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Chap. 2 Force Vectors
2
Chapter Outline
Scalars and VectorsVector OperationsVector Addition of ForcesAddition of a System of Coplanar ForcesCartesian VectorsAddition and Subtraction of Cartesian VectorsPosition VectorsForce Vector Directed along a LineDot Product
3
Scalar and Vectors
4
Vector Operations Multiplication and Division by a Scalar
Vector Addition
5
Vector OperationsVector Subtraction
* draw the vector from the head of B to the head of A
6
Vector Operations
Resolution of a Vector
1. RAB
2. RAB
7
Vector Operations
Vector Addition of Forces
p.29, Problem 2-17Resolve