Solution Manual Mechanics J L Mariam

University of Lahore. Cell#03338189587
By Khalid Yousaf
Solution anual by Khalid Yousaf BS(Continue)Civil Engineering The University of
Lahore.fb!"halid#$%&'%''*+yahoo.,o
What is Mechanics? Mechanics is the physical science which deals with the effects of forces on objects. he s!bject of "echanics is loically di$ided into two parts% statics&which concerns the e'!ilibri!" of bodies !nder action of forces& and dynamics& which concerns the "otion of bodies. ()*+, ,-,/* he followin concepts and definitions are basic to the st!dy of "echanics& and they sho!ld be !nderstood at the o!tset. Space is the eo"etric reion occ!pied by bodies whose positions are described by linear and an!lar "eas!re"ents relati$e to a coordinate syste". or three2di"ensional proble"s& three independent coordinates are needed. or two2di"ensional proble"s& only two coordinates are re'!ired. Time is the "eas!re of the s!ccession of e$ents and is a basic '!antity in dyna"ics. i"e is not directly in$ol$ed in the analysis of statics proble"s. Mass is a "eas!re of the inertia of a body& which is its resistance to a chane of $elocity. Mass can also be tho!ht of as the '!antity of "atter in a body. he "ass of a body affects the ra$itational attraction force between it and other bodies. his force appears in "any applications in statics. Force is the action of one body on another. ) force tends to "o$e a body in the direction of its action. he action of a force is characteried by its magnitude& by the direction of its action& and by its point of application. h!s force is a $ector '!antity.. ) particle is a body of neliible di"ensions. +n the "athe"atical sense& a particle is a body whose di"ensions are considered to be near ero so that we "ay analye it as a "ass concentrated at a point. We often choose a particle as a differential ele"ent of a body. We "ay treat a body as a particle when its di"ensions are irrele$ant to the description of its position or the action of forces applied to it. Rigid body. ) body is considered riid when the chane in distance between any two of its points is neliible for the p!rpose at hand. or instance& the calc!lation of the tension in the cable which s!pports the boo" of a "obile crane !nder load is essentially !naffected by the s"all internal defor"ations in the str!ct!ral "e"bers of the boo". or the p!rpose& then& of deter"inin the e4ternal forces which act on the boo"& we "ay treat it as a riid body. *tatics deals pri"arily with the calc!lation of e4ternal forces which act on riid bodies in e'!ilibri!". eter"ination of the internal defor"ations belons to the st!dy of the "echanics of defor"able bodies& which nor"ally follows statics in the c!rric!l!".
PROBLEMS Introductory Problems 2/1 he force F has a "anit!de of 800 . /4press F as a $ector in ter"s of the !nit $ectors i and j. +dentify the x and y scalar co"ponents of F.
Soln. Ste-! /ree body 0iagra
Ste-!M agnitud e of for,e *##1 2
,o-onent of for,e3 /245/sin$&o
/ 245 *##sin$&o
/ y4*## ,os$&o
/ y4'&& 1 /or,e ve,tor3 /4(5%&i5'&& j)1

Ste- ! /ree body diagra
Ste-$! /or,e ve,tor F4'##,os$#i 5 '##sin$# j F4&#i 7 $## j
Ste- %!S,alar ,o-onents of for,e 8long 25a2is /24&# lb 8long y5a2is /y4 5$## lb

2/3 The slo-e of the %.*5"1 for,e F is s-e,i:ed as sho;n in the :gure. E2-ress F as a ve,tor in ters of the unit ve,tors i and j.
Soln. Ste-!/ree body diagra
Unit ve,tor of for,e3 n4 5 3
5 i5
<)
4(5.**i5$.*%<)"1 2/4 The line of a,tion of the .'5 "1 for,e F runs through the -oints A and B as sho;n in the :gure. 0eterine the x and y s,alar ,o-onents of F.
Soln. Ste-! Magnitude of for,e /4%*## lb =osition of -oint 84 5&i5# j =osition of -oint B4$#i># j Ste-! /ree body diagra
St-e$! =osition ve,tor of 8B3 8B4?B 7 ?8 8B4%&i>$# j
Magnitude of 8B3 8B 4 √ 45 2+30
2
AB
AB
4 45 i+30 j
54.08
2/5 8 ,able stret,hed bet;een the :2ed su--orts A and B is under a tension T of ## lb. E2-ress the tension as a ve,tor using the unit ve,tors i and j3 :rst3 as a for,eT A a,ting on A and se,ond3 as a for,e TB a,ting on B.
Ste-! Magnitude of tension in ,able 8B3
T 4 ## lb
√ 22+3 2 i
√ 22+3 2 j
n8B 4 #.*$i 5 #.&& j T84 T8 n8B T84##(#.*$i 5 #.&& j) T84(@%i5 % j)lb But T84 5 TB 45(@%i5% j)lb 4(5@%i>% j)lb 2/6 The *##51 for,e F is a--lied to the end of the 6 bea. E2-ress F as a ve,tor using the unit ve,tors i and j.
Soln. St-e! Magnitude of for,e /4*## 1
5 i5
4
5 <)
45#*#i15%%# j1 2/7 he two str!ct!ral "e"bers& one of which is in tension and the other in co"pression& e4ert the indicated forces on joint -. eter"ine the "anit!de of the res!ltant R of the two
forces and the anle A which R "a6es with the positi$e 42a4is.
25,o-onents of resultant for,e
24 ∑ /2
24 5$.$ "1 y5,o-onents of resultant for,e
y4 ∑ /y
y4 5.&* "1 Magnitude of the resultant for,e
4 √ R x
2+ R y
A4tan5( R x
4'o
8ngle A ade by D ;ith -ositive 25a2is A4*#o>'o
4#'o
8 wo forces are applied to the constr!ction brac6et as shown. eter"ine the anle which "a6es the res!ltant of the two forces $ertical. eter"ine the "anit!de R of the res!ltant.
Ste-!
/4*## lb /4%& lb iven that the resultant for,e is noral to 25a2is.Therefore the 25 ,o-onent of resultant for,e is Fero.
24 ∑ /2 4 #
CosA4 800cos 70
y4 ∑ /y
y4 5#@# lb Magnitude of the resultant for,e
4 √ R x
2+ R y
4#@@ lb
Representative Problems
2/9 +n the desin of a control "echanis"& it is deter"ined that rod AB trans"its a :02 force P to the cran6 BC . eter"ine the x and y scalar co"ponents of P. Soln. Ste-! /ree body 0iagra
Ste-!Magnitude of for,e =4'# 1
13 i5
5
13 <)
45%#i15## j1 S,alar ,o-onent of = along 2 =245%# 1 S,alar ,o-onent of = along y =y45## 1 G# /or the e,hanis of =rob. G3 deterine the s,alar ,o-onents Pt and Pn of = ;hi,h are tangent and noral3 res-e,tively3 to ,ran" BC. *oln. *tep1%/ree body diagra
Ste-!agnitude of for,e /4'# 1
θ=¿ tan5( 5
A4.'o
S,alar ,o-onents of for,e = along nD =n4'# ,os($#o5.'o) =n4&* 1 S,alar ,o-onents of for,e = along tD =t4'# sin($#o5.'o) =t4$$. &1 2/11 he t 2co"ponent of the force F is 6nown to be 75. eter"ine the n2 co"ponent and the "anit!de of F.
*tep% he t 2co"ponent of the force &t;75 .<et n be the n2co"ponent of the force t;cos=0o >1 n;2sin > i$idin e'!ation > by >1
F n
75; cos=0o
;97.9 2/12 ) force F of "anit!de 800 is applied to point C of the bar AB as shown. eter"ine both the x-y and the n-t co"ponents of F.
Ste-! /ree body diagra
Ste-! Co-onents of for,e / along 2D /24 5*## ,os##
/24 5@& lb Co-onents of for,e / along yD /y4 *## sin##
/y4 @% lb Ste-$! Co-onents of for,e along tD /t4 5*## ,os%##
/t4 5'$ lb Co-onents of for,e along nD /n4 5*## sin%##
/n4 5&% lb 2/13 The t;o for,es sho;n a,t at -oint A of the bent bar. 0eterine the resultant R of the t;o for,es.
Ste-! /ree body diagra
24 ∑ /2
y4 ∑ /y
y4 5@sin'#o>$ sin$#o
y4 5$.%& "i-s The resultant of the t;o for,es is @;@4iA@y j
4 (.$&i5$.%& j)"i-s 2/14 To satisfy design liitations it is ne,essary to deterine the eHe,t of the 5"1 tension in the ,able on the shear3 tension3 and bending of the :2ed 65bea. /or
this -ur-ose re-la,e this for,e by its eIuivalent of t;o for,es at A3 F t -arallel and F n -er-endi,ular to the bea. 0eterine F t and F n.
*oln. *tep1%ree body diara"
Ste-! Co-onents of for,e along tD /t4 ,os(##>$##) /t4.*' "1 Co-onents of for,e along nD /n4 sin(##>$##) /n4.&$ "1 2/15 eter"ine the "anit!de s of the tensile sprin force in order that the res!ltant of Fs and F is a $ertical force. eter"ine the "anit!de R of this $ertical res!ltant force.
Ste- !Magnitude of for,e /4# lb
Tensile s-ring for,e be /s
*ince the horiontal co"ponent of B@C is ero ;s cos:0o
;10 cos:0o
@;10sin:0o
@;103.9 lb 2/16 The ratio of the lift for,e L to the drag for,e D for the si-le airfoil is L/D 4#. 6f the lift for,e on a short se,tion of the airfoil is ## 13 ,o-ute the agnitude of the resultant for,e R and the angle θ ;hi,h it a"es ;ith the horiFontal.
Soln. Ste-! /ree body diagra
Ste-! Lift for,e L4&# lb atio of the lift for,e to drag for,e L
D 4#
√ L2+ D2
4 √ 52+502
4&#. lb *tep3% )nle "ade by the res!ltant with BC
θ=¿ tan21>
θ=¿ 8=.3o
2/17 eter"ine the co"ponents of the 26 force alon the obli'!e a4es a and b. eter"ine the projections of F onto the a2 and b2a4es.
