&,2.s ,n)&
Ao= -ft,8tr',til '*)i + (ts?,sil' ,)i +(tl1'ttN,n) !
f i"t Ao: Hrj+\i+t4'?!
7jh'r"Fre: Pl : -1283 N,n, ry 
=770 N,*, Mn=fiilil,n d
Wil*l iotr l t cov'h,l ' haw' betn oblarnr' l
Ul ^fFlYttS tl"' fwo, nt E a"/ w''!:_'j-,in+
Ll 
l-A:i,, 5 
-l-]-1-ff1- 
, ,
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l t t l
Problem 3.41
t t t l
3.41 ,
knou
davit
A small boat hangs from two davi
n that the moment about the e a
at I must not exceed 279 lb . I
able tension in line ABAD when.r
ts, o
ris r
t i n
r : l
rll€ (
lf tl
abs
t f t .
D.
4l
rfw
l e n
rolu
hich is shown in the fig*". I
ssultant force R7 exerted on
le value. Determine the larg
t i s
the
est
T
oll
t
al
L
l.
L
I ';,.
low
En, = 9%,* F*
X-
:r' a f
v .
n . l -L
h D V e 'r -+ A1_ \+ a A n4 r + o t ) W( La e
I
l"l. 4
'!o X !-t o ) & Lx, l) E lr + 5 If ' z l 1l l l l l l l
l4!e= fr ry (
l l t t r t l
)b, ( x. F., A 7 I
nt:
W.
dn
.--J
--l
erl
27
? t
t
i;,
WI
wr
l-l
( , :
, ,1
nr
:,,fe
, 
'Glu, ' 
t!-t ='219 lh''ftr'i 
's/bsift'nng' tr.g )r
r = (i t+)11[ t 
'El = t't4,, lt
< l ; Y ' - , '
,Fe fto = Wf+ ftrt#)3+ (sir1' = lo,zs ftr
I
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Problem 3.42 3.42 For the davit of Prob. 3.41, determine the largest allowable distance x when
the tension in line ABAD is 60Ib.
3.41 A small boat hangs from nvo davits, one of which is shown in the figure It is
known that the moment about the e a:ris of the resultant force R"l exerted on the
davit at I must not exceed 279 lb ' ft in absolute value. Determine the largest
allowable tension in line ABAD when r : 6 ft.
| | | | | I | | | I | | r I I I I I | | | l_
TEru
J J, '1" | , ,JJ. f, l lh oL', ),o A
J
4 ) A (
'-Lt x ,) G 4 /L x " Fl,*4
- ( x ..-l-l
(
8,t
)u
El7[[ ]++-i-l+- +-f4--lllo l l l l l l l l l l l r r
"'^ I lla vl ='{ ?9 
lt" -++, w.' w'riha
]c lqf = il9 t6,+t
l- lq/= F CP*);- -(t0u)&-frD 
lx]+(7,71)'
bsh'fuhkg in Cd ad gva'rihg bo rA wt'
,: ggT,',1\=, = (a7q)".w
(, ieai ) 'z" J' xn + (7,7i) '+ ( 3 )"
*lql+-+rf
=6,23ft
t_
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3.43 A force P of magnitude 25 lb acts on a bent rod as shown. Determine the
moment of P about (a) a line joining points C and F, (b) a line joining points O and
C.
cF= fffi, =,/iFF= n,ol
12,0 t+ 12,04
lrtr^- =- zgglb-;n,
CF
abaot 0C i
= tTI*8i+9t 0C=t7
- O? I y ' t . , t r o - l ' . t 
^ \
- 
o€ 
b 
n(tz!+dj*g&)
lic-'in <
\
J
-9
0
l ,
0
e&
0
0
t0
q4-2 , lbo- l
+o
Mr, =
*(alx 
p)=
f 60+ , r ,
i7
2
fg ) rylor?ren F
a!
dE,
)-0a.
Mbo =
floo =
lem 3.43
l l l l
E
t,
A
l2 in.
v
B
/
8 ln .
F'
/
t7,ln.
(,
I
- 8i +6 & ?F = toin.
= |Ftl(si + 6'te) in,
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3.44 A force P of magninrde 25 lb acts on a bent rod as shown. Determine the
moment of P about (a) a line joining points A and C, (b) a line joining points A and
D.
Problem 3.4
A
v
l2 in.
B
,/ I In:
F"il
v,in.
fr a-8i+6& Bf =toin,
f;= Pffi--ffi(-si+6&)in.
ahoul
A9 =
At= h( : l ,
)0,= 
*= # 
(rz!+e &)=t' 0ir3&)
fb) Pfomenf
M = ) . /oox' 
A D - A D .
= z 1,60
,7
* lZ1.l lb- in.
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3.45Two rods are welded together to form a T-shaped leverthat is acted upon by a
650-N force as shown. Determine the moment of the force about rod AB.
Problem 3.45
l l
= l(d:-ti$g
, , ' | | | I | | t ' , I I l_ l__t__T__f_tt6 _?. 3 l
M^s= io6 (rl x I) = + [ 9^^ :g i[= += * 
58,5 N'r^ _ il,.u=+s7'6 /('n' l -enzso 
0 l | | | L | | I I
3.46 The rectangular plate ABCD is held by hinges along its edge AD 3nd 
by the
wtre BE. Knowing that the tension in the wire is 546 N, deterrrine the moment
about AD of theforce exerted by the wire at point B.
t --"
! = Ag =(o,VS^),
, _ f . r | | | | I | | | | | | |
Ano = )no.(Zxf)
l^o = #= 
-(o,"ri,!r!:(u3^)&
125 mm
J-
l l
F=
300 mm '. '. ' ' ;3 b1lsm) i+fu,zzfnn)2'+(o,E'nF )^_= F ?Z=!S+td-88 gE. \ : . - ' ; ' 0 ,525m
F = -(+mN)l + \e,t N)! t (i:6 N)&
f.4; = -t- 12U,2 N'fi1
A D
d
125 mm
] - '
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3.17Ttte 23-in. vertical rod CD is welded to the midpoint C of the 50-in. rod AB.
Determine the moment aboutr4B of the 235-lb force P.
-r-_l-_-T---r--r-r
'(zv ;^)! , AB = sDin
so j - o,qv!
t l l t t l l l l l l l l l l l l r - l
W* tt o'l[ ̂ prrlv +h" fwce ? af- rtoint $ :
r f t , 
- I - - '.:- I
?oin)Q
+ (ta ;nS +
= (n s/b)
, l l l l l
f = ( rcr n)i 
-( tqo h)i (fo tb)&
The n1$wtevt | 't'F ? abo"f ft$ it
fraa= \ts'(9r7,tx f)' Igf; 
-o',
IrorD 
-Y
= o- lgfD+?S6 -0 + jc+91270=
Nnn= J
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3.48 The 23-in. vertical rod CD is welded to the midpoint C of the 50-in. rcdAB.
Determine the moment about AB of the |74-lb force Q.
(
l
30 ir
t, t
I'
)u -16
Y
I
17 ir
n
)
Ir
X
,,
-2r
7
n
T
v
\
\
$-t-
12 in
{
v
3
l8 ir
AB '(sy.)i- QoiM-(zv t^)*, AB=
A^-=B = o3Hj - o,6oj- 0, q||g-frb- ft8
r r l l l l l . l l l l l l r l
W" Shq ll appl,l lh, trnr' -4. it Eoin'l
!nl u= 
-(sZin') i+(tt t^;
c? = - ( 16'" ):+'35f ;n') !
gt-+#+f
; Q fr '(tlu tb) 
-tbi ::i
5F-,t" ' , ' , , ,7,x
Q= - (qL 0i - (rce tb)i 
-0t /L) &
wlol1*rt , of Q al to, : t f r f i i1
.A , ( t 
' \ [ o ' ,6V 
-o '60 :o 'vB
6 
-n6 ,- qd 4 ) = [ 
-3tin. l7 in. ow 
l - fbt t -n-bl /o". . -72 i |5
= -7EJJ( + o - n3r,34 -w3.Je -o + t3?1,v
g - Jt | 1.7 lb. in,
H lrs 
- 176,6 l l ' l
5Din.
H,.
