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•
CH"'PfE~
2
FORCE VECTORS
PROBLEMS
.2-1. If 8 - 30" and T - 6 kN. determine the magnitude
of Ihe rC\$ullant force acting on Ihe 1.')'1.'0011 and its dir.::clion
measured clockWise from the posi tive .( axis..
2-2. If (J _ fI.'f' and T _ 5 tN . determine Ihe magniludl.'
of the resultant force acting on the eye boll and its direction
meflSurcd clockwise from Ihe posi li\'c .J axis..
2-7. If foB - 2 kN and the re~ultanl force acts along the
positive" axis. delermine Ihe magnitude of the resultant
force and the angle 9.
"2-3.
If thc resultant force is required to aet along Ihe
posilh'e" uis and ha"c a magnitude of S kN.dclermine the
required magnilude of f Band ils direc.ion 9.
2-3. If the magnitude of the resultant forre is 10 be 9 kN
directed along Ihe posit;"c xaxis,octcrminc the magnitude of
force T ~Iing on the eyebolt and ils :Ingle O.
T
~-IJI-~-··'
Probs. 2-718
('robs. l- l/lIJ
. 2-4. Determine Ihe magniUlde of the resultant forcl.'
acting on
Ihe bracket and ils direction measured
oo\mtcrdod:wisc from Ihe po\$it;I'C II
a_~is.
·Z- 5. Rcsoll'c FI inlo components along llie" and v axes.
and determine the magnitudes of these components.
2-6, Rcsol,·c F! ;n10 components along Ihe " and 1/ axes.,
and delCrnl;ne Ihe nlaJllillldcs of Ihcse oo"'pQnenl\$.
' 2-9. 1lIc plDIC is subjttled to the t,,·o forces at A and 8
as shown. If 9 - «1'. determine Ihe magnitude of the
resultant of these two forces and its direction measured
clock,,·ij.C from the horizontal.
2-10. Determine the angle of 9 for connecting nlcnlbcr A
to lhe plate 5() Ihat the resultant force of f A and f a is
directed Iiorizontally to.he right. Also.. wll." is the nlagnitude
of lhe rcsUI13n! force?
•
~ F. _
f>kN
l' rob!<.l-4I5I6
I'ml~
2-9/10
2.3
2- 11. If the tellsioll in the table is 400 N. detennine tbe
tn:Igllitudc and di rection of the resultant fon:e acting on
the pulley. This angle is the same angle 0 of lille AB on the
tailboard bind:..
29
V{CTOR AOOl110N 01' FO!tC€s
!-14. [ktennine the design angle 0 to" s 0 s 90") for
strul JI B so that the 4OI).Ib horizontal force has 3
component of 500lb directed from A to ..... ards C. Wh at is the
component of force acting along member AB? Take
<b - 4Q".
!-15. Delennine the design angle <b to" s <b s 90")
bet"'ecn StrulS AB and lie so thai the 4OI).Ib horizoncal
force has a tomponem of 600 lb which aclS up 10 the left. in
Ihe same direction as from B to,,·ardsA.Takc 8 - 30".
,
.100 Ib A
"
c
I' roo. !-Il
!'rob!>. !-14I15
*2- 12. The devke is used for surgical replacement of the
knee jomt, If the force 3"illg along the kg is 360 N.
de tennine its componen ls along the .f and y ' axes.
*2-16. Resolve t'l inca components along the" 8nd u axes
and (\ctermi ne the magnitudes of these tOnlponents.
oZ_IJ. The device is used for surgical replacement of the
knec joint. If the force acting alollg the leg is 360 N.
de termme it5 components along the.t· and axes.
02-17. Resolve f ! into components along Ihe" and u axes
and dete rmine the magllHudes of these componenls.
r
F, - ZSON
!'rob§.. 2- IUIJ
!'robs. 2- 16117
30
CH"'PfE~
2
FORCE VECTORS
'l'he truel: is to be lowed using IWO ropes. Delcnn;nc
the magnitudes of forces f " and FII acting on each rope in
order 10 dC"clop a resultant force of 950 N directed along
2- 1 ~.
lite positive:r axis. ScI 6 .. 50".