*tep% (y the law of sineCs
F a
sin15 o ;
sin120 o D
b;1.:33 6 *tep3%
<et a and b be the projections of force alon a and b. a4 cos=5o
a41.=1= 6 b4 cos15o
b4.$ "1 2/18 0eterine the s,alar ,o-onents Ra and Rb of the for,e R along the non re,tan 5gular a2es a and b. 8lso deterine the orthogonal -ro<e,tion Pa of R onto a2is a.
Ste-! (y the law of sine
Rb
sin 40
@a; 1170 *tep3%<et a be the orthoonal projection of onto a2a4is
*tep=% a;@ cos300
;800 cos300
;:93 2/19 eter"ine the res!ltant R of the two forces shown by >a applyin the parallelora" r!le for $ector addition and >b s!""in scalar co"ponents.
*tep%>a ro" the law of cosineCs @;=00A:00 E >=00>:00cos10
@;7:0000 @;87 <et be the anle "ade with the $ertical&then by the law of sineCs
600
24 ∑ /2
4 √ R x
2+ R y
A4tan5( R x
4tan5( 700
4&$.%o
So the angle ade by resultant ;ith the verti,al A4#o5&$.%o
A4$'.'o
2/20 6t is desired to reove the s-i"e fro the tiber by a--lying for,e along its horiFontal a2is. 8n obstru,tion A -revents dire,t a,,ess3 so that t;o for,es3 one .' "1 and the other P3 are a--lied by ,ables as sho;n. Co-ute the agnitude of P ne,essary to ensure a resultant T dire,ted along the s-i"e. 8lso :nd T .
*oln. *tep1% /ree body diagra
Ste-! /ro :gure
A4tan5( 4
(y the law of sine P
sinθ 2
sinα ; P
sinθ 2
; 800.5=1 lb 2/21 )t what anle θ "!st the 8002 force be applied in order that the res!ltant R of the two forces has a "anit!de of 000 ? or this condition& deter"ine the anle θ between R and the $ertical.
Ste-! Magnitude of the resultant for,e 4### lb ro" the law of cosineCs 000;1=00A800 E >1=00 >800cos>180o2 But ,os(*#o5A)45,osA Therefore
000;1=00A800 A >1=00>800cos A4&.$o
/ro the la; of sineDs
800
800
4*.o
2/22 he !nstretched lenth of the sprin of "od!l!s k ;1. 6" is 100 "".When pin P is in the position θ=30o deter"ine the x 2 and y 2 co"ponents of the force which the sprin e4erts on the pin. >he force in a sprin is i$en by F=kx & where x is the deflection fro" the !nstretched lenth.
Ste-! Gi$ent hat sprin "od!l!s 6;1. 6"
Hnstretched lenth of sprin l ;0.1"
=osition of -oint 84#.#'i>#.#% j =osition of -oint =4#.#*sin$#i5 #.#*,os$# j =4#.#%i5 #.#'$ j =osition ve,tor =84?85?= =84(#.#'5#.#%)i5(#.#%)5(5 #.#'$) j =84#.#i>#.#$ j
Magnitude of =84 √ 0.02 2+0.1093
2
=84#.
0.02 i+0.1093 j ¿
n=84#.*i>#.*% j
Magnitude of s-ring for,e /4 K2 Nhere 2 is ,hange in length.
24=85 l 2 4#.5#. 24#.# /4. "1J#.# /4$.$ 1 *tep3% *prin force $ector
; n=8
4$.$(#.*i>#.*% j) =(.%i>$. j) 1
2 and y ,o-onents of for,e are /24.% 1 /y4$. 1 2/23 @efer to the state"ent and fi!re of rob. . When pin P is in the position θ=0o&deter"ine the n2 and t 2 co"ponents of the force F which the sprin of "od!l!s k ;1. 6" e4erts on the pin.
*tep1%Gi$en that
*prin constant of the sprin 6;1. 6" Hnstrectched lenth of the sprin&
l40.1"
he anle "ade by the pin with the $ertical ;0o
<et the centre of the coordinate syste" be positioned at 0. *tep%he arrane"ent of the sprin syste" at the present instant is as shown by the free body diara"
*tep3% he coordinates of points ) and are );>0.0:i&0.0= j ;>0.08 sin0oi20.08cos0o j ;>0.07=i20.075 j Ience the position $ector ) is i$en by );-)2- );>0.0:20.07=i A J0.0=2>20.075K j );0.03:iA0.115 j he "anit!de of the $ector is i$en by
=84
0.0326
¿ ¿ ¿ √ ¿
);0.1197" he "anit!de of the sprin force is i$en by ;64 Where 4;) 2 0.01 4;0.119720.01
4;0.0197" ;1.D0.0197
;0.037 6 ;3.7 6
2/24 The ,able AB -revents bar OA fro rotating ,lo,";ise about the -ivot O. 6f the ,able tension is @&#1.0eterine the n5 and t 5 ,o-onents of this for,e a,ting on -oint A of the bar.
Soln. *tep1% /ree body diagra
Ste-! Tension in the ,able T4@&# 1 Let Tn and Tt be the ,o-onents of for,e TD along nD and tDa2es res-e,tively By the ,osine la; 8B4
√ OA 2+OB2−2(OA)(OB)cos 120o
8B4
Ste-$! 8--lying sine rule for triangle B?8
sinθ
Tn4T Sin θ
Tn4@&# Sin'.$@ Tn4$$$. 1 Co-onents of for,e T along tD
Tt45T ,os θ
Tn45@&# ,os'.$@ Tn45'@ 1 2/25 )t what anle θ "!st the =002 force be applied in order that the res!ltant R of the two forces ha$e a "anit!de of 1000 ? or this condition what will be the anle θ between R and the horiontal?
*tep1%ree body diara"
*tep% Manit!de of the res!ltant force @;1000 lb <et be the anle between the res!ltant force B@C and horiontal force 700 lb. ro" the law of cosineCs 1000;=00A700 E >=00 >700cos>180o2 cos;0.:5 ;51.3o
(y the law of sineCs
400
o) 1000
4*.o
2/26 6n the design of the robot to insert the sall ,ylindri,al -art into a ,lose5:tting ,ir,ular hole3 the robot ar ust e2ert a #51 for,e P on the -art -arallel to the a2is of the hole as sho;n. 0eterine the ,o-onents of the for,e ;hi,h the -art e2erts on the robot along a2es (a) -arallel and -er-endi,ular to the ar AB3 and (b) -arallel and -er-endi,ular to the ar BC.
Soln. *tep1% Gi$en orce e4erted by robot ;90 <et parallel force ; t
Let -er-endi,ular for,e 4 =n
Ste-! (a) Co-onents of for,e ;hi,h the -art e2erts on the robot along -arallel and -er-endi,ular to the angle ar 8B.
By resolving for,es3 =t45#,os$# =t45@@. 1 (-arallel for,e) =n4#sin$# =n4%& 1 (-er-endi,ular for,e) Ste-$! (b) =arallel and -er-endi,ular for,es to the ar BC.
By resolving for,es =t4#sin%& =t4'$.' 1 (-arallel for,e) =n4#,os%& =n4'$.' 1 (-er-endi,ular for,e) 2/27 he !y cables AB and AC are attached to the top of the trans"ission tower. he tension in cable AC is 8 6. eter"ine the re'!ired tension T in cable AB s!ch that the net effect of the two cable tensions is a downward force at point A. eter"ine the "anit!de R of this downward force.
*tep%
ension in the cable ),&ac;8 6 <et the tension in the cable )( be ab
<et the "anit!de of the res!ltant force be @ ro" the fi!re
A4tan5( 50
A$4&o
T ac
sinθ1 4
T ab
sin51.3 o
@;10.1 6 2/28 The gusset -late is sub<e,ted to the t;o for,es sho;n. e-la,e the by t;o eIuivalent for,es3 F 2 in the x 5 dire,tion and Fa in the a5dire,tion. 0eterine the agnitudes of Fx and Fa. Solve geoetri,ally or gra-hi,ally.
Soln. *tep1%Geo"etric sol!tion
*tep% <et 4
and a be the forces alon 4 and a2a4es respecti$ely. <et @ be the res!ltant force. ro" law of cosine 4##>*##5(##)(*##),os@&o
4#@@$##.&@& 4#$@.$ Ste-$! By the la; of sine
sinα
*#o4>%&o>'&o>%*.o
sinβ
1037.93
/24&%@.# 1 Ste-$! 1o; ,onsidering triangle ?CaD By the la; of sine
sin (α +65) F a
4 sin 45
4 sin 45
/a4$%.'$ 1
PROBLEMS Introd!tor" Pro#$%&' 2/29 he 10 6 force is applied atpoint A. eter"ine the "o"ent of F abo!t point . eter"ine the points on the x 2 and y 2a4es abo!t which the "o"ent of F is ero.
*tep% he "anit!de of force ;10 6 <et ( and , be the points on y and 42 a4es respecti$ely. ,o"ponents of force alon 4 and y2 a4es.
4;10D 4
4;8 6
y;10D 3
5
y;: 6 a6in "o"ent abo!t B-C >,W A Mo;4D5 2 yD= Mo;8D5 2 :D5 Mo;10 6" *tep3%,onsiderin si"ilar trianles
3
4
y; " point (>0& ,>.:7&0 2/30 eter"ine the "o"ent of the 8002 force abo!t point ) and abo!t point -.
*tep%Manit!de of force ;00 lb a6in "o"ent abo!t B)C M );00cos30oD35 M );:0:. lb2in >,W a6in "o"ent abo!t B-C M- ; 00cos30oD35 E 00sin30oD35 M- ; :0:. E 500 M- ; 35:. lb Ein >,W 2/31 eter"ine the "o"ent of the 50 force >a abo!t point by LarinonCs theore" and >b abo!t point C by a $ector approach.