)a
TA'
rlt
t: a
t
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--t
-l
9?
-t
_-l
5 N
I
lff
l
l-,T:
Probl
4,
I
;
em
=
D
l m n
3.4
=
.q
' o
-TE
r9
T
-90
"ET
D m n
1,
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t-
3.49 A couple formed by two 975-N forces is applied to the pulley assembly
shown. Determine an equivalent couple that is formed by (a) vertical forces acting
at A and C, (b) the smallest possible forces acting at ̂ B and D, (c) the smallest
possible forces that can be attached to the assembly.
(a\
F6l = ll
F(o.86m) ; Q?s r!,r4
F=?7lN
Srr*Wrsf Anrrr F and .. F
qt & e,nl D rn;$+ he
Per7endie,u lan ta BD.
Fcl =M
P(o,?sn)=g- t . .s N,M
F a 390 l'/
S^ailo{ possibh f"r.*, rfi,usl
be locofeJ acrna* larrqas.t
tfrr'le,alionr An o;d I* rt, AD.
Fd=f1
F ( uae n) = 17.s N,tr
F = ?sbN
( rr)
(c)
r
0 , t56
L
t '
I
fl
F
t I I l - | | : -
T
I-T
P
2 in.
- t l l l l l l l
- 
3.50 Four l-in.-diameter pegs are attached to a board as shown. Two strings are
- passed around the pegs and pulled with forces of magnitude P = 20 lb and Q= 35
lb. Determine the resultant couple acting on the board.
, l _ | I I I I | | | | | | |
Diif'qnce d. betw""a ho"tgonfi{
Disto^nt
4,=t-t4+grt= 7im-
dL bet**n vertlcal forces:3
dt= *rS 
+ t : 1i^.
Pd ,- a. 4z= - (ao uXz i*,)- (gsfix 
tt in.)
= - t ,#O- l 'Qo= -?EOlb* in ,
ry = Leo lb,ln,?
P1+J
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t t l-l--l -l-Problem 3'51
_-l-l--T--, 
| | |
l)
5(X) rnnr
i
I
H 3.51 Two 80-N forces are applied 
as shoum to the corners B and D of arectangular
plate. (a) Determine the moment of the couple formed by the trvo forces by
iesolving each force into horizontal and vertical components and adding the
moments of the trvo resulting couples. (b) Use the result obtained to determine the
perpendicular distance between lines.BE and DF.
r C€'ho pone.4fu6
P=(do r't)sinsoi= 6 Lu N
Q = ftoirlc"sso"; fl,f 2 N-
+rrq _(i;Si[ii;]u
f"f = + 7, 91.6 N,n t4 P 7,1 3 N' n)
,' (b) Eisl-an.- fuh*e^ /a"s 8E a^/ DF.
=tV /"tr= Ft'7 ,376 Al ,nE(narr l l ;
)
/= o,oql6 n
,{- 71,6 nn {
I I I I I I r l__l__L__L
Problem 3.52
WK.Y
, rl l-f
3.52 A piece of plywood in which several holes are being drilled successively has
been secured to a workbench by means of two nails. Knowing that the drill exerts
a 12-N . m couple on the piece of plywood, determine the magnitude of the
resulting forces applied to the nails if they are locate d (a) at A and .8, (b) at B and C,
(c) at A and C.
@) M-
12 N'rY1 =
fA) M =
lL t' l 'rn =
trd
f (0,4e n)
F= 2e .7 A/
J
d.=
0,2H 1n-T
d o,'r5)7+(oJ a)'
Q, sto m
, 12 N,tn = f (o,ft o*)'
f= "?3,5 N, 4,
( ' )
Whe ve
{r,l =
T
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Problem 3.53
3.53 Fou lf,in.dialneter p€gs arc attached to a board as shov*'n. Two sfrings are
passed around the pegs and pulled with the forces indicated. (a) Determine the
resultant couple acting on the board. (b) lf only one string is used, around which
pegs should it pass and in what directions should it be pulled to meate the same
couple with the minimum tension in the string? (c) What is the value of that
minimum tension?
(*) ReE-.ffo"f 4,e<tpl-
+) M = (60 hf,to,s n,I
+ 1+oit)( B,rinJ
= 6Jo lb. in. * Sfo./b, i f?,
/4 -, lt.70. lb,ra.5
b ireofto, o f, hr.cc s. t,
Wfr, ffSs ft=qd n:
,ttlirh ffi! J ryqt: 1:
G) The d,rshlnc< beil'rcen thg
Tlr" re .frsr.* , ffi"
ls
ld. *'15t -h4v z
fi = Fd..) '
g- f , , ' lo {
h
Ib ril.,l"' {
.Cen.fzr,s o !, t't^.r,. .fW-PffS. i:
= /g ' ih .f|ziiitr + (q'in,)"
perp<rwL,'cn,la,r d,.t.lj.anc.e d il+rn. th< f'rxs
d *' llin. + z, (fr i") 3' J$,5 irt'
I t7 0 lb-in.= F(t6.s in) F = 70' 1' 
tb{1
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3.54 Fow pegs ofthe same diameter are attached to a board as shown. Two strings
are passd around the pegs and pulled with the forces indicated. Determine the
diameter of the pegs, knowing that the resultant couple applied to the board is
| 132.5 lb'in. countercloclnvise.
Mrohng .b" 1." d fhe cl,'ae1*o
- 4otut7,in,*d
= /leQ lh,ilt + IOO d
Thrs: lUl,5 =
d = /, /25 in,
Jl.J;5 The a:des and drive shaft of a rear-wheel drive automobile are acted upon by
the three couples shown. Replace these three couples by a single equivalent
couple.
Fl = - (tro N,^) ! +(3 ro ̂ r,-) lg + Gsl ,t,n)&
lv = -(ro N.*): + (aoo rl,-) &
lv l=M = c l f ,sN'nn 1v! =5t81/ ,m
co.r?*=k = 
#= 
-o,Z+Ls 0*=/o+,0"
cufl , f =4l#:0 f l=qo,o?
d' f4
cos 0, = 
# 
= 
##-- o^(iot tE.= t4,o'
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f l= 14+M
f l f 
- l - -
l3,oo lb,l't
| - t
1,4 =
fan 0^- t = E, = 2,V{ O*= 67 SA"7 -Nr ,
on"=f0". q= qo"-67,J8't12,62'o'-a I ?
[\ = t 3,oo lb,+f ; q = 67 +: Of =?o) 0r= 
I L,6o
I I 
- l 
| | | - l - - l I
{
l t l
t4
I
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3.58 Solve Prob. 3.57,assuming that rwo l0-N vertical forces have been added, one
acting upward at C and the other downward at ̂ 8.
r l l l l l
3.57 Replace the two couples shown by a single equivalent couple, speciffing its
magnitude and the direction of its a:ris.
l l l l l l l l r l l
Cowlc in Plan< A9CJ.
- t
lAe. I ax'ples s hown in .skdch
? =60 ilt?= 6o 0!#,=$oN
CE 200
Un
+,80N,n-
T\e rr,ovnenl U, a* th, od.cilf ton a I rnvytlc\
is obte"inrtr L"l a-rtputing fhe rnovnent abort C o{ Itht
/D - N 
(are it 8." Wt ha'IL ;
J I {ilil 
guL= (r.16'ij +Pllvli-(o't4/ft)&
/-40 [r] lutrx(ro'v)d
E itntl^rsiqls I r' tnr",
ft ' !,r/o N,
d{so r}ff=(roil #=3oN
= - 2, lF} N,ttt,
{ofr)
1.60 nrrn
e,BB 9) -2,4 D1i'+ Ft,ry.i-
tl't N,ilJ 17,10 N. m) &
fi-= X,72 N,m
l,ttt+/l,i2 t htq= 1,28/,1
58,0"? 0o= 61,1'
y=y
M
14 =,ft
ht o^ t
van
+ (t,
1,
0-
g=
t,6! )
{
,71
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t'
- 3.59 Shafts I and B connect the gear box to the wheel assemblies of a tractor, and
shaft C connects it to the engine. Shafts A mdB lie in the verticalye plane, while- 
shaft C is directed along the x axis. Replace the couples applied to the shafts by a
- single equivalent couple, speciffing its magrritude and the direction of its anis.