2-ZJ. If 0 - 30" and 1-2 - 6 kK (\ctermine the magnitutk
of the resultant fom: acting on the plate and its dircrtion
measured dock"isc from the positi"exaxis;
.·R
2- 19. The truck is \0 be lowed using two ropn. If the
resultant fOIT(: is 10 be 950 N. directed along the posit'I'c:r
axis. determine llie magniludcs of forces FA and FJI aCling
on each rope and the angle II of Fs so thallh c magnitude of
FII is a m;II;"'''"" FII acts at 20" from the ~' :uis as showo.
is directed along a
· 2-24, If the resultant force
line mea su red 75' clockwise from the positi"e x axis and
the magnitude of F2 is to be a minimum. determine the
magnitudes of F R and ••~ and the angk 0 s 90".
,
-
-
...
~
Prob. 2- 18I19
I'robs. 2- lJI24
· 2-20. If 4> _ 45°, Fl .. 5 ~N. and the rcsuhaot (orIX is
6 tN dira:lcd along the Jl'O\$ith'c )' axis.. determine lhe required
magnitude of Fz and ils direction 11.
' 1-21. If It> .. 3l)" and the !'e.,ullan! fon:c is 10 be 6 k.N
directed along !he positive y axis.. determine the magnilUOCs
of F , and Fz and the angle 0 if '"1 is required 10 be a minimum.
If <I> - 30", FI - 5 kN. and the resultant force is 10
be direcled along the positil'e y a~i!. delenninc the
magnitude of the resultant force if '"i is to be D minimum.
Also. what is '"1 and the angle 8"!
2- 22.
"robs.. 2-211/1 1/21
°z.-2S, Two forces f j and f t act on the Kreweye. If their
lin..-s of aenon ar..- at an angle 9 apart and the magnitude
of each force is FI - 1-i - F, determine the magnitude of
the resultant force FR and the angle bel,,'eCn . '/1 and .'\'
I'rob. 2-25
2.3
l-16. 'Ibe log is being towcd by two tractors , I and
n,
(}ctcrmine Ihe magniwdes of the \1'10 10"'ing forces t"" and
) "/1 if it is required that the resultant force ha"c a magnitude
1'" '" IOkN and be direCled along the.l axi'!. Sct 0 _ ISO.
2-Z7, 111e resultant .'" of tbe two foren acting on the log is
to be directed along the positive.l a.~is and rum: a magnitude
of I0 kN. determine the angle tl of the cable, aHaehcd 10 B such
Ihal lhe In.1gnitude of foTCt Fa in this (;lblc is a minimum ,
What is tl\o;> magnitude of the foTCt in each (;lblc for this
siluation'!
l-3Il, 'iliree chains aCI OII lhc bracke t slICh thatlhey create
a resultant force ha,ing a magnitude of 500 Ib, If two of thc
chains are subjected to knO"ll forces. as shown. determine
the angle Ii of the third chain measured d oc kwis<: from the
positive A' a~is. so that the magnitude of force .' in this chain
is a ",;",'mllm , All forces lie in the .t-)' plane. What is the
magnitude of .~! lIillr. First find the rcsulta nt of the 11'10
known forces. Force )" acts in this direnion .
,
I
'"
I'robs. 2-21>127
0l-l8. l'hc heam is to be hoisted using twO chain'!. Deter,
mine the magnitudes of foTCts t"" and .'~ acting 00 each ell-1in
in order to de,'clop a re:lultant fom: of 600 N directed along
the posilive)' a.~is. Seltl '" -15°.
02-29. 111(' beam is to be hoisted using t"'O ehains. lf the
resultanl force is to be 600 N directed along the positi,"C "
axis. determine the magnitudcsofforecs F" and .'"acling on
each chain and thc angle Bof . "/1 so thaI the magnitude of ) "/1
is a minimllm. FA aCtS M 3fr from the)' uis. as shown.
,•
31
V{CTOR AOOl110N 01' FO!tC€s
I' rob. 2- 30
""'"
2-J I, Three cableli pull on Ihe pipe such thatlhe}' create a
rcsul\3nt for~e havi ng a magnitude of 900 lb. If two of the
abies arc subjeetcd to known forces. assho"'n in the figure.
determine the angle /I of the third cable so thm Ihc
m.1gn itude of force F in this ~able is a m;lIjlll"'". All forces
lic in the x-y plane. Whm is the magnitude of F? "ml: Firsl
find the resultant of the two known forces.