Ste-! iven that /or,e a,ting along line 8B3/4&# "1 /ro :gure. =osition ve,tor of 83O(45&i5# j =osition ve,tor of B3OB4%#i># j =osition ve,tor of C3O 4& j 9e,tor for of 8B3(B=OB-O( (B4(%#>&)i>(#>#) j (B=&&i>$# j /or,e ve,tor along line 8B3F4/ AB
AB
√ 55 2+30
62.65
F4%$.*i>$.% j Ste-$!(a) Ta"ing oent about ?D Mo4O(JF Mo4(5&i5# j)J( %$.*i>$.% j) Mo4(5&J$.%)* >(5#J%$.*)(5 * ) Mo4&*.@* Magnitude of oent about ?D is Mo4&*.@ 1(CCN) Ste-%!(b) Moent about CD M!4(JF =osition ve,tor of C83(4O(5O (45&i>(5#5&) j
(45&i5%& j M!4(5&i>(5#5 &) j)J( %$.*i>$.% j) M!4(5&J$.%)* >(5%&J%$.*)(5 * ) M!4'&.&* Magnitude of oent about CD is M,4'&.& 1(CCN) 2/32 The for,e of agnitude F a,ts along the edge of the triangular -late. 0eterine the oent of F about -oint O.
Ste-! =oint 8(#3h) =oint B(b3#) 9e,tor 8B4bi5h j
Unit 9e,tor 8B4n8B4
9e,tor ?84h j Moent about ?D M?4 ?8 J /
M?4 h j J+ Fb i− Fh j
√ b2+h 2
M?4 5 Fbh
√ b2+h 2 "
√ b2+h 2 (CN)
2/33 +n steadily t!rnin the water p!"p&a person e4erts the 10 force
on the handle as shown. eter"ine the "o"ent of this force abo!t point .
*tep% Manit!de of force ;10 Mo"ent of force abo!t B-C >,W A Mo; cos>0oA15oD0.15 Mo;10cos35oD0.15 Mo;1=.7= " 2/34 The throttle5,ontrol se,tor -ivots freely at ?. 6f an internal torsional s-ring e2erts a return oent M 4.* on the se,tor ;hen in the -osition sho;n3 for design -ur-oses deterine the ne,essary throttle5,able tension T so that the net oent about ? is Fero. 1ote that ;hen T is Fero3 the se,torrests against the idle5 ,ontrol ad<ustent s,re; at R.
*tep% <et the tension re'!ired be BC a6in "o"ent abo!t B-C >)nticloc6 wise A M-;0 1.8 2 D0.05;0
; 1.8
0.05
;3: 2/35 ) force F of "anit!de :0 is applied to the ear. eter"ine the "o"ent of F abo!t point .
*tep% Manit!de of force ;:0 y; cos ,os#o
y;:0 cos ,os#o
y;5:.38 a6in "o"ent abo!t B-C >,WA Mo;yD0.1 Mo;5:.38D0.1 Mo;5.:= " 2/36 Cal,ulate the oent of the &#51 for,e on the handle of the on"ey ;ren,h about the ,enter of the bolt.
Ste-! iven Magnitude of for,e /4&# 1 2 and y ,o-onents of for,e /D /y4&# ,os&o
/y4%.%* 1 /24&# sin&o
/24'%.@ 1 Ste-$! Ta"ing oent about ?D Mo 7 /y(#.)>/2(#.#$)4# Mo 7 %.%*(#.)>'%.@(#.#$)4# Mo 5 %*.'>. 4# Mo4%'.$' 1 2/37 ) "echanic p!lls on the 132"" co"bination wrench with the 1=02 force shown. eter"ine the "o"ent of this force abo!t the bolt center .
Ste-! Manit!de of force ;1=0 Ta"ing oent about ?D (CCN>) Mo4/,osAJ#.#& Mo4/,os(&o5#o)J#.#& Mo4%#,os#oJ#.#& Mo4$. 1 2/38 8s a trailer is to;ed in the for;ard dire,tion3 the for,e F 4&## 1 is a--lied as sho;n to the ball of the trailer hit,h. 0eterine the oent of this for,e about -oint O.
Ste-! iven that for,e /4# lb Let /2 and /y be the ,o-onents of for,e along 2 and y5a2es res-e,tively. Ta"ing oent about ?D
>)nticloc6 wise A M-;4 D1.5Ay D 11 Mo; 10 sin30o
D1.5A10cos30o D 11 Mo;90A11=3.15 Mo;133.15 lb2in >,W 2/39 ) portion of a "echanical coin sorter wor6s as follows%ennies and di"es roll down the 0o incline&the last trian!lar portion of which pi$ots freely abo!t a horiontal a4is thro!h . i"es are liht eno!h >.8 ra"s each so that the trian!lar portion re"ains stationary& and the di"es roll into theriht collection col!"n. ennies& on the other hand& are hea$y eno!h >3.0: ra"s each so that the trian!lar portion pi$ots cloc6wise& and the pennies roll into the left collection col!"n. eter"ine the "o"ent abo!t of the weiht of the penny in ter"s of the slant distance s in "illi"eters.
Ste-! g,os## and gsin## are ,o-onents of ;eight -er-endi,ular -arallel to the in,lined -lane res-e,tively.
Ta"ing oent about -ivot ?D (CN >) Mo4 g,os##Js> gsin##(.&>$.&) Mo4$.#'J#5
$J.*,os##Js>$.#'J#5$
J.*sin##(.&>$.&) Mo4(#.#*s>#.$$&) 1 (s in ) 2/40 Eleents of the lo;er ar are sho;n in the :gure.The ass of the forear is .$ "g ;ith ass ,enter at G. 0eterine the ,obined oent about the elbo; -ivot O of the ;eights of the forear and the $.'"g hoogeneous s-here. Nhat ust the bi,e-s tension for,e be so that the overall oent about O is FeroP
Ste-! iven that Neight of the fore ar N4& lb Neight of the s-here N4* lb Let TD be the tension in the bi,e-. Moent about ?D due to the ;eights & lb and * lb. >,loc6 wise A Mo;8D13A5D:sin55o
Mo;10=A=.57 Mo4*.&@ lb5in (CN) The ,obined oent due to the all for,es is Fero. >,loc6 wise A Mo;0 18.572D;0 ;:=.9 lb
2/41 ) 3 lb p!ll T is applied to a cord&which is wo!nd sec!rely aro!nd the inner h!b of the dr!".eter"ine the "o"ent of T abo!t the dr!" center C . )t what anle θ sho!ld T be applied so that the "o"ent abo!t the contact point P is ero?
*tep% Manit!de of force ;3 lb a6in "o"ent abo!t B,C>,WA Mc;D5 Mc;3D5 Mc;1:0 lb2in or the "o"ent abo!t the contact point to be ero&the applied force sho!ld pass thro!h point . <et be the anle of with the horiontal s!ch that
ro" the trianle ,(
8
;51.3o
2/42 ) force of 00 is applied to the end of the wrench to tihten a flane bolt which holds the wheel to the a4le. eter"ine the "o"ent ! prod!ced by this force abo!t the center of the wheel for the position of the wrench shown.
*tep% Manit!de of force ;00 -);=502:.5 cos0 -);391.7 "" Moent about ?D >,loc6 wise A Mo;D-) Mo;00D391.7 Mo;7853.8= "" 2/43 +n order to raise the flapole -,& a liht fra"e -)( is attached to the pole and a tension of 780 lb is de$eloped in the hoistin cable by the power winch . ,alc!late the "o"ent Mo of this tension abo!t thehine point -.
*tep% Manit!de of tenion ;780 lb

a6in "o"ent abo!t B-C >,,W A Mo;780cos0oD10cos30o2780sin0oD5 Mo;:3=7.:21333.88 Mo;5013.7= lb2ft 2/44 he !nifor" wor6 platfor"& which has a "ass per !nit lenth of 8 6"& is si"ply s!pported by cross rods A and B. he 9026 constr!ction wor6er starts fro" point B and wal6s to the riht. )t what location Bs" will the co"bined "o"ent of the weihts of the "an and platfor" abo!t point B be ero?
*tep% Gi$en that Mass of the platfor" M;8 6" Mass of the wor6er ";90 6 Weiht of the platfor" W;8D>1A=A3D9.81 W;197.== Mo"ent of all force at B(C is ero. M(;0
"D* 2WD1;0 gJS4N
90 × 9.8
*;.=9 " 2/45 +n raisin the pole fro" the position shown& the tension T in the cable "!st s!pply a "o"ent abo!t of 76" . eter"ine T .
Ste-! Gi$en&"o"ent abo!t B-C Mo;7 6 <et BC be the tension in the cable ,onsiderin triable -(,
sin:0o; h
h;5.98 "
cos:0o; x
tan; h
12+ x
tan; 25.98
12+15
tan; 25.98
1800;FA=3.897o A10o
F;1:.103o
*tep3%
a6in "o"ent abo!t B-C Mo;0 Mo2sinFD30;0 sin1:.103oD30;7 ;8.:53 6 2/46 The for,e e2erted by the -lunger of ,ylinder AB on the door is %# 1 dire,ted along the line AB3 and this for,e tends to "ee- the door ,losed. Co-ute the oent of this for,e about the hinge O. Nhat for,e FC noral to the -lane of the door ust the door sto- at C e2ert on the door so that the ,obined oent about O of the t;o for,es is FeroP
*tep% Gi$en that orce e4erted by the pl!ner ;=0
tan; 100
;1=.03:o
a6in "o"ent abo!t point B-C Mo;=0cos1=.03:o>75A =0sin1=.03:o>=5
Mo;910.=3A=13.035 Mo;7033.=:5 "" >,,W <et BcC be the reaction force at B,C a6in "o"ent abo!t point B-C Mo;0 c (*&) 5 Mo;0 c (*&) 5 7033.=:5 ;0 c (*&) 4 7033.=:5 c 4*.&& 1 2/47 he lb force is applied to the handle of the hydra!lic control $al$e as shown. ,alc!late the "o"ent of this force abo!t point .