) i *= . 'B4o a )so- - 
- lD2 '6 tda l J
ztuzr- 
- - ' t - f lTm +sD{=26
4=2tf o/b,-f f ; 0r=113.0", l r=72.7i0r= 13,2o<J
--r-+- -[p4-#-#--] l--l-
Zi
,tr
W
\
#
l,00llr.ft
: l
I
c,
I
-l
B
a
M 2
3a
3.60The couples Mr and M2 represent couples that are contained in the planes ABC
and ACD, respectively. Assuming that Mr = Mz: M, determine a single couple
:o't*","':*:*:*i"**:ot:'' r | | | | | r | | r | | |
H,- -M"=A
a,= rF 
n4I uE'-
=hHtrrtyi
S,hgle er1 uivalent corp lqwi l l hale 'Mu*, nt
U,+ Az=
= 
f.rt-C-,*)n-r ++ryt
l l l l t r r r r r , , r r r r r t l
I_[I]f 
v1e -- o, sss,'1! + 1,2'79 H 
i 
+ 0,8i{ Mk-l--T--T-n
flt- z
lc t
I
xl^
E
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3.61 A 60-lb vertical force P is appliedatAto the bracket shown, which is held by
scnews at I and C. (a) Replace P by an equivalent force-couple system at B. (D)
Find the two horizontal forces at B and Cthat are equivalent to the couple obtained
t t t t t r t t ' l l l l
F= 60tb
E=6olb
( {ou)(7,s in,),
4Eo lb,in,
!b = 450 $';n,) {
W= too l[
tfr,
9 !; l0o lb '* 3 g= loolb 
+{:
3
= 4,5in,'f
ProblEm 3.61
t l l l l l l
r r t t l l l t t r
3.62 The force and couple shown are to be replaced by an equivalent single force.
Determine the required value of aso that the line of action of the single equivalent
force will pass through point B.
l l l l l l l l l l l l l l l l l l r r r r l
tt lh._ ltne of actiory o{ I he sr'n3/rl T Tne t tn ( o r ,
€ Jvt v rt l€ rtl f 
wr( (6 sses I h ' :ni,? 
8 ,
;l;;""H Au= a', 
u/e rhoil cotrtpvte *tn'
n4 o?lrrl1/s o/ 
-"n, 
(otl'ple a*'d of th " 
760- $/
fowe, a.fJr,",u,g' fn{t ' ihrc /ntlp, isqfylielal 
0'
z L1s= 0":o!,{':il 
!,x(q6oN)(cosdr.j + s,^41 )
= !'rl N,n 'k - (q e il 'm) si n e( E
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-l
I
1
I
I
I
- l
i
3.63 Knowing that a= 6Ao, replace the force and couple shown by a single force
applied at a point located (a) on line AB, (b) on line CD. [n each case determine
the distance from the center O to the point of application of the force.
l
lrlt f t: + fe o l , e I l-' e c Je + a 4r v i e
2l ol ata U I )+ ,f a
1t o N A 6t o a o +l e 1- 12) - l
+r tA e s C ) )
,4 ( , T L N\1O n i " v , ) V 7_4N W.
a
?, t l (- I \
5l l c ( \ s r a'
?.\ Fa r e A f i n T I)/ J o tA ? A f
o
,16'tl?M nVe n + J ] , t^f4 K ? ll" 4
t a N t4l1 ( kr\ l\ S )C- ) K
d zr N,nr i Lt N 33l . t-> ^ a) O
rv')
\ ; ,1 ta N A 10 ',9 ?̂ n Yvl +, he r Jh l I D I ) 1 {
t-t t-Tt-T-[I-l .r) F ) n e n t O tt O A j y\ r( -1:-_r--i'--1_-l-_r-
tl (Y lu e- n t ll i n + In
('l /V Vvl, 1ac) S r
'1, o i ) ̂
L_(
Itl u W L+4c
,ff (J . t J : n \> n n
-l 
r]lt]-ll]
-T-T;I:
?
ft
ft fc -g ,1
' , f
N ,1 )o 5c YAv o l w D I ) I I
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2.5 in.
A
rl---r-Problem 9.64
t t l
t -T - t r r r r l
I
I
4 in.
260lb
-
2 in.
l
t t t r r l r r l r r l t l
3.61 A260-lb force is applied at A to the rolled-steel section shown. Replace that
force by an equivalent force-couple system at the center C of the section.
I | | | | | | I | | I I I I r I r r r r - r - | r
2,,f in,f l= 260 lb ftB =l(i , r)"+ (qn = 6,f i,t.
T'
tf irt,
. s i f q , =4, I=J,CoSA=6 =&
d= 2I,6o 6,5 13 
' 
i,3 t9' 
f =.-Fs ind ! Fcoso(, i
=-(zooh)fr!*(teo,t)# A
f '= -(too'tb)L-(zuoU i
Ur= 2oo lb,irr, = t fao+- 6m&
U. = *&uo lb;i^.-)&
Uo= ?oolb ' ;n , )f= z60 fb y 6Tl'3
L
t t-
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Problem 3.65
' r t t t t t t t l l l
3.65 Force P has a magnitude of 300 N and is applied at I in a direcJion
perpendicular to the handle (a= 0). Assuming f = 30", replace force P by (a) an
equivalent force-couple system at B, (D) an equivalent system formed by two
parallel forces applied at B and C.
t t l
l l l l l l l t t t t l l l l l l l l l l l l
affi
Ne ha ut
- Pzro
(a) Ee u i v^,lcnf f,n.te - o>qo l< gc1 .S'tewr
- t
I
-
?0
c I= C
Sinre ti^e f,rtct
M
t ' b
i s g<rpenr l l sv t . l a r fo AB,
( t* *)(0,2s,^) ' 7T N'nr
F=lffN
We
= 500N
8 =000 N;rlo' {
C=5nN43ao
futr*corple;syste,*r at I :
F= 3oo tt fi/10"/ f;l =7f ll'm) <
,b) Efr,r^ P^f f r* tlrl fircrs af I ,.nd C
W" regre5enf ffu c,)'Ple Ue b7 fi'''o fivres 
Q' av(
-A at 8u, ,dC:
r= 3rV
C
harc. Ns=8(0,t,s^)
75f l '1Y1 €e (0 ' t f^) 4
-0
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3.66 A force and couple act as shown on a square plate of side a = 25 rn. Knowing
that P = 60 lb, Q= 401b, and &= 50" ,replace ttre given force and couple by a single
force applied at a point located (a) on hne AB, (D) on line lC. In each case
determine the distance froml to the point of application of the force.
l l l l l l
, Thr g irnr
e.qvitalen,f
Fcw6s ano( eouple are replacrol
f ,nq - co v .ple Sysfu,a at 
' 
fr i
l l l l l l l , l l l
n $- *l tr,:lp' 
O= 40lbi- Zfrn.$l **
a" -^7f in. sA
= ( +r tu t$(zs;") - ( + o lb)(zi in, )
M = 141, oJ [b,ir,)
41q63
F*=7fr,r6'7 h n
B5
d
- l
a !
60 th
A
l ig , ig l14 = ltfqi
' lb, in
A (LJ I
= 45,qn lh
futua lthq 11s3vr+n {e libo,i A , ,t, lr,ave I
f \ f
For f,osi l isn (^)' : 141,O7 lh,in,= (+S,qel b) x, L 
= 3,24 i n
f = 5 a lb E so? 6't prinL 3,?+, /nfrom, A, :{
FJ, yusr-f ron (q, l+q,07 lb-in = (3 g,rtl /5)d , U 
= 3, 87 in,
P = 60 tb K so" at p:tvtt 3,87in, hlod A { i
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3.67 Replace the 250-kI{ force P by an equivalent force-couple system at G.
t t l l l
l:--(rroLN)i
3o **t
ryL -- !*/r\
:hroo N"'t & rQ5 oc'o N'^)i ,
The orluivaie',t 
tnr 
- toltle
V.nr)l Sls l-rlr,, at G ie'F=-(zro.hili; 
{
M-=( ls&N,r) i +0,s4[linr) ,fg 1-G
l l l l l
3.68 A 4-kip force is applied on the outside face of the flange of a steel channel.
Determine the components of the force and couple at G which are equivalent to the
[=(i k,Vnt
frcs A 9* E = V * 
- ( t'37''o''8)'&^]'( 4 i
-- - (lXD/ -prqs)(q$
= -3,lBi-16&.