,,
,
'9-----.
Probs. 2- 2lII!9
l'roh. 2- J I
38
•
."2-7.
CH"'PfE~
2
FOR CE VE CTORS
FUNDAMENTAL PROBLEMS
RC50I,'c each fOTCC acting on the post into ils.r and
y romponcnls.
,
I
."2- 111.
If the resultant force aCling on the bracket is \0 be
750 N directed along the
1', .. ..100 N
posi' h~
x axis. dcu:rmim: the
magnitude of F and ils direction O.
,
Fl-'
t'241. Dc\cmunc Ihe magnilud<: and direction of Ille
resultant force.
F'2-11. If Ihe magnitude of Ihe resultant force aCling on
lhe bmekcl is 10 1M: 80 lb direclcd along Ihe /J axis.
determine the magnitude of F and ils direction O.
15c>...J~"-N-'---'
Fl- I I
"2- 9. O"lCmlinc the magnitude of llie rcsultant forre
aCling on Ihe corbel and its direction II measured
counterclockwise from Ilte.r nis.
,
I
!-"2-12. Dclcnninc the magnitude of the rcsullam force
:md its dire<:lion 0 mcasur... d rounlcrclockwise from lhe
positive .f axi5.
Fl - 4001b
Fl - I!
2.4
•
AoolllON OF A SYSTEM OF CoPlANAR FO!tC€s
39
PROBLEMS
*2- 32. Determine llie magnitude of llie rcSUllan! force
acting on Ille pin and ils di rection measured dock " 'isc from
the positil·"' .r axis.
2-35. The oontac\ IlOint bel""Cen the femur and tibia
bonn or the leg is 3\ A. lf a ,·crlie.. l forr:e of 1751b isllpplicd
al this poin!. determine the romponcnls along the .r and y
axes. NOle Ihal tile y component rcprcscnl~ the noroml
force on Ihe load·bearing rcgioll of Ille bones. BOlh Ihe x
and y componenls of Ihi5 forcc cause s)'lIo\;al nuid 10 be
squeezed OUI ofille bearing space.
"'''
1"011.2-32
I' rob. 2-35
02-.B. If 1', _ 600N Dnd 4> - 30". deteml;nc the
magnitude of the resuhant force :lCling on the eyebolt and
its dircClion measu red clod\\isc from Ihe posith'c x axis.
l-J4. If the magnitude of Ihe resultant force actin g on
lite cycboll is 600 Nand ilS direction measured clockwise
from Ihe positive ,r 3.'(;\$ is 8 - W. determine Ihe magnitude o f .'\ and Ihe angle <1>.
0Z-J6. If <b - 30" and F! .. 3 kN.dclcmlinc: the magnitudc
of the resultant force OCIing on the plate and ilS direction 6
measured clockwise from the posilivcx axi5.
' 2-37. If thc magnitude for Ille rcsultant force actillg on
lite plale is required to be 6 kN and its d ireel;oll measured
clockwise from Iltc posi li,'c x axis is 6 - JO".de lermille lit e
magnit ude of F2 and il\$ dire<:tion 4>.
Z-38. If 4> - 30" and tile resultant fortt acting on Ihe
gusset p13tc is directed olollg Ille positi>'c x axis. determine
lite magnitudes of t"2 and Ihe resultant fortt.
~
Prohs. l-3J1)..1
l'rOIlS.2- J6f37/JIS
kN
40
CH"'PfE~
2
FOR CE VE CTORS
If d> - 30" and Fl - 250 lb. determi ne Ihe
magnitude of the resultant force acting on the bracket and
2-39. DClcmlinc the magnitude of FI and its direction (I
so Iha\ the resullanl forte is din:clcd \'crlicalir upward and
has a magnitude o(SOI) N.
ils diredion measured clockwise from the positi,'e of axis.
*2-40. Determine the magnitude and direction measured
coun terclockwise from the posit;,·c x a:cis of the re\$uhanl
force of the three forres acting on the ring A . Take
. 2-44. If the magnitude of the rC\$ultant force acting on
the bradel is 400 Ib directed along the positive x axis.
determine the magnitud~ of F t and its direction </>.
f '\ - SOONandlJ - 20· ,
2-43.