*tep% Gi$en&applied force ; lb a6in "o"ent abo!t B-C >,,W A Mo;2cos0oD>10sin:0oA1.52 sin0oD>1ocos:0o Mo;219.09523.= Mo;2.5 lb2in 2/48 Cal,ulate the oent M A of the ##51 for,e about -oint A by using three s,alar ethods and one ve,tor ethod.
Ste-! Method Ta"ing oent about 8D M84/,os&oJ*#> /sin&oJ%## M84##,os&oJ*#> ##sin&oJ%## M84@%@@.% 1 (CN) Ste-$! Method
Considering the for,e a--lied at 0D C04%##tan&o
C04#@.* 0848C>C0 084*#>#@.* 084$*@.* Ta"ing oent about 8D M84 ##,os&oJ08 M84 ##,os&oJ$*@.* M84@%@@.% 1 (CN) Ste-%! Method$
/inding the -er-endi,ular distan,e 8E 084$*@.* 0E408sin&o
0E4$*@.* sin&o
2
8E4$@% Ta"ing oent about 8D M84##J$@% M84@%@@.% 1 Ste-&! 9e,tor ethod r4##i>%*# j /45## ,os&oi>## sin&o j M84 r J / M84(##i>%*# j) J (45## ,os&oi>## sin&o j) M );7=797.=k " 2/50 >a ,alc!late the "o"ent of the 902 force abo!t point for the condition ;15o. )lso& deter"inethe $al!e of for which the "o"ent abo!t is >b ero and >c a "a4i"!".
Ste-!
Elasti, odulus of band k 4'# 1G Unstret,hed length of band 2o4#.@% -,;-)A), -,;0.:35A0.7= -,;1.375 "
;tan21 1.375
*tep3% ,hane in the lenth of band 4;(,24o
4;1.51520.7= eflection of sprin 4;0.775" *prin force ;64 ;:0D0.775 ;=:.5 a6in "o"ent abo!t B-C >,W A Mo;sinD-( Mo;=:.5sin:.5oD0.535 Mo;:.8 " 2/50 (a) Cal,ulate the oent of the #51 for,e about -oint O for the ,ondition θ=&o. 8lso3 deterine the value of θ for ;hi,h the oent about O is (b) Fero and (c) a a2iu.
*oln. iven that 8--lied for,e /4# 1
a ;15o
Ste-! >)nticloc6 wise A Mo;cosD0.: 2 sin D 0.8 Mo4#,os&oJ#.'> #sin&oJ#.* Mo4$$.& 1 Ste-$! b) Moent of the for,e about ?D is Fero. Mo4# cosD0.: 2 sin D 0.8;0 cosD0.:;sin D 0.8
tan; 0.6
0.8
;3:.9o
*tep=%
c or the "o"ent to be "a4i"!" the applied force sho!ld be perpendic!lar.
tanF; 0.6
;90AF ;90A3:.9o
;1:.9o
2/51 he s"all crane is "o!nted alon the side of a pic6!p bed and facilitates the handlin of hea$y loads. When the
boo" ele$ation anle is θ==0o&the force in the hydra!lic cylinder BC is =.5 6& and this force applied at point C is in the direction fro" B toC >the cylinder is in co"pression. eter"ine the "o"ent of this =.526 force abo!t the boo" pi$otpoint .
Ste-! <et the anle "ade by the force with the horiontal be F
tanF; 360+340sin 40−110 cos 40
340 cos 40+110sin 40
tanF; 494.283
F;5:.o
a6in "o"ent abo!t B-C >,W A Mo;cosFD-( Mo;=.5cos5:.o D0.3: Mo;0.90 6" 2/52 0esign ,riteria reIuire that the robot e2ert the #51 for,e on the -art as sho;n ;hile inserting a ,ylindri,al -art into the ,ir,ular
hole. 0eterine the oent about -oints A3 B3 and C of the for,e ;hi,h the -art e2erts on the robot.
Ste-! iven that /or,e e2erted by the robot at 0D =4# 1 Ta"ing oent about CD QMC4# MC 7 =(&#)4# MC4$&## 1 MC4$.& 1 Ta"ing oent about BD QMB4# MB 7 =(E/>/B)4# MB 4 =(E/>/B) MB 4#(&#>%&#sin$#o) MB 4#($@&) MB 4$$@&# 1
MB 4$$.@& 1 Ste-$! Ta"ing oent about 8D QM84# M8 7 =(E/>/B>B)4# M8 4 =(E/>/B>B) M8 4#(&#>%&#sin$#o>&&#sin%&o) M8 4#($@'$.) M8 4'*@&. 1 M ) ;:8.75 " 2/53 The asthead :tting su--orts the t;o for,es sho;n. 0eterine the agnitude of T ;hi,h ;ill ,ause no bending of the ast (Fero oent) at -oint O.
*tep% <et be the tension in the strin. a6in "o"ent abo!t B-C Mo;0
5cos30oD90A5sin30oD902
5
√ 22+5 2 ;0
389.71A150211.=2.8;0 133.7;539.71 ;=.0= 6 2/54 The -iston3 ,onne,ting rod3 and ,ran"shaft of a diesel engine are sho;n in the :gure. The ,ran" thro; OA is half the stro"e of *in3 and the length AB of the rod is %in. /or the -osition indi,ated3 the rod is under a ,o-ression along AB of $&&#lb. 0eterine the oent M of this for,e about the ,ran"shaft a2is O.
Ste-! ?84* in 8B4% in Co-ression in the rod 8B3 T4$&&# lb. 804?8 ,os$#o
804%(#.*'') 804$.%'% in ?04?8 sin$#o
?04% sin$# ?04 in
B04 √ AB 2− AD
;tan21 AB
A4%.$$o
Ste-$! a6in "o"ent abo!t B-C >,loc6wise A Mo;sinD-( Mo4$&&# sin%.$$oJ&.&' Mo4$'@.* lb5in Mo;1139 lb2ft 2/55 he 102 force is applied as shown to one end of the c!r$ed wrench. +f F;30o& calc!late the "o"ent of F abo!t the center of the bolt. eter"ine the$al!e of F which wo!ld "a4i"ie the "o"ent abo!t O state the $al!e of this "a4i"!" "o"ent.
Ste-!8--lied for,e /4# 1 a6in "o"ent abo!t B-C>,WA Mo;10cos30o>70A150A70A 10cos30o>5A70A70A5 Mo;=1537.:8""
*tep3% or "a4i"!" "o"ent Mo the force sho!ld be perpendic!lar to the line joinin )(.
tanF; 25+70+25+70
70+150+70
√ (25+70+25+70) 2+(70+150+70)2
-);3=:.7"" a6in "o"ent abo!t B-C Mo;10D3=:.7 Mo;=1:03.8= "" 2/56 6f the ,obined oent of the t;o for,es about -oint C is Fero3 deterine (a) the agnitude of the for,e P. (b) the agnitude R of the resultant of the t;o for,es. (c) the ,oordinates x and y of the -oint A on the ri of the ;heel about ;hi,h the ,obined oent of the t;o for,es is a a2iu. (d) the ,obined oent M A of the t;o for,es about A.
Ste-! a) Ta"ing oent about CD. QMC4# =J*>##,os'#oJ%5 ##sin'#oJ*4# =4'.' lb Ste-$! b) Q/242
24 5 ##,os'#o5'.' 245.' lb Q/y4y
y4## sin'#o
A4tan5 86.6
24%. in
y4*,osA 24*,os$@.*o
24'.$ in Ste-&! Moent about 8D M84JB8 B84?8>?B B84*>% ,osA B84*>%,os$@.*o
B84.' in Sin,e M84JB8 M84%$.$J.' M84&@@ lb5in (CN)
University of Lahore.
PROBLEMS Introductory Problems 2/57 he caster !nit is s!bjected to the pair of =002forces shown. eter"ine the "o"ent associated with these forces.
2/58 ) force F=# 0 acts alon the line AB. eter"ine the e'!i$alent forceE co!ple syste" at point C .
2/59 he top $iew of a re$ol$in entrance door is shown. wo persons si"!ltaneo!sly approach the door and e4ert force of e'!al "anit!des as shown. +f the res!ltin "o"ent abo!t the door pi$ot a4is at is 5 "& deter"ine the force "anit!de F .
2/60 he indicated forceEco!ple syste" is applied to a s"all shaft at the center of the plate.@eplace this syste" by a sinle force and specify the coordinate of the point on the x 2 a4is thro!h which the line of action of this res!ltalt force passes.
2/61 he brac6et is spot welded to the end of the shaft at point .o show the effect of the 9002 force on the weld& replace the force by its e'!i$alent of a force and co!ple M at . /4press M in $ector notation.
Solution Manual by Khalid Yousaf BS(Continue)Civil Engineering The
2/62 )s part of a test& the two aircraft enines are re$$ed !p and the propeller pitches are adj!sted so as to res!lt in the fore and aft thr!sts shown. What force F "!st be e4erted by the ro!nd on each of the "ain bra6ed wheels at A and B to co!nteract the t!rnin effect of the two propeller thr!sts? elect any effects of the nose wheel C & which is t!rned 90 and !nbra6ed.
2/63 @eplace the 1026 force actin on the steel col!"n by an e'!i$alent forceEco!ple syste" at point .his
replace"ent is fre'!ently done in the desin of str!ct!res.
2/64 /ach propeller of the twin2screw ship de$elops a f!llspeed thr!st of 300 6. +n "ane!$erin the ship&one propeller is t!rnin f!ll speed ahead and the other f!ll speed in re$erse. What thr!st P "!st each t! e4ert on the ship to co!nteract the effect of the shipCs propellers?