Problem 3.68
t r t l l
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3.69 The l2-ft boom AB has a fixed end I and the tension in cable ̂ BC is 570 lb.
Replace the force that the cable exerts at Bby an equivalent force-couple system at
A .
' 
fc= -(n@i +(+,s ft1ft- (B ft)&
BC: / l ,E- f t '
. ___s
570lb
F; - \sott') i+(rouh)!-Unt\&
+e#b
3.70 Replace the 150-N force by an equivalent force-couple system at l.
F = ,*(lro N) a#s? j -( t So N)t i?t 3 S'4
=-(22,q N) j - (a6,oLr N) g
9rro= (o, | 
.B^)i 
- ( o,tz,",,) (' + (0, to n) oe
=&-..^xF=fo,,ir -o!?* o,tf* II -DfA"- -1"; _tzz,qN _ii,orrf
=[21.n, i l ' ,^): n (s.v1 N,-l LF?,t N, ̂ ),t,
60 nrrn
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3.71 The jib crane shown is orientated so that its boom AD isparallel to the x a,xis ;
and is used to move a heavy crate. Knowing that the tension in cable AB is 2.6
kip*, replace the force exerted by the cable at A by an equivalent force-couple
system at the center O of the base ofthe cftme.
L,6k I tir,
/ 6,25lV
J.
ftB = - (/5ff) i +6,2ftr) .t
ftg = 
-/ 
6,2f{+
= - (2, akip') j +'( tktp)&
\L = 9A x f
where jA = Q o ft)1 +Q s ft)_i"
--b xF=
quivd [e"F hrce-o O ,
$=( 
'i[ 
f;fr)frqF
Problem 3.72
l l l r t
I;ZLAIZOO-N force is applied as shown on the bracket ABC. Determine the
comaonents ofthe force and couple 
"j, l"i 
*: *:TTI"':. 
l r I
f = -(Zoor,/)c"s 30" j + \oo,v)cos 6f &
7 = -filzN)jf r,,nr)&.
M = At ' x F-R
vthe re Lt = A? = (o,oe o *)I+ fo, oTS"n ) X:*"
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Problem 3.73
l | | l l l l l l l l l l l l l l l l t t t l l l _
u*to
3.73A lz-ft beam is loaded in the various ways represented in the figure. Find
two loadings that are equivalent.
r-**--l l-
I I-1801 500 r I
200!b s00lb
lb.ft
(c)
200Ib
lb.ft lb.ft
(b)
200Ib
(c)
z00lb
l l l l l l l l l
Answer 1 Loadir 'rgs
lb'ft
(d)
Foreach Ioad;nX w€ l ino l an €4v iva l tn t
l ;;rr- ioupt* s1su1e ,r't , ?', 
r,^', 
,+ ,tlo , '' r , t t ! r . ^ l l l l l l l l l l i
It-J-l #niyj'.; , r. \
lb-ft lb.ft
(el ffi
i i l l l l l l l i l l l l l l
(e)
l l l l
2,oo lb
A$=+ I Rrvl-, i/nnl/rr) l i n.,^ ,
r r t ! l
fn \ LwlDt 14 . = -
i'/-I / r--------r rWH nl=-r 1oa-Qo,o\rz)
tWl ; - 4 Loo tb:lf \-/ 4 Zoo b-ft
v 5s6h.rc \7ffiFF
/r \ Izoo tu \_ I zoo lbt- > -
t - l -
vl 8oo tb't? \-/ | Bao'tb-ft
(d) _4,^tii,f;,;#, W
t Sootb-tfL/ \,/ 4 zoo tb. ft
ro ) Zoob$ fAf=+4 8oo -(zoo)(re) fzoo ,u
- r . = - L t r n I r ' . F r i - i=-6oo l b ' f t f f
L/ 6oo lb-ft
ff) zooul ML- t6oo-(z*)01 W
=- / Eootb ' i t r - l
\-/ I 8oo lb-tF
mf'= -t Boo+(zooltz) i zoots
(il '" :-4 
'= 
t 6oo ta'tt
eoo tb-ft
(h) W r$f=*6oo h-u I,t"'+ ,
\y'' 600 tb-tt
and f
| 8oo
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3.74 A l2-ft beam is loaded as shown. Determine the loading of Prob. 3.73 that is
equivalent to this loading.
3.73 A l2-ft beam is loaded in the various ways represented in the figure. Find
fwo loadings that are equivalent.
(b)
200lb
rb.ft
(a)
I - lb.ft lb.ft
-i - (,) (f,
L l l
:---l-.+---r
<ffi U-J6or m
r- \ | zoou(c) L'"" -r r e)
\-rt I 8oo tu-r? \- ,l
200Ib s00lb
rzft -l 
t1- 4
r -Tffr 
trft
(d)
l* ,q
For each load;r5 w€ F;n, l an 91r;val tnt
lorre - eo,tple s7 ite rn th e lfr e n ol ,
(a),ffi nl=-r,soo-(zooNrz) W!- 
-l] 
crl
@J 
: - 4 zoo tu'lt \J + zoo tb'fP t
@ :'4 
nl=+M,,.';#'o1g 
-JI r'l W,
ts.''ufl 
=+1200 
\-r+. rrffi T 
" 
%T'rtJt
Ml=+sootb'tr Ly
l l l l l l l l l l l l r r l
We re Ptace 
tht 6 
r'vt n t 
?:o'," 1,.ol :
[rr- Jo,rp t" s1 e te"*" 
al the te rf enoL i -]+
, , , l t | | 
' , 
, I I | | | | | | I | | |
Eo ht foll l F 
-- Zoo lb I 
t rr
,ffi;l=6oxrz) .,%'L* - )e l io tb ' l t \TGootb ' f t
m[=+r Eoer'-(2oo)(rz]
= - 6oo lb.ff
\-r'6oo lb-ft
d/'6oo lb-ft
200lb
6m lE'
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3.75 3.?5 By driving the tnrck shown over a scale, it was detennined that the loads on
the front and rear axles are, respectively, l8 kN and 12 kl{ when the truck is empty.
Determine (a) the location of the center of gravity of the truch (D) the weight and
location of the center of gravity of the heaviest load that can be carried by the truck
if the load on each axle is not to exceed 40 kll.
l l l l l l
rSkN12 kN
l t t t t rL t l l l l l l
t*) Ce"rler o1 gg af lrvck
5 m
fl" syslcm b,
GtIe7 oiua lenf
IB KM
7rl
ZHe
t 'LkN+tokN--uJr W = Sokhl
?t*.v)(s*') = QotcN)r, Et = /,oo rn
-fhvs, 
Gt, is locat d 2,00 rn (r,rn (ronl- axle
(6) \fuB@;rn at mcvintrtw
'[-h" 
s7 sle,m carnS itl' r.3 o+ yt an/
'eclaiua ldn r- +u 'lhe rf ̂ y i*ru.* 4O-kN,
ltoEu
C
ffih
2 q = t F, ;Tic*io t,u = 4otN + vo 11N y/ =soft,\/
Trlrg - Ztle: (fik N)a [ + Go rlr[t n) =(t1 oh, AGD, r,r=2,80w1
Hwvidt lool is 50kN, /xot,ol l,,B\n fr,'n f"nr 6,X/ev u l , ' - . I 
,
1 r /
ALs*i9
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3.76 Four packages are transported at constant speed from Ato B by the conveyor.
At the instant shown, determine the resultant of the loading and the location of its
line of action.