.2-45. If the resul tant force act ing on the bracke t is to be
directed along the positive.T axis and the magnitude of . ' 1 is
required to be a minimum . determine the magnitudC\$ ofche
rcsultam force and F L'
,
I'robs. 2- 39/40
'2-4 1. Determine the magnitude and direction /I of F8 so
thaI Ihe rC\$ullam force ;s dirc(lcd along the posilj"C y axis
lind has a magniHldc of ISO) N.
2- 12:. Iklcrminc the magnilUdc and anste measured
counterclockwise from lhe positil'c )' uis of the resultant
force acting on the bracket if F . - 600 Nand fJ - 20' ,
F, .. 260 Ib
2-46. The three concurrent forces acting on the sere'" e~'e
produce a resultant force 1'/1 - O. lf " 2 F Land t't is to
be 90" from . '2 as shown. detumine the required nmgnilude
of Fl expressed in terms of FLand the anll-Ie 8.
i
"
I',eobs. 2..... 11-'2
I'rob. 2-46
2.4
t·"
2_ 17. Dctcnnine the ma" utude of
and 1t5 direction 6
so that thc resultant force is dm~ctcd alon, the posiu,'~ .l
.lXI' and has a rnapitude of 1250 N.
0l-4ll.
Dt:1~rminc
the magrutude and dn«tlon measu red
f,om the posiu,'c x u's of the resullant
force acting on the ring al 0 if 1'" _ 750 Nand tJ '" 45".
AooInof.I Of,. Sm£M Of' C()IV.N.tJt FOIIICES
41
2-50. The thrcc forces are apphed to the brackct.
DelumlOc thc runge of values for thc rnagnitu<tc of force P
so thai the resultant of the three forces docs nOl c:c~ecd
2400N.
~ollntcr~lock ... ,se
,
l'rub. 2-511
2-51. If 1'1 - ISO N and ~ - 30". det~nnl~ the rnagmludc
of lhe resultant force acting on the brnctet and ilS direction
mcasUt«! dod: ..-I:SC from thc pailti'~!l a.ris.
' ~'.I.
Dcl~nnine
thc magnitude of Ih~ r~sultanl force
and liS dlr«1ion measured OOIIn terdockwise from th e
poslti'·c .r a:..1
,
°1_5l. If the m3'"'tude 0( thc resultant fOKe aclln, 01\
the brxkc l IS to be:.ISO N direcled alon, the posiu,'c I/.IX~
de tcnnmc the magnitude of " , and its directlon~.
02-53. If the resultant force 3("\ing on thc brKket IS
rcquir.::d to be: a minimum.detcrminc the magnitudes of F,
and 11K: resultant force. ScI ~ '" 30".
"
"'F. '" SO Il!
"rob. 2---4'.1
Probs. 2- 51/SU.!iJ
42
CH"'PfE~
2
FOR CE VE CTORS
l_~ 'Ibree forces act on tile brad;ct IXtcrminc th~
magnitude and dircClion 6 of t'l so that the resultant force is
directed along the p<l6iti\"c II axis and has a magnitude orso lb.
2-55. If f : '" ISO Ib and II .. 55~. determine the
magnitude and di reCTion measured clockwise from the
positive .r a~is of the resultant force of lhe three forces
acting on the bracke t.
' 2-57. Determine tbe magnitude of force f so tllat Ille
resultant [orce of llie three forces is as small as possible.
What is Ihe masnitudc of Ihis smallest resultant force'!
,
IHN
,
F, _ SOII>
--,
I'rob.2-57
'"
•
"
"
I' robs. 2-5-1155
2-58, Expre§5 each of Ihe Ihree forces aCling on Ihe
brackC I 11'1 Carlcsi an ,-celor form wilh .('spectto Ihe;t and }'
:ues. Determine Ihe R'l3gninlCie and direction 0 of 10-1so Ihal
Ihe re~ulmnl force is dircded along the posilh"e;t' axis and
h3S a magniludc of Fit - 600 N.
*2-56. 111c tllrc~' concurrent forces acting on the post
produce a resullam force t'R .. O. If F: .. F ,. and f l is \0
be 90" from .' z as shown,determine the rcquir... d magniwdc
of F} expressed in terms of F, and the angle 6.
J
,
I"~
II' '--,
",
l'rob. 2- 56
Prob. 2- 511
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