Representative Problems 2/65 ) l! wrench is !sed to tihten a s'!are2head bolt. +f 502 forces are applied to the wrench as shown& deter"ine the "anit!de F of the e'!al forces e4erted on the fo!r contact points on the 52"" bolt head so that their e4ternal effect on the bolt is e'!i$alent to that of the two 502 forces. )ss!"e that the forces are
perpendic!lar to the flats of the bolt head.
2/66 !rin a steady riht t!rn& a person e4erts the forces shown on the steerin wheel. ote that each force consists of a tanential co"ponent and a radiallyinward co"ponent. eter"ine the "o"ent e4erted abo!t the steerin col!"n at .
2/67 he 1802 force is applied to the end of body AB. +f θ= 50o& deter"ine the e'!i$alent forceEco!ple syste" at the shaft a4is .
2/68 ) force F of "anit!de 50 is e4erted on the a!to"obile par6in 2bra6e le$er at the position x ;50"". @eplace the force by an e'!i$alent forceEco!ple syste" at the pi$ot point .
2/69 he tie2rod AB e4erts the 502 force on the steerin 6n!c6le A as shown. @eplace this force by an e'!i$alent forceEco!ple syste" at .
2/70 he co"bined dri$e wheels of a front2wheel2dri$e a!to"obile are acted on by a 70002 nor"al reaction force and a friction force F& both of which are e4erted by the road s!rface. +f it is 6nown that the res!ltant of these two forces "a6es a 15o anle with the $ertical& deter"ine the e'!i$alent forceEco!ple syste" at the car "ass center $. reat this as a two di"ensional proble".
2/71 he syste" consistin of the bar A&two identicalp!lleys& and a section of thin tape is s!bjected to the two 180 tensile forces shown in the fi!re. eter"ine the e'!i$alent forceE co!ple syste" at point .
2/72 ,alc!late the "o"ent ! B of the 9002 force abo!t the bolt at B. *i"plify yo!r wor6 by first replacin the force by its e'!i$alent force2co!ple syste" at A.
2/73 he brac6et is fastened to the irder by "eans of the two ri$ets A and B and s!pports the 26 force. @eplace this force by a force actin alon the centreline between the ri$ets and a co!ple. hen redistrib!te this force and co!ple by replacin it by two forces&one at A and the other at B& and ascertain the forces s!pported by the ri$ets.
2/74 he anle plate is s!bjected to the two 502 forces shown. +t is desired to replace these forces by an e'!i$alent set consistin of the 002 force applied at A and a second force applied at B. eter"ine the y 2 coordinate of B.
2/75 he weld at can s!pport a "a4i"!" of 500 of force alon each of the n2 and t 2directions and a "a4i"!" of 1=00 " of "o"ent. eter"ine the allowable rane for the direction θ of the 7002 force applied at A. he anle θ is restricted to 0 PQ90o.
2/76 he de$ice shown is a part of an a!to"obile seatbac62release "echan2 is". he part is s!bjected to the =2 force e4erted at A and a restorin "o"ent e4erted by a hidden torsional sprin. eter"ine the y 2intercept of the line of action of the sinle e'!i$alent force.
PROBLEMS Introductory Problems 2/77 ,alc!late the "anit!de of the tension T and the anle θ for which the eye bolt will be !nder a res!ltant downward force of 15 6.
2/78 eter"ine the res!ltant R of the fo!r forces actin on the !sset plate. )lso find the "anit!de of R and the anle θ x which the res!ltant "a6es with the x 2a4is.
2/79 eter"ine the e'!i$alent forceE co!ple syste" at the oriin for each of the three cases of forces bein applied to the ede of a circ!lar dis6. +f the res!ltant can be so e4pressed&
replace this forceEco!ple syste" with a stand2alone force.
2/80 eter"ine the heiht % abo$e the base B at which the res!ltant of the three forces acts.
2/81 Where does the res!ltant of the two forces act?
2/82 Hnder non!nifor" and slippery road conditions& thetwo forces shown are e4erted on the two rear2dri$e wheels of the pic6!p tr!c6& which has a li"ited2slip rear differential. eter"ine the y 2intercept of the res!ltant of this force syste".
2/83 +f the res!ltant of the two forces and co!ple ! passes thro!h point & deter"ine ! .
2/84 eter"ine the "anit!de of the force F applied to the handle which will "a6e the res!ltant of the three forces pass thro!h .
2/85 eter"ine and locate the res!ltant R of the two forces and one co!ple actin on the +2bea".
2/86 ) co""ercial airliner with fo!r jet enines& each prod!cin 90 6 of forward thr!st& is in a steady& le$el cr!ise when enine n!"ber 3s!ddenly fails.eter"ine and locate the res!ltant of the three re"ainin enine thr!st $ectors. reat this as a twodi"ensional proble".
2/87 @eplace the three forces actin on the bent pipe by a sinle e'!i$alent force R. *pecify the distance x fro" point to the point on the x 2a4is thro!h which the line of action of R passes.
2/88 he directions of the two thr!st $ectors of an e4peri"ental aircraft can be independently chaned fro" the con$entional forward direction within li"its. or the thr!st confi!ration shown& deter"ine the e'!i$alent forceEco!ple syste" at point . hen replace this forceEco!ple syste" by a sinle force and specify the point on the x 2a4is thro!h which the line of action of this res!ltant passes. hese res!lts are $ital to assessin desin perfor"ance.
2/89 eter"ine the res!ltant R of the three forces actin on the si"ple tr!ss. *pecify the points on the x 2 and y 2a4es thro!h which R "!st pass.
2/90 he ear and attached L2belt p!lley are t!rnin co!ntercloc6wise and are s!bjected to the tooth load of 1:00 and the 8002 and =502 tensions in the L2belt. @epresent the action of these three forces by a res!ltant force R at and a co!ple of
"anit!de ! . +s the !nit slowin down or speedin !p?
2/91 he desin specifications for the attach"ent at A for this bea" depend on the "anit!de and location of the applied loads. @epresent the res!ltant of the three forces and co!ple by a sinle force R at A and a co!ple ! . *pecify the "anit!de of R.
2/92 +n the e'!ilibri!" position shown& the res!ltant of the three forces actin on the bell cran6 passes thro!h the bearin . eter"ine the $ertical force P. oes the res!lt depend on θ ?
2/93wo interal p!lleys are s!bjected to the belt tensions shown.+f the res!ltant R of these forces passes thro!h the center & deter"ine T and the "anit!de of R and the co!ntercloc6wise anle θ it "a6es with the x 2a4is.
2/94 While slidin a des6 toward the doorway& three st!dents e4ert the forces shown in the o$erhead $iew. eter"ine the e'!i$alent forceEco!ple syste" at point A. hen deter"ine the e'!ation of the line of action of the res!ltant force.
2/95 Hnder non!nifor" and slippery road conditions& the fo!r forces shown are e4erted on the fo!r dri$e wheels of the all2wheel2dri$e $ehicle. eter"ine the res!ltant of this syste" and the x 2 and y 2intercepts of its line of action. ote that the front and rear trac6s are e'!al >i.e.)(;,.
2/96 he rollin rear wheel of a front2 wheel2dri$e a!to"obile which is acceleratin to the riht is s!bjected to the fi$e forces and one "o"entshown. he forces A x ;=0 and Ay ; 000 are forces trans"itted fro" the a4le to the wheel& F ;1:0 is the friction force e4erted by the road s!rface on the tire& & ;=00 is the nor"al reaction force e4erted by the road s!rface& and ' ; =00 is the weiht of the wheeltire !nit. he co!ple ! ;3 " is the bearin friction "o"ent. eter"ine and locate the res!ltant of the syste".
2/97 ) rear2wheel2dri$e car is st!c6 in the snow between other par6ed cars as shown. +n an atte"pt to free the car& three st!dents e4ert forces on the car at points A& B& and C while the dri$erCs actions res!lt in a forward thr!st of 00 actin parallel to the plane of rotation of each rear wheel. reatin the proble" as two2di"ensional& deter"ine the e'!i$alent forceEco!ple syste" at the car center of "ass $ and locate the position x of the point on the car centreline thro!h which the res!ltant passes. elect all forces not shown.
2/98 )n e4ha!st syste" for a pic6!p tr!c6 is shown in the fi!re. he weihts ' %& ' m& and ' t of the headpipe&"!ffler& and tailpipe are 10& 100& and 50 & respecti$ely& and act at the indicated points. +f the e4ha!st2 pipe haner at point A is adj!sted so that its tension FA is 50 & deter"ine the re'!ired forces in the haners at points B& C & and ( so that the forceE co!ple syste" at point is ero. Why
is a ero forceEco!ple syste" at desirable?
,h#3 /'!ilibri!"
1.5
=?BLEMS Introductory Proble! 3/1 eter"ine the force P re'!ired to "aintain the 0026 enine in the position for which θ=30o. he dia"eter of the p!lley at B is neliible.
*tep% Mass ";006 Weiht W;00D9.81
;19: ,;sin30o
,;1 " );cos30o
);1.73 " (;2)
(;0.7 "
F;tan21 DC
o)
sin75 o
;1759 3/2 he "ass center $ of the 1=0026 rear2enine car is located as shown in the fi!re. eter"ine the nor"al force !nder each tire when the car is in e'!ilibri!".*tate any ass!"ptions.
*tep% Mass of car ";1=006 Weiht W;1=00D9.81 W;1373= y;0 *ince there are two front and two rear tyres of a car&therefore @ )A@(21373=;0 >i a6in "o"ent abo!t B)C M );0 @(>1.38:A0.9:=21373=D1.38:;0 =.7@(;19035.3= @(;=050 >ii Hsin the $al!e of @( in >i
@ )A>=05021373=;0 @ );817 3/3 ) carpenter carries a 1 lb !nifor" board as shown.What downward force does he feel on his sho!lder at A?