Lhc e7 uiva Ie nf force -
fhen t!5e u 
7riva 
leo{
+t K = 4oo t zsort5n+ rr lo=l3oolb
i t t l l l l l l l l l l l l
rt l4t= (u oo[e)+(e 5oX6) t(sdo )+(coo)( t r)
' J a \ n = 60o + 15.00'+ /9oo+ 7f,rrJ : lllr.to(b'(t
r l l l l l l l l l l l l l l
E= B&tb I F [^* 
= lt Soob'# )'
K : l30o /b
-
-
g= l3oo lb
/V l= rt 300 b'ff 
-A
A4R= t t f rolb, f t=( lsoolb)d d=8,61 {t
:- '6 -
Rd:
= l3oo lb
B',6h +:t fo 'ghi' if fi'
3.77 Determine the distance from pointz4 to the line of action ofthe resultant ofthe
three forces shown when (a) a: 1 m, (b) a= 1.5 m, (c) a -2.5 m.
,{ }rN
t3 D
l t t i l l
VJe havc; + t R=-l l -(QN -B4N+ Z&,N =-to+N
R, = tlfrN I
,r*= B(t,r)-!= ta t( = 
ffi, 
= ln,l
H: =ahr)-t;tet d =ffi=/,8n<l
+) nf = (z& u{a *) -(q+p)(t^)- (a *u) a = t&N,rt -(a ftt th
Mf = a(D-t =g d= ffi=abn11
NR:-A
= l rnZ(a) a
( i l ds tsm:
I (c) d:2, fn:
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3.78 Two parallel forces P and Q are applied at the ends of a beam'{8 of length I.
Find ttre distance x from I to the line of action of their resultan[ Check the
formula obtain by assumingl= 200 mm and (a) P = 50 N down, 0 = 150 N down;
(b) P = 50 N dowrs Q= 150 N up.
+lz[=zt:
+') Z H^= Zlto:
P+Q = R
Q,L = Rx T = qL x =/\ - ""?-
t<
Ch eck f o. L = ?oo m- nNo
(a) P=gN t , P=*€oN
q = tSo t l { . t 0 l = l tg1 t , t
r r t l l l l l
fb ) P= to lv l , p=*Fo i l
Q = /€o,vN, Q =-/So,{;
/: t{9t4q x. = tto m,n
.SO+ tgo
X. = :lSIt (zoo) :
'o- r f i xs3oonn
A ( nF*arv,r€(: K = ]BTB lbVn+-7d, M9 = i otu/b-ft\ {Answtr :8=t878 lby?Srr .7 ' ) Uf = Zqsolb- f t }
S _ 2 4solb-ft= l4,oftoL=- 
t
f rrs; R=tB78 16
14, o f tr qbove A
j.79 Tltree forces act as shown on a traffic-signal pole. Determine (a) the
equivalent force-couple system atA,(b)the resultant of the system and the point of
intersection of its line of action with the pole.
, r t t l l l l l l l l l l l l l l r r r r l
Le) ETuivaknt ftce <upk t1sttn" of ft ,
1 lzs.6<-1-41 8r= 
t75 lb +
| +zo6lF-, $ = 1870 I? +
l- ,''otLootb ,;=;f#,il?#, f#hhuft*
)z utn
l*4*
hv
I
1,200lh
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Problem 3.80
l l l l r l
i- l-]_-+--, I I ,
3.80 Four forces act on a700 x 375 mm plate as shown. (a) Find the resultant of
these forces. (b) Locate the two points where the line of action of the resultant
intersects the edge of the plate.
(ioon)j
@) f = Z!=(+ooN+r60N-76on/)-, t ({oolt*300t!14ili)i
= -(towN,lj +(tuooil)i
R- = 1562 AJ t To,Lo
il y: = 7 9x! = (0,5,n) !x(6o0N)!' = (7oo N' n) &
(tlmt,tli
-Qnqai,
(1oo tt,,i g. = 7! x (rcoo r'/7i
=(zoo0r t
pu>N.*) & = g i * ( royov)ii
=(/wMJ t
[nlrfsrrlip,r .?52&o,rn * lv ri ght- of C dn/ 3 D\mnt 
a bove C '
i i , l0 N 
'-
5oo tr
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0,r", ts tt) i
tbJn)t:
3.81 The three forces shown and a couple of magnitude M= 80lb . in. are applied to
an angle bracket. (a) Find the resultant of this system of forces. (D) Locate the
points where the line of action of the resultant intersects line AB and line.BC.
-(o R.r d
12in
(t7,5 lb) i
Pl B in . =
, t-lffit
@Nered{uce STsfem fu a fore ilrd a coul.tle a-f f i t
8 = Tf = ( n,s tL- tt 0 tl,) ! + (t,,6s tb- to ftA'
= _(rr,s /r)f +( tt,6t l!)n &, J1,? f6E23,oo.d
ryE = (aorb.inJ &,( tz in)i x(-tltb)i - ftnJjx (- H 0 tb)j
-- ( B0 lb,irt. t lL| lb,in, - J z0 lb,,rrfte r -- (tto /b,in,/e
{b)
d.L: /0,J0 in,.
z= lZ in .^
,ts tt)4
10,30 in. . .= / ,70 in.
[=-gtzotti,)g J*--lr
Je-
SE=-(tt/.th,in,)!r= tl *( z7,rriliu^
(Uo tb.in.)! = -gz7,s/6)* t
l t f : 4 ,96 in ,
U 
= Oin, - 4,36;n,= 3,64 in ,
b th" r ighf 
" 
F A and 3,64 in, a lcore C ,
_(tAotb,;n. \ k =_(t t ,6sl | ) .u &
1,70 in ,
8 in.
l,
I
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resultant of the system and the point of intersection of its line of action with (a) line
AB, (b) line ̂ BC (c) line CD.
ft = 30!+5D ! - sli + ?5j = (80/b)i 
- (t'U
I | | | | | | I I I 1 . : - . 1 - l , - l - . I 
- 1 - I " l 
- r
1 . ' 
I I | | I I I t - I I I I
ttt = 6;J y 30! + tLi x?Sj -2a0.k - 40o.le i
= - l8o& + Soat -Zoot - t l1o! =-( ' tB0 lb ' ;n . )k
- t r 804 =-BoA&
4i:W= 6in,
- ( +go lb,in )'k YVe ltw J have
- 480{8 = .zt'x (- 6od)
-+sot= - 6ox &
*=+- !=o = Ei r t ,-50
?oinf af irtlernrt,'o', wt'fh BC ts 8in' 6 rgAto{ B
g
G
'Botb\
Wemusl have f'd = ?c--Fto,tb,in,l 
g =gru,A3t U j]x[(8olb[- (6otb)iJ'
l l l l l l l l l l l l l l l l l l l l
Point. of rn{erseefrbn, wilh LD rs 3 in- belu L
6otb)j
- (oo ruli
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3.83 The roof of a building frame is subjected to tlre wind loading shown.
Determine (a) the equivaleni force-couple system at D, (D) the resultant of the
loading and its line of action.
(a) tVe rr7b rc ltae
oftkroo+ bY a
load;rtg on €acl, s.ide
single 
- 
60 zN f 61rt& '
f =(t,o*n) +'WD
F = (r,anu/j + (sdte uDa
D | = (r,att*)1 
- (ser z uu) 1
R = [ tF' = (3,7rru) i
I I I | | | r r I r ' ' - i l - t - t \ t - t i - t
| : = 1i,r^) gr blitbil + fui.1i x (anr)X: a fus*tix(st..,-)i-'c)
= -(zg,t6ry,*) t -(?56 1w,a)*, + (T(,gs La,n) k
= (?A,a I kll^) p
n 4 - ^ - t r ' - n t 0Ftxce-couple systen : Et 3JIsLN+, ry::Yl!, )
lf l-.l I | | | I I | | I I
(il Sing lr e7i;vale"l ftrce is /5is rl oriZyrhl favce I
A:- riffi = -dzn
S,'ngle ftrte rS R =3,? ? kt'l* t E ni be lnw -DE {
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3.84 Two cables exert forces of 90 kI',{ each on a finrss of weight IP: 200 k}.1. Find
the resultant force acting on the truss and the point of intersection of its line of
action with line.,{^8.
0t,qq r.l.r)!
E.6n 7,qr kN) i
7.?t*r
- (,tsN) f-
'-itooxntli =
-8q
l*- x
ft, = Z != (77,q+kr+7l,qvk{); 
-(+st'nt+qst rl + 2w kr{) i
)kN)i N ft=l2qkVss 51,74<
j -+til *V,q !)xkoqL)
- { f d ) 
n r t t . a n - t . . \ D- \ gttt,E 4€ -ttga & -?,t,r + -i{t,Z t = 
- ]qTI,FkAJ. ,t *
Wo'** P tr thr .Bhl sa ffi
Hf =7F 8,
- fii,fi,r,t-=rj* Ftql i)
= - fl,q\fr' &
6,82n fo.nhtof f $
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3.85 Two forces are applied to the vertical post as sho'vrm. Deterrrine the force and
couple at O equivalent to the two forces.
8 = EI= - looj-3ooj t3ooi - 6ooJ 
- 2oo&
4: =E941= 9^xf*+ es
= gJ x (-+oo-i - luo,i )
+ I ,-1iii 
- 6ooi *2oo!)
=tlrr lo!- t8oo/4- t2ooi
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dimensions in meters
S.E[Inorder to move a 70.Gkg crate, two men push on it while two other men pull
on it by means of ropes. The force exerted by manr4 is 600 N and that exerted by
man B is 200 N; both forces are horizontal. Man C pulls with a force equal to 320
N and man D with a force equal to 480 N. Both cables form an angle of 30' with- 
the vertical. Determine the resultant of all forces acting on the crate.
- l 
t | - r - - [ - t - i | | I | | | |
C =320t/.
?'tZ -tr tlt6 i
B = -(zaoil) i
A =(too a)j
o,Bzt'+{ . - 2'1"-.f 'f 
o'?zn
W=nJ =Qo,6$(t,Bl- i . i=)= d?s N
l J l l t l
8 = Z I = 6ooi-eooj 
-/60 i-2,,v! +?7, !,r 
4t.6 
i 
-6?3i
R'=0
r;,
+(o,ies1,ti) X (- ,roj + z.77 il
t 1s i x 1- ttt x')
/= (- 3601z.60 .*1gz f 59? f 83,/ +?s6 
- Vat )*
Ltf = (trr n.^)&
SIa ten redurcs lo a covyle ,
r / 
7r" 
s< nted an1*h+re on
E=0, ["4= t
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3.87 The machine component is subject to the forces shown, each of which is
parallel to one of the coordinate axes. Replace these forces by an equivalent
force-couple system at l.
| | | | I I l l t l - L l I I I I | | | I
Tlre fot rt a/^d fhe,,y posih'on tecfurs Vlru_
rcsorcf tT fl o.re ag fo/low s:
l ' : ' : : : r | | I I
;
l r - n -
50mm. Y 
'
ilF-l-; -l-_l
fi= 9.En* ! r 9q^x9 + grnx D
r r t t t t l l l l
' +
I I I I I | | I | | | I l - t i t | | I I i I I
Tf,,* (wrc.-couptc s'lsk,r. a,f 0 is: 
P ' -isoo il. - ftv.ou)3 +f 21 N)!=, , 1 , rfri :Yf;llh;'i ?' {!a*: }; f l'fr u ̂u "il- ' , 
-(3?o N)! - (7v-, N)j +lz7 rr )Y,
fr ,J(tuli)i + (rs,lsnrr,i lI +'U',1,d!
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J.88 In drilling a hole in a wall, a man applies a vertical 30-lb force at B on the
brace and bit, while pushing at C with a l0-lb force. The brace lies in the
horizontal r plane. (a) Determine the other components of the total force that
should be exerted at C if the bit is not to be bent about the y and a axes (i.e., if the
system of forces applied on the brace is to have zero moment about both they and z
a:res). (b) Reduce the 30-lb force and the total force at Cto an equivalent force and
couple at,4.
l t r l t l l l t l t t t t l
vekavt Q= ( to lb)r +td*C.
R= g-3oj=-loi *(cl ,-30)j+er& (r) '
l
&
R
= (Bj '6*)x C3oj )
, t s - /
A - b
=Zyxl F = (ai '6+)x (-3
+ tbi / ( - to! rcyiv1+
=-Z+0&- tsa!t16cr&.-r &: t(qi
_T
@
R
c, t)
98)
{
)and ft) :
Afr = - t loi - t6cri +QGct-1,q0)
cr-- o
cJ -- | 5.00 lL
(u) Srtsh'luhrrg valves, furnl ForeJ r'n hr flat"d e+
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3.89 tn order to unscrew the tapped faucet l, a plumber uses two pipe wrenches as
shown. By exerting a 40-lb force on each wrench, at a distance of 10 in. from the
ucis of the pipe and in a direction perpendicular to the pipe and to the wench, he
prevents the prpe from rotating, and thus avoids loosening or furlher tightening the
joint between the pipe and the tapped elbow C. Determine (a) the angle dthat the
wrench at I should fonn with the vertical if elbow C is not to rotate about the
vertical, (D) the force-couple system at C equivalent to the two 40-lb forces when
this condition is satisfied.
ts ' im.#_
E=
AR
- t
(u, tb)( t - h'q : - (qo Il s;n 
g j
= U n+As + (trin,) & r?$otila'sl! 
-(.+ oh)s;^o di
+ (7,s in,\lex (r+o tb) y 
'
a 0 -50)cosf.7*6Jssin01+ioan ,
fl : = (sootb,in)=sin s r' + 
(lio lb'j'l(l - 2ax 0)i G)
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3.90 Assuming 0:60" in Prob. 3.89, replace the two 40-lb forces by an equivalent
force-couple system at D and determine whether the plumber's action tends to
tighten or loosen the joint between (a) pipe CD and elbow D,(b) elbow D and pipe
DE. Assume all threads to be right-handed.
3.89 In order to unscrew the tapped faucet l, a plumber uses two pipe wrenches as
shown. By exerting a 40-lb force on each wrench, at a distance of l0 in. from the
axis of the pipe and in a direction perpendicular to the pipe and to the wench, he
prevents the pipe from rotating, and thus avoids loosening or further tightening the
joint between the pipe and the tapped elbow C. Determine (a) the angle dthat the
wrench at / should form with the vertical if elbow C is not to rotate about the
vertical, (D) the force-couple systemat C equivalent to the two 40-lb forces when
this condition is satisfied.
\ . 
. t 
- 
_ f , 
' n - . . ' . - L r r . . t { . v t . r , t I u . , u , 5 
} " t F t s 
1 y , v , , '
y,llnlifh.' ftiilien no'tilAre,J oin t betw;&f 
fif, CDqnol dhofi/.,q
G) Sinq. x, ooo'1ona'.fof AI ;t 6-. , plvvrtb<r'sach.ort
hl;ll fe,-,ct fu, f,Il"t" join f A7twren elboq'on/ f 
il* DE, 4
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3.91 A rectangular concrete foundation mat supports four colnmn loads as shown.
Determine the magnitude and point of application of the resultant of the four loads.Pro
R- (tookNh -l
-@okv)t
-
l-
fn
El"ivaftny furc- couyi ld S t s len at D t
R = -(s0kN)i - (ttaLN)0 -'qzoo kN)i -(tookM)/=- goatu)y
8l' 6^)&x (*aok$ +fft*)j *(s^)ri lx(- rzokv)j ': '
,= (+, o k ',, . rn) r. I+ o o( rr,,r.,) iiil# 6! :&Z: ryr!^> t
A; (tooo kn/ m) !- (lzaokrV,nr ) g
l tk wnte ; Al - ft: +z &) x (*toou)i
(t Ooo k r/,') d - (tzao kN,rn),lS = -\ou t)r&{noufu y
EP nf,n$ ** f{ic: e nt,,gorrrtf# ==(W 
i-,r;:N?*r-
500 k t / ' rL, fb m fronr Ac nnal 2,o0n f io, DC - {
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3.92 A concrete foundation mat in the shape of a regular hexagon of side l0 ft
supports forn column loads as shown. Determine the magrritude and point of
application of the resultant of the forn loads.
c 
l,/ 
lo rt
: ("lo tra;edL
f
tt:,rl 
l,
- (,+3,3 kip,{f}i
A
t
fr=-(ts r;fljat L; Qo*) i
f = - (1s k;p,,) i ot l,t ' FE=-[f5k;fs)i"t !u=$ti l i -(0,66f:r)t
Elriue.<^t !: '.",:-"'^l ' ', ,t-t,t lt i : i:"i .* , ,g =- (zo+lo+25+ t5') i : 
- 0or';ydi
Y: = L ! x ! = 
- to*i x f-r i l}r+ (5)+8,66 +)"(-to; ' )
+ toi r (25j) +(s! - 8,# !) r (-rt i)
= 2ooh -,03 + 86,L L - l f l ! ' 
-75* - lL1,l i
{t f 
= - (41; kip,fI) ! -(7s kt?. lt) &
!'( f 7s k;p,Ilp
E|unry c+e ffic:ertts:' 
- | 3, 3, k;p.f t -- (7o k;ys) *
- 17 5 k;p,. lt -- (to kr yts)
7o k ips t at x = +Z'5a1+ ?