*tep1%ree body diara"
*tep%Weiht of the board W;1 lb <et ) and ( be the reactions at points ) and ( respecti$ely.a6in "o"ent abo!t (& M(;0 WD>A12 )D;0 1D32 );0 );3: );18 lb
3/4 +n the side $iew of a 706 tele$ision restin on a=6 cabinet& the "ass centers are labeled $1 and $. eter"ine the force reactions at A and B. >ote that the "ass center of "ost tele$isions is located well forward beca!se of the hea$y nat!re of the front portion of pict!re t!bes.
*tep% Weiht of L& W1;70D9.81 W1;:8:.7 Weiht of the cabinet W;=D9.81 W;35.== a6in "o"ent abo!t )& M );0 ( D0.72W1D0.:2WD0.35;0 0.7( 2:8:.7D0.:235.==D0.35;0 0.7(2=1.028.=0=;0 0.7(;70: ow y;0 )A(2W12W;0 )A70:2:8:.7235.==;0 );1:
3/5 he roller stand is !sed to s!pport portions of lon boards as they are bein c!t on a table saw. +f the board e4erts a 52 downward force on the roller C & deter"ine the $ertical reactions at A and (. ote that the connection at B is riid& and that the feet A and ( are fairly lenthy horiontal t!bes with a nonslip coatin.

@;1:.55 ow y;0 @ )A@25;0 @ )A1:.5525;0 @ );8.=5 3/6 he =5026 !nifor" +2bea" s!pports the load shown.eter"ine the reactions at the s!pports.
*tep% Weiht of the 62bea" W1;=50D9.81 W1;==1=.5 Weiht of the dr!" W;0D9.81 W;158. @ )A@(;==1=.5A158. @ )A@(;:57.7 >i a6in "o"ent abo!t )& M );0 >,WA ==1=.5D=A158.D5.:2@(D8;0 @(;3718 !ttin this $al!e in >i
@ )A3718;:57.7 @ );85=.7
3/7 ,alc!late the force and "o"ent reactions at the bolted base of the o$erhead traffic2sinal asse"bly. /ach traffic sinal has a "ass of 3: 6&while the "asses of "e"bers C and AC are 50 6 and 55 6& respecti$ely.
*tep% Weiht of each traffic sinal is 80 lb. Weiht of "e"bers -, and ), are 110 lb and 10 lb respecti$ely. <et -4&-y and Mo;0 be the reactions and reacti$e "o"ent at B-C respecti$ely. ,onsiderin forces alon 42a4is 4;0 -4 ;0 ,onsiderin forces alon y2a4is y;0 -y2802802110210;0 -y;390 lb a6in "o"ent abo!t B-C
Mo;0 >,,WA 80>15A3A1A80>3A1A10D12Mo;0 Mo;50=0 lb2ft >,W
3/8 he 026 ho"oeneo!s s"ooth sphere rests on the two inclines as shown.eter"ine the contact forces at A and B.
*tep% Mass of the sphere& ";06 Weiht of sphere& W;0D9.81 W;19:. ,onsiderin forces alon 42a4is 4;0 @ ) cos1502@( sin30o;0 0.9:: )20.5(;0 >i ,onsiderin forces alon y2a4is y;0 @ ) sin150A@( cos30o219:.;0 0.9:: )A0.5(;19:. 0.59 );19:.20.8::(
);758.0:23.3==( >ii !ttin abo$e $al!e in >i 0.9::>758.0:23.3==(20.5(;0 73.923.3(20.5(;0
3.73(;73.9 (;19:.3 Hsin this $al!e in >i 0.9:: )20.5>19:.3;0 0.9:: );98.15 );101.: 3/9 ) 10 lb crate resets on the :0 lb pic6!p tailate. ,alc!late the tension T in each of the two restrainin cables& one of which is shown. he centers of ra$ity are at $1 and $. he crate is located "idway between the two cables.
*tep% Weiht of the crate W1;10 lb Weiht of the tailate W;:0 lb
tan; 12
9.5+2.75
;==.=o
a6in "o"ent abo!t B-C Mo;0 >,,WA 10>1=A:0>9.52sin==.=o>1.5;0 1:80A570217.1=;0 ;131.7 lb 3/10 ) portable electric enerator has a "ass of 1:0 6 with "ass center at $. eter"ine the !pward force F necessary to red!ce the nor"al force at A to onehalf its no"inal >F ; 0 $al!e.
*tep% Weiht of the enerator W;1:0D9.81 W;15:9.: <et be the force re'!ired. <et @ ) and @- be the reactions at ) and - respecti$ely. or ;0
a6in "o"ent abo!t B-C Mo;0 >,,WA 2@ )D0.=:A159:.:D0.135;0 @ );=:0.:=
or 1
2 @ );30.3
a6in "o"ent abo!t B-C Mo;0 >,,WA 15:9.:>0.135230.3>0.=:2>0.7;0 20.7;2105.95 ;151.3: 3/11 With what force "anit!de T "!st the person p!ll on the cable in order to ca!se the scale A to read 000 ? he weihts of the p!lleys and cables are neliible. *tate any ass!"ptions.

/12 he de$ice shown is desined to aid in the re"o$al of p!ll2tab tops fro" cans. +f the !ser e4erts a =02 force at A& deter"ine the tension T in the portionBC of the p!ll tab.
*tep% orce applied ;=0 <et the tension in portion B(,C be BC <et @o be reaction at B-C a6in "o"ent abo!t B-C Mo;0 >,,WA cos=5o>3Asin=5o>3: 2 =0cos10o>78A32=0sin10o>7;0 ;9=.0:
3/13 ) woodc!tter wishes to ca!se the tree tr!n6 to fall !phill& e$en tho!h the tr!n6 is leanin downhill.With the aid of the winch ' & what tension T in the cable will be re'!ired? he 100lb tr!n6 has a center of ra$ity at $. he fellin notch at is s!fficiently lare so that the resistin "o"ent there is neliible.
*tep% Weiht of the tr!n6 W;100 lb <et BC be the tension in the cable a6in "o"ent abo!t B-C Mo;0 >,,WA 100>1sin5o2>10A=cos15o;0 ;81.1 lb 3/14 o facilitate shiftin the position of a liftin hoo6 when it is not !nder load& the slidin haner shown is !sed. he projections at A and B enae the flanes of a bo4 bea" when a load is s!pported&
and the hoo6 projects thro!h a horiontal slot in the bea". ,o"p!te the forces at A and B when the hoo6 s!pports a 30026 "ass.
*tep% Weiht of the bloc6 W;300D9.81 W;9=3 <et @ ) and @( be reactions at ) and (. a6in "o"ent abo!t B)C M );0 >,,WA 9=3D0.= 2 @(D0.:;0 @(;19: ,onsiderin forces alon y2a4is y;0 @ )2@(2W;0 @ );19:A93 @ );=905 3/15 hree cables are joined at the j!nction rin C . eter"ine the tensions in
cables AC and BC ca!sed by the weiht of the 3026 cylinder.
*tep%
<et ),& )( and , be the tension in the cables )(&(, and , respecti$ely. Weiht of the cylinder W;30D9.81 W;9=.3 Iere ,;9=.3 4;0
2 ),cos=5oA9=.3cos15o2(,cos:0o;0 0.707 ),A0.5(,;8=.7 >i y;0 ),sin=5oA9=.3sin15o2(,sin:0o;0 0.707 ),;0.8::(,27:.17 ),;1.5 (,2107.7= >ii Hsin >ii in >i 0.707>1.5(,2107.7=A0.5(,;8=.7 0.8::(,27:.17A0.5(,;8=.7 1.3::(,;3:0.== (,;:3.87 Hsin abo$e $al!e in >ii ),;1.5>:3.872107.7= ),;15.=9 3/16 ) 7002 a4ial force is re'!ired to re"o$e the p!lley fro" its shaft. What force F "!st be e4erted on the handle of
each of the two prybars? riction at the contact points B and ) is s!fficient to pre$ent slippinOfriction at the p!lley contact points C and F is neliible.
*tep% otal a4ial force BC re'!ires is 700 *ince two prybars are actin at both sides of the p!lly&the a4ial force by each prybar wo!ld be 350. a6in "o"ent abo!t B/C M/;0 >,,WA
! F
2
D0.038Acos5>0.5Asin5>0.031;0 213.3A0.=9A0.007;0 0.5;13.3 ;5.8= 3/17 he !nifor" bea" has a "ass of 50 6 per "eter of lenth. ,o"p!te the reactions at the s!pport .he force loads shown lie in a $ertical plane.
*tep% <et -4 and -y be the reactions at B-C alon 4 and y directions respecti$ely. <et Mo&be the reaction "o"ent at B-C *elf load of the bea" portions.i.e 1;50D9.81D>1.8A0.: 1;1177. 1;1.117 6 ;50D9.81D>0.:A0.: ;588.: ;0.589 6 *tep3% a6in "o"ent abo!t B-C Mo;0 Mo21D1.23D.82>.=A0.:cos30oA= E 1.=>.=cos30oA1.;0 Mo21.177D1.25.=20.589D.91:A=2 1.=D3.78;0 Mo21.=125.=21.7A=2=.59;0 Mo;9.1 6" 4;0 -4A1.=sin30o;0 -4A0.7;0 -4;20.7 6 y;0 -y2122321.=cos30o;0 -y21.17720.5892321.1;0 -y;5.98 6
Representati*e Problems
3/18 ) pipe P is bein bent by the pipe bender as shown. +f the hydra!lic cylinder applies a force of "anit!de F ; = 6 to the pipe at C & deter"ine the "anit!de of the roller reactions at A and B.
*oln. *tep1%ree body diara" *oln. *tep1%ree body diara"
*tep% )pplied force = 6 <et @ ) and @( be the reactions at ) and (. Iere @ );@(
y;0 =2@ )cos15o2@(cos15o;0 @ )cos15o;= @ );1.= 6 @ ); @(;1.= 6 3/19 he !nifor" 15" pole has a "ass of 150 6 and is s!pported by its s"ooth ends aainst the $ertical walls and by the tension T in the $ertical cable.,o"p!te the reactions at A and B.
*tep% <enth of the pole& <;15" Weiht of the pole& W;150D9.8 W;1=71.5 <et BC be the tension in the cable. <et @ ) and @( be the horiental reactions at ) and ( respecti$ely. -(A1;15
-(; √ 15 2+12
ro" si"ilar trianles ), and )(-
A
15
)/;: " a6in "o"ent abo!t B)C >,,WA M );0 @(D921=71.5D:A1=71.5D=;0 @(;37 @ );@(;37 3/20 eter"ine the reactions at A and ) if P ; 500 . What is the "a4i"!" $al!e which P "ay ha$e for static e'!ilibri!"? elect the weiht of the str!ct!re co"pared with the applied loads.
*tep%
<et )4&)y and /4 be the reactions at ) and / respecti$ely. 4;0 )4A/42=000sin30o;0 )4A/4;000 >i y;0 )y2=000cos30oA500;0 )y;3=:=.12500;0 )y;9:=.1 a6in "o"ent abo!t B)C M );0 500D8A/4D32=000cos30o>=;0 3/4;1:000cos30o2=000 /4;385.=: Hsin abo$e $al!e in >i )4A385.=:;000 )4;2185.=: or "a4i"!" & /4;0
a6in "o"ent abo!t B)C M );0
D82=000cos30o>=;0 8;1385:.= ;173 3/21 While diin a s"all hole prior to plantin a tree& a ho"eowner enco!nters roc6s. +f he e4erts a horiontal 52 force on the prybar as shown& what is the horiontal force e4erted on roc6 C ? ote that a s"all lede on roc6 C s!pports a $ertical force reaction there. elect friction at B. ,o"plete sol!tions >a incl!din and>be4cl!din the weiht of the 186 prybar.
*tep% <et >c4 and >cy be the horiontal and $ertical reactions at B,C. <et ( be the force e4erted by roc6 at B(C >a ,onsiderin the weiht of the prybar. a6in "o"ent abo!t B,C M,;0
50D:8A=0D3=tan0 o2( ×
8
cos 20 o ;0
3=00A=9528.51(;0 8.51(;3895 (;=57.7 lb a6in force alon 42a4is >c4A502 (cos0o;0 >c4; =57.7cos0o250 >c4;380 lb *tep3% >b /4cl!din the weiht of the prybar. a6in "o"ent abo!t B,C M,;0
50D:82(
× 8
cos 20 o ;0
(;399 lb a6in forces alon 42a4is. 4;0 ,2 (cos0oA50;0 ,2 399cos0oA50;0 ,;35 lb 3/22 eter"ine the force P re'!ired to bein rollin the !nifor" cylinder of "ass m o$er the obstr!ction of heiht %.
*tep%considerin the trianle )(,
(,; √ ! 2+(!−h)2
2
! ;0
2
!−h
3/23 ) 352 a4ial force at B is re'!ired to open the sprinloaded pl!ner of the water nole. eter"ine the re'!ired force F applied to the handle at A and the "anit!de of the pin reaction at . ote that the pl!ner passes thro!h a $ertically2elonated hole in the handle at B& so that neliible $ertical force is trans"itted there.
*tep% <et BC be the force applied at handle. <et -4 and -y be the horiontal and $ertical reactions at point B-C. Gi$en that 35 of force is the reacti$e force at B(C a6in "o"ent abo!t B-C Mo;0 35D182cos10o>382 sin10o>==;0 =5.0:;:30 ;13.98 *te3% 4;0 -42cos10o235;0 -4213.98cos10o235;0
-4;=8.8 y;0 -y2sin10o;0 -y;.=3 Manit!de of reaction at B-C
-; √ O x
2
-;=8.8: 3/24 ) person holds a 3026 s!itcase by its handle as indicated in the fi!re. eter"ine the tension in each of the fo!r identical lin6s AB.
*tep% Weiht of the s!itcase W;:0 lb !"ber of lin6s is =. <et BC be the tension in each lin6&so there are force in the left and riht. y;0 :02=sin35o;0 sin35o;:0 ;:.15 lb 3/25 ) bloc6 placed !nder the head of the claw ha""er as shown reatly facilitates the e4traction of the nail. +f a 002 p!ll on the handle is re'!ired to p!ll the nail& calc!late the tension T in the nail and the "anit!de A of the force e4erted by the ha""er head on the bloc6. he contactin s!rfaces at A are s!fficiently ro!h to pre$ent slippin.
*tep% orce applied on the handle ;50 lb <et BC be the tension in the nail. <et )4 and )y be the horiontal and $ertical reactions at B)C respecti$ely. a6in "o"ent abo!t B)C M );0 50D82;0 ;00 lb 4;0 cos0o 2)4;0 )4 ; 50cos0o
)4 ; =:.98 lb y;0 sin0o A)y2;0 )y ;2 50sin0o
)y ;002 50sin0o
)y ;18.9 lb he "anit!de of the force B)C e4erted by the ha""er head of the bloc6.
); √ A x
);188.8 lb
3/26 he indicated location of the center of "ass of the 3:00 lb pic6!p tr!c6 is for the !nladen condition. +f a load whose center of "ass is x ; =00 "" behind the rear a4le is added to the tr!c6& deter"ine the "ass m+ for which the nor"al forces !nder the front and rear wheels are e'!al.
*tep% <et the load weiht be W<
Weiht of the tr!c6 W;3:00 lb <et the nor"al forces at ) and ( be @ )
and @( respecti$ely. Gi$en that the nor"al forces !nder the front and rear wheels are e'!al. @ );@(
a6in "o"ent abo!t B)C M );0 3:00>=5A W<>=5A:7A1:2 @(>=5A:7;0 1:000A 18W<211@( ;0 >i ,onsiderin forces alon y2a4is. y;0 @ )A@(23:002W<;0 @( ;3:00A W< >ii Hsin >ii in >i 1:000A 18W<25:>3:00AW< ;0 1:000A18W<210:0025:W<;0 7W<;39:00
W<;550 lb !ttin abo$e $al!e in >ii @( ;3:00A550 @( ;=150 @ );@(;075 lb 3/27 he wall2"o!nted .526 liht fi4t!re has its "ass center at $. eter"ine the reactions at A and B and also calc!late the "o"ent s!pported by the adj!st"ent th!"bscrew at C . >ote that the lihtweiht fra"e ABC has abo!t 50 "" of horiontal t!bin& directed into and o!t of the paper& at both A and B.
*tep% Weiht of the liht W;.5D9.81 W;=.53 <et )4&)y and (4 be the reactions at ) and ( respecti$ely. a6in "o"ent abo!t B)C M );0 >,WA (4D302=.53D300;0 (4;3 ,onsiderin fores alon 42a4is 4;0 )42(4;0 )423;0
)4;3 *tep3%ree body diara" of fi4t!re only
a6in "o"ent abo!t B,Cd!e to weiht BWC M,;0 M,;=.53D100 M,;=53 "" M,;=.53 " 3/28o test the $alidity ofaerodyna"ic ass!"ptions "ade in the desin of the aircraft& its "odel is bein tested in a wind t!nnel. he s!pport brac6et is connected to a force and "o"ent balance& which is eroed when there is no airflow. Hnder test conditions&the lift +& dra (& and pitchin "o"ent M$ act as shown. he force balance records the lift& dra& and a "o"ent ! P . eter"ine ! $ in ter"s of +& (& and ! P .
*tep% a6in "o"ent abo!t BC M;0 M2MG2<d2h;0 MG;M2<d2h;0
3/29 he chain binder is !sed to sec!re loads of los&l!"ber& pipe& and the li6e. +f the tension T 1 is 6 when θ=30o& deter"ine the force P re'!ired on the le$er and the correspondin tension T for this position.)ss!"e that the s!rface !nder A is perfectly s"ooth.
*tep% ension 1; 6 ;30o
<et )y be the reaction at s!pport B)C a6in "o"ent abo!t B)C M );0 >,WA D:002D100sin30 o;0 ;0.1::7 6 ,onsiderin fores alon 42a4is 4;0 21A sin30o;0 25A0.1::7D0.5;0 ;1.9 6 3/30 he de$ice shown is desined to apply press!re when bondin la"inate to each side of a co!ntertop near an ede. +f a 102 force is applied to the handle& deter"ine the force which each roller e4erts on its correspondin s!rface.
*tep% orce applied at handle ;30 lb <et the reaction forces at ( and , are @(
and @, respecti$ely. ,onsiderin trianle (,-
-(;3.5tan=5o
(,;=.95 a6in "o"ent abo!t B,C M,;0 (D3.5230D5.5cos=5o;0 3.5(;11:.:7 (;33.=3 lb ,onsiderin fores alon y2a4is y;0 ,2(230;0 ,233.3=230;0 ,;:3.3= lb 3/31 he two liht p!lleys are fastened toether and for" an interal !nit.hey are pre$ented fro" t!rnin abo!t their bearin at by a cable wo!nd sec!rely aro!nd the s"aller p!lley and fastened to point A.,alc!late the "anit!de R of the force s!pported by the bearin for the applied 6 load.

325
;.:o
a6in "o"ent abo!t B-C Mo;0 >,,WA D002D15;0 ;3. 6 ,onsiderin fores alon 42a4is 4;0 3.cos.:oA@4;0 @4;2.95 6 ,onsiderin fores alon y2a4is y;0 @y223.sin.:o;0 @y;3.3 6 Manit!de of the reaction force B@C
@; √ R x
@; √ (−2.95)2+(3.23)2
@;=.37 6 3/32 +n a proced!re to e$al!ate the strenth of the triceps "!scle& a person p!shes down on a load cell with the pal" of his hand as indicated in the fi!re. +f the load2cell readin is 1:0 & deter"ine the $ertical tensile force F enerated by the triceps "!scle. he "ass of the lower ar" is 1.5 6 with "ass center at $. *tate any ass!"ptions.