I t
/ : - Q,6n +r
f t
x- T -72,50 +f
z = - 0,611 f t :
7
ff = (x! rL & ) r (- rokfs) g
- (lt,J krp,!t) i- (tls k;p,+t) & = -(tokrps ), &{tot;S,s) zy
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3.93 Deteimine the magnitudes ofthe additional loads that must be applied at B and
Fif the rezultant of all six loads is to pass through the center of the mat.
f;-@:,rlit et' Lno'0 o+t)i
f ;* (tP krfj i 
'aE
x. 
v 
L.u=$Et i'! (8,66 f{)'&
.8 . = - (tr r;y)jnL L; ( to rl i
Fu=-fis k,f,)i"t !r= (Stili-(0,6e l0&
! x ! =- /0'*d x (- 20i l+ (5j+8,66'k)x(- Oj)
+ loi / (-251) +$i - B,# !) r ( .rsj}
= 200/e - F03 + 86,1 L -7fr ! ' 
-7s! - lL1,1j
A, o= 
- (49; ktp, +'!) ! -(tI5 k,F, f t) g--r-r-FF-r-__r--.--r-r-r--I-r-T-l 
I I I | | I I I I r i I I
let fu=- 6i be oppliect *t b 
attd fF =' - F, i. 
b, opl,liecl a.t F.
, - | | I I I I I | | | | I - l L l I I I I Ir r r , - r l l l l t l '
horc la=-(st r ) j + (8,664t) ,&
I | | | t , - | | I l l 
I l ' l l l t I - l L l
t(" ho,r" !a= - (s tr)j + (9,664t),& dnol & r = 
- 
ftlt)j - (6,65+t) *
!r= L
Fo, (he resvlb.,,nL Fo paes [h"o,4h 0, ule rrtvsf
lrave Ao*= 0 , fldotng fhe ,n,xt'nk of IAM fr
'tb l'h" Suv,ra u f rno*r+nf-J rf fhe e,*t*i"3 ,!ooJ,a,
We wrile I
ML= -(43.3kir.$L)j- (t7s k;p,{r)&
+ [- (s il! + ( s,66ft) g] x (- Ci)
+ F$t)j -(8,66+P).el x(- 1il = o
-+zJr?, tgFo-8,66F," =e Fu_Fr= Sk;S,s
*17,5 | fF6+Tfr =o 
FatFr- j f rk ip.r
ftdtt;nt ll. 2 e7 s: Z Fu = 4o k;y:
5 ub sf i /u tir'g inlo tAu ,f;nst e1u ution ,,
20k;ys- Fo= 5 f ;ps
E=20 lr ;ys {
Fr= lf kiTs <
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s.g4lnProb. 3.gl,determine the magnitude and point of application of the smallest
additional load that must be applied to the foundation mat ifthe resultant of the five
loads is to pass through the center of the mat.
3.91.A rectangular concrete foundation mat supports four column loads as shown'
Determine the magnitude and point of application of the resultant ofthe four loads.
W be ffre o44ilimal loa$ anol i& and aLcEW0
lhc htordinalL<t 0f i fJ point oF oPPl ic*Fion,
fu fte resoll-c,,nt oF fhe 5 lodls f+ pass 
purouSh O,
r,/e mus1 haft M\ = 0.. mtts| haft 14\ = 0. Tli, /oafu,; are
;^ . 
- lco4 l{;,, 
- '-(ro 
i -so1 nt,g-o= -.2 ! +z.s &'A- '-;
- ts.oj a,t 96 = & i +1,, 'lf
T,s^ -t ooj * ge= zL .-2,r,l i
1,,^ 
-z - lool * !o=-Li -Z.s t
-W,etf 
\=n!+zk
Af :29x | 
= (-ei +zst )x( soj) * (zirz.rk) r (- tzo,,)
+ (z! - Lf.te)x [- zo ai) + (?!-L"rt)x (- roul )
+(*i*+&)x(wi)=o
I6ole ILo,o! -Zt+ot f 3oa!- l too&-foaj
fzoo&-ZroL- Wr& +\ i l+ ( , =O
Efuaft"g the co6ftr.cfe.r.|s 0F ! and & fD tseru;
7 .0o+300- foo-Zfo+Vt /+=o Wt=Zf0 Ct )
l6o: | r fo -4ooteoA-WV=ct Wr=-ZgO (Z)
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3.95 Four horizontal forces act on a vertical quarter-circular plate of radius 250
nrm. Determine the magnitude and point of application of the resultant of the forn
forces ifP = 40 N.
I t l l r
2 f= ltzo N)! - (Bou); r (?00 N)i - (tro N'j | =(zcor)!
Zt=x E= (o,zs*,,)d. r (lzov)i
+[( o.zsa )co* u"j 
.? (0,?5;r ) Sin3T't] x (-aoil) '
+ (o.zSn') t x Qoa N) 1
=" - (3oM.n) fi + (n.stu.z) f - (loru,nr) !
+(so v,n )1
f l- ]yro[/,^4',1{' - (tz.6s N .^) E- 0
C,": Q-ir* 4) x (zotory) 
j
tto N,a =Qao N)u
- 12,68N,n = - (20o4# 
T 
= 0,011y m;,63, L{ffi'ftt
Hognifvde af, r€sultanf - 2oAN
?otnt ,l 
"ppltcat.ie,t 
at
#=63,4uryn,2=t00mw,
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3.96 Determine the magnitude of the force P for which the resultant of the four
forces acts on the rim of the plate.
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3.97 Aforce P of magnitude 520 lb acts on the frame shown at point E. Determine
the moment of P (a) about point D,(b) about a line joining points O and D.
3.98 A force P acts on the frame shown at point E. Knowing that the absolute
value of the moment of P about a line joining points Fand B is 300 lb ' fq determine
the magnitude of the force P.
z
\
? = ? Lrn= P U,qut j - o,zio1J' + o,vo77 &)
r r l t t l l t l t t t l
lhe rh orncnF aF f abovl t l & is
Hru =Irs'(?,rx t) ( l)
W" have 7t: po;n) , +.(tri,)i -0oir)d , f B=3s)n,
I ft=F*= 
roi r!!i -tog - 
f i rii:f t
A lt,
5 
"bs f i lu 
t ;n3 i4
A[^=I .( t xrb -F&'w
= 1.1 "1 B ? +
Thust P = ., ,4f8,
3,156
lqr = (7,5 in,)i
x1 ,(t \
l ,(78 f =Q,tl5e ir,)
?=110 lb
We ha{e p = S to lb
fr =Qo in,) j - (t,s ;n) i s(t o i nyp
, r r l l l l l
t r t l t l l l l
!=lbrr= (Fzotb) \n
b ro = # 
= ioitl,,!,t-t to& 
'' 
= ),+'lsii-0uo9jr'r,ji7v &
fol'uilJ-ob.Fd
lrl :,o = 4r, ro= ry - ( ny; 
r r'_ooi,
M = (30 in,) i + (t5 in,li + ( lo in,) &
r, ^^' lo i ' i tsi + l0 k / to,ra t ! vN,3oj {tsi +to&
35
Trso t, + rao o i + 7 too h)= 3 otd
Nn^= loqo [b' in,
'oD= 
loqo lb' in.
- - ] r , L - - t - l I I i
(4aotD: - (rto tb)i+060U!
oD=35 in.
x
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3.99 A crane is oriented so that the end of the Zi-mboomlO lies in the yz plane.
At the instant shown the tension in cable AB is4 ktl. Determine the moment about
each of the coordinate axes ofthe force exerted on I by cable AB.