*tep% Weiht of the lower ar" is 3. lb <et BC be the force in the triceps "!scle load at the pal" is 35 lb a6in "o"ent abo!t B-C Mo;0 >,,WA 35D>:A:23.D:2D1;0 2; 2 =01 ;=01 lb 3/33 ) person is perfor"in slow ar" c!rls with a 1026 weiht as indicated in the fi!re. he brachialis "!scle ro!p >consistin of the biceps and brachialis "!scles is the "ajor factor in this e4ercise. eter"ine the "anit!de F of the brachialis2"!sclero!p force and the "anit!de ) of the elbow joint reaction at point ) for the forear" position shown in the fi!re. a6e the di"ensions shown to locate the effecti$e points of application of the two "!scle ro!psO these points are 00 "" directly abo$e ) and 50 "" directly to the riht of ) . +ncl!de the effect
of the 1.526 forear" "ass with "ass center at point $. *tate any ass!"ptions.
*tep% Weiht of the lower ar" is 3. lb <et BC be the force is the (richiates "!scle load in the pal" is 0 lb.
tan; 2/8
;1=.0=o
a6in "o"ent abo!t B/C M/;0 >,,WA cos1=.0=o>2GD:20D1=;0 1.9=23.D:280;0 ;15=.3 lb ,onsiderin fores alon 42a4is 4;0 /42sin1=.0=o;0 /4215=.3sin1=.0=o;0 /4;37.= lb ,onsiderin fores alon y2a4is y;0 /y23.20Acos1=.0=o;0 /y;21:.= lb
/; √ (37.4)2+(126.4 )2
/;131.8 lb 3/34 ) wo"an is holdin a 3.:26 sphere in her hand with the entire ar" held horiontally as shown in the fi!re. ) tensile force in the deltoid "!scle pre$ents the ar" fro" rotatin abo!t the sho!lder joint O this force acts at the 1o anle shown.eter"ine the force e4erted by the deltoid "!scle on the !pper ar" at A and the x 2 and y 2co"ponents of the force reaction at the sho!lder joint . he "ass of the !pper ar" is m, ;1.9 6& the "ass of the lower ar" is m+ ; 1.1 6& and the "ass of the hand is m ; 0.= 6O all the correspondin weihts act at the locations shown in the fi!re.
*tep% Weiht of the !pper ar" W!;1.9D9.81 Weiht of the lower ar" Wl;1.1D9.81 Weiht of the hand Wh;0.=D9.81 Weiht of the sphere W;3.:D9.81 <et BC be the tensile force actin in the deltoid "!scle. <et -4 and -y be the reactions at joint B-C a6in "o"ent abo!t B-C
;
125sin 21 o
;710 ,onsiderin fores alon 42a4is 4;0 -42cos1o;0 -4 ;::.8 ,onsiderin fores alon y2a4is y;0 -yA710sin1o21.9D9.8121.1D9.812 0.=D9.8123.:D9.81;0 -y;2185.7 3/35 With his weiht ' e'!ally distrib!ted on both feet& a "an beins to slowly rise fro" a s'!attin position as indicated in the fi!re. eter"ine the tensile force F in the patellar tendon and the "anit!de of the force reaction at point & which is the contact area between the tibia and the fe"!r. ote that the line of action of the patellar tendon force is alon its "idline. elect the weiht of the lower le.
*tep% <et BC be the force in the patellar tendon. <et -4 and -y be the reactions at B-C.
<et W /2 be the weihton one le.
a6in "o"ent abo!t B-C Mo;0 >,,WA
D50 2 W
2 D5;0
;.5W ,onsiderin fores alon 42a4is 4;0 cos55o2-4;0 .5cos55o2-4;0 -4;1.9W ,onsiderin fores alon y2a4is y;0
sin55oA W
2 A-y;0
.5sin55oA 0.5WA-y;0 -y;2.3=W Manit!de of the reaction at B-C
-; √ O x
-; √ (1.29 W )2+(−2.34 W )2
-;.:7W 3/36 he ele"ents of an on2off "echanis" for a table la"p are shown in the fi!re. he electrical switch . re'!ires a = force in order to depress it. Whatcorrespondin force F "!st be e4erted on the handleat A?
*tep% he force at B*C is 0.9 lb <et -4 and -y be the reactions at B-C. he force has been replaced by a force2 co!ple syste" at (.
Where M;>.=cos15o a6in "o"ent abo!t B-C Mo;0 >,WA 0.9D1.2 D>.=cos15o2D>3.:cos15o;0 1.82.31823.=77;0 ;0.18: lb 3/37 he !nifor" 1826 bar A is held in the position shown by the s"ooth pin at and the cable AB. eter"ine the tension T in the cable and the "anit!de and direction of the e4ternal pin reaction at .
*step% Weiht of the bar B-)C W;18D9.81 <et BC be the tension in the cable )( <et -4 and -y be the reactions at B-C.
tan; 1.5 sin 60
tan; 1.299
sin33.7oD1.5cos:0o2cos33.7oD1.5sin:0o
0.=21.08A::.;0 0.::;::. ;100 *tep3% ,onsiderin fores alon 42a4is 4;0 -42100c0s33.7o;0 -4;83.=5 ,onsiderin fores alon y2a4is y;0 -y218D9.812100sin33.7o;0 -y;3 Manit!de of the reaction at B-C
-; √ O x
-; √ (83.45) 2+(232)2
-;=: 3/38 ) person atte"pts to "o$e a 026 shop $ac!!" by p!llin on the hose as indicated. What force F will ca!se the !nit to tip cloc6wise if wheel A is aainst an obstr!ction?
*tep% Weiht of the cart W;=0 lb <et )4 and )y be the horiontal and $ertical reactions at B)C respecti$ely. <et BC be the e4tended force. he reaction B@C at ( is ero at the point treppin. a6in "o"ent abo!t B)C M );0 =0D72cos15oD>15A8Asin15oD3;0 8021.==;0 ;13.0: lb 3/39 he e4ercise "achine is desined with a lihtweiht cart which is "o!nted on s"all rollers so that it is free to "o$e alon the inclined ra"p. wo cables are attached to the cart2one for each hand. +f the hands are toether so that the cables are parallel and if each cable lies essentially in a $ertical plane& deter"ine the force P which each hand "!st e4ert on its cable in order to "aintain e'!ilibri!" position.he "ass of the person is 70 6& the ra"p anle θ is 15o& and the anle / is 18o. +n addition& calc!late the force R which the ra"p e4erts on the cart.
*tep% Weiht of the person W;70D9.81 +nclination of the ra"p ;15o
<et B@C be the force e4erted by the ra"p on the cart. <et BC be the force e4erted by each hand. ,onsiderin the forces alon the inclined plane. 4;0 70D9.81Dsin15o22cos18o;0 177.7323.90;0 ;=5.: ,onsiderin the forces perpendic!lar to the plane. @270D9.81Dsin15o2sin18o;0 @;::3.3AD=5.: sin18o
@;:91 3/40 he de$ice shown is !sed to test a!to"obile2enine $al$e sprins. he tor'!e wrench is directly connected to ar" B. he specification for the a!to"oti$e inta6e2$al$e sprin is that 370 of force sho!ld red!ce its lenth fro" 50 "" >!nstressed lenth to = "". What is the correspondin readin ! on the tor'!e wrench& and what force F e4erted on the tor'!e2wrench handle is re'!ired to prod!ce this readin? elect the s"all effects of chanes in the an!lar position of ar" B.
*tep% orce e4erted by the sprin on B-(C;83 lb <et -4 and -y be the reactions at B-C. a6in "o"ent abo!t B-C Mo;0 M283D:;0 M;=98 l.in ,onsiderin free body diara" of B-)C
a6in "o"ent at B-C M;cos0oD15 =98;1=.095 ;35.3 lb
r!ss%) fra"ewor6 co"posed of "e"bers joined at their ends to for" ariid str!ct!re is called a truss. e.O r!sses of roofin syste"& bride tr!sses& trans"ission towers etc.
ypes of tr!sses% here are two types of tr!sses. i lane tr!ss ii *pace tr!ss
lane r!ss% When the "e"bers of the tr!ss lie essentially in a sinle plane& the tr!ss is called a plane truss.

Si"#$eT%&ss' ) tr!ss which is constr!cted
fro" a basic trian!lar str!ct!re to s!ch a
"anner that to increase new ele"ents&
two "e"bers and one joint is added& is
6nown as si"ple tr!ss.
for"ed by connectin two or "ore si"ple
tr!sses.
be classified as si"ple or co"po!nd.
)ss!"ptions in the analysis and desin of
tr!sses.
12Me"bers are joined toether by s"ooth pins& altho!h the "e"bers are ri$eted& bolted and welded. 2he centre line of all the "e"bers are conc!rrent at the joint. 32)ll the loads are only applied at the
joints and the weiht of the "e"bers is
ass!"ed neliible so there will be only
a4ial forces >tension or co"pression.
*tability and eter"inacy of r!sses. *tability% /4ternal *tability% ) tr!ss will be e4ternal !nstable if all the reactions are conc!rrent or parallel. +nternal *tability% ) tr!ss will be internally stable if it is not liable to collapse. ) si"ple tr!ss is always internally stable. +ndeter"inacy%) tr!ss will be indeter"ina2 te in which all s!pport reactions and

is con$enient to consider stability and deter"inacy as follows% 12With respect to reactions. e.O e4ternal stability and deter"inacy. 2With respect to "e"bers. e.O internal stability and deter"inacy. 32he co"bination of internal and e4ternal conditions. E+,e%na$ -e,e%"inac.' ) deter"inate str!ct!re sho!ld ha$e at least three reactions. e;r2 Where r;n!"ber of reactions ;n!"ber of e'!ilibri!" e'!ations a$ailable.
n,e%na$ -e,e%"inac.' +t can be chec6ed if "ini"!" n!"ber of reactin co"ponents necessary for the e4ternal deter"inacy and stability are 6nown. "Ar;j where &qu

Solution Manual Mechanics J L Mariam

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