Wehave . . *
oc =rl@n)'-(nc)'!"/r/ 
='rcffi=/om
fhe ̂ r f { i r rcn ls oF ! , X, ! ,
A4 = +(3.lur,)(zo)
Mu = r 0,tste)(u)
f r* = -(o,r , i t6)( ls)
= (t5 ̂ )j | (20 n) &
-'(2,srh)'+ 
(s'n)'= lt2o7 m
. /,1 f i l i-(t{i l i
= (4ktv) J-!- tq,zo7 w,lS '207 h
| = (0,6576 k N)!-(3,1 
+&kN)t
rcp r.Sc'r t /h" morrre^fr N, , Hl, Hr 3
Mn = 7B'1 kN'n
l\g = q, /5 l'ril,u {
3.100 The 25-m crane boom AO lies in the yz plane. Determine the maximum
permissible tension in cable AB if the absolute value of the moments about the
coordinate axes ofthe force exerted onl by cable ABmust be as follows: lM'l < 60
kN' m, lUrl t 12 khl 
' m, and lM) t 8 kN 
' m.
riPlPlflg
- = lq,l L0 Pi. +3,28E Pl' -2,+6 L P le
l i l r lSiakN.m: tq,UBp:to ps J,0+ktl
f4J I t 12 kN,nr' S,ZBSFI tZ p g 5,65 kN
lrurl : gkN,,rt; l . |ogP4B ?*s,?+k/v
Flalimvn y;v.rnt'ssible tenSr'o'r is 3,0+kN
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3.101 A single force P acts at C in a direction perpendicular to the handle .BC of the
crank shown. Detemine the moment M* of P about the x axis when 0 = 65",
knowing *rat/,4n= - 15 N'm andM": '36 N'm.
9r= (0, z r*) : t^ (o,z ^) sln e! +@,2") @se4
P: - Fsi 
"+i 
l Pcos+k
: - , | | , i | | | i f I
l l r l l l l l i l l i l
l t t l
N, -(0,Zn)P(s;n 0 ,os6+ cot#s,l',*)'
Nr=0,2f t , \Ps, i r . (0++)
h, =-(o,zr^) ? c'rs *
fr"* = -(o,z5m)F s;nQ
STva.rS tnen!<rs of (2)c*l 1j1 a,nr( ada{,hg :
( , )
(z)
(3)
(+)
Hr'n fqL"=p, z s nfr. f = (+n;\ltt 'ffi 6)
f,{ = (o,zrxt s6, o n/) s;n (e ;"+ 67 JB)
t i4 .k
t u 
- rs r r rT rT - |
, 
E,xpandir^5 fA< de{e.rninan ti vrc f inal
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3.102 A multiple-drilling machine is used to drill simultaneously six holes in the
steel plate shown. Each drill exerts a cloclnrise couple of magnitude 40 lb . in. on
the plate. Determine an equivalent couple formed by the smallest possible forces
acting (a) at,{ and C, (b) at A and D, (c) on t}re plate.
^ l--16 
in.
c
oo
oo
O O
o
./
\o;'o2
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T.
lJ5 ttr-
3.103 A 500-N force is applied to a bent plate as shown. :Demroiae (a) an
equivalent force-couple system at B, (b) an equivalent system fonned by a vertical
force atA and a force at B.
(a) 6rr< -cotryle slsteu at B 
"
500N [= (sOoN) sin 3 oJ - (roo N)c"5ro"i
=(zso*)i-(+s3N/j
A*= ( 0,3 L- o,nfj) x (zra r' -+ lri )
= - 12q,1l/r.q 4 3.75fr = -86.1f &
F= foonf
f = foo^l<( 60", /u = 86,L'N,r) {
Verfialforrc a.t A ama( .(arre af I :
d
F
THu= Z f la i
B6,tSl'/ '-^ = n(o, l25n)
A=(Gs? N)i != 6aq N | {
f = .nrE:
B = f - 0 = (zron]1 - (tt3 N) i 
-lstt u)X
=(Zrofl3 -( l tz 'zN)X
9: t$qN \77,+'
0, l7 5n
3.104 A 100-k$,1 load is applied eccentrically to the column shown. Determine the
components of the force and couple at G that are equivalent to the 100-ld\i load.
100 kN
Ac--.9x E , ..," =Kr, tzs'{ i -(o,oso't'g] x (- toa a N) i
=-- Qz,s .(4 N'^) t 
- (r l<N''Fr)!
The eQvivalenf {ort€-(D.'tple sJ {te'n i5
F= -(t00 &N)_{ , A*= 
-(s4ev,*).r -02,s+tl^)&.
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3.105 The speed-reducer unit shown weighs 75 lb and its center of gravity is
located on they axis. Show that the weight of the unit and the two couples acting
on it, of magnitude Mr= 20 lb . ft and Mz:4 lb'ft respectively, may be replaced
by a single equivalent force and determine (a) the magnitude and direction of that
force, (D) the point where its line of action intersects the floor.
F= - (tr h)i
7
( 2o tu,Frl ,&
Mo= Nu,l rMit = (+lb,ft) i t-(uz/i,ft)&
3.106 For the tnrss and loading shown, determine the resultant of the loads and the
distance from pointl to its line of action.
q &=tzh;ps
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3.107 A force P of given magninrde P is applied to the edge of a semicircular plate
of radius a as shown. (a) Replace P by an equivalent force-couple system at point
D obtained by drawing the perpendicular from B to the x axis. (b) Determine the
value of d for which the moment of the equivalent force-couple system at D is
maximum.
(a-) {orc<-couple <7sft*r df D :
P=Pa,st j+Ps;"06
F= E R=pE s
U!=3^,x7=rtxf-'D' -o/o' -
^1\ere fg = e ein 2 a) i
Thrs, ryf = @sa z 0)t x( Pc,oslt 1.Psinl d)
- + aPs in7o c-os0 &
NE = Pa sin 29 oosfl) {:'
U Valug oF 0 h , ^ax i ,nvm va lue o f n5,
#ufuE) 
=q 
A G;nro aso) 
= o
2 acszocoso - l in2 'o s ino =
Z(c .os l .e -s in" 0)oos0 - 2s inh
Cos6=0 ' oR ?cos l0-
o
cos g Sin0 = D
A s;nt | = t )
(uE= o) fan" 0 = ,.
fant=t / r , 0=3
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center of the mat.
3.108 A concrete fogndation mat of 5-m radius supports foru equally spaced
columns, each of which is located 4 m from the center of the mat. Deterpine the
magnitude and point of application of the smallest additional load that'must be
uppti.a to the foundation mat ifthe resultant of the five loads is to pass through the
l l l
rr 
'3?5 
hN u
-(roe4t+
+-x
l -T- [ l I I I i I I I I I
!: = 
- + t x (- to?A' ) + tt "4 ff:tj),-o + u1v1-2ti i:+!* ttz{i)
:4oo& +3001Jtoo t- tuoi
yf =- (loo kN,,n)j +QffitN.')&
t
Lrt ? tre fhe r^eyuirel (m^ll"sf ar?if;"^"!-Fe' 
loart , onrf , 'L and , , lh 'o?oordin*ks
fr, tht ..tL tlr,r,{-af ftie' S loads fb
r.ass fhrorts h A, w, mvsf haveil{r,.=0'
zt4 : - (t00k/v, ^) i +pooril,m) k + (r:,rz ilv (- pX') , 0-o ' 
-rcoj+soa'!-PrtrPzir o
ETvo fin$ h* ff t'i. tf the unif rrc firrs.:
- f loOt7z=0, z=?,oo/? (D
3oo-Pz:0, x*Joo/p G)
{ inc" 
' fh,e 
radt 'us is grur.t+/e har.e
lL= 
#=1, 
16n
. n'ieZ (r^):
{ t ) orr l (z):{ ubstifuh:n, nh e fturn
/ zy= (s),
t / 
L '
P'= (ry oo, (
Subrlitfrvg fu ? in ( t) anct (z):
7t,, ll e l/
l l l t l l l l l l l l l l l r r r l
PsTx,, lkdt ; a l f r=4, lbn, ?=2,77n
t t t l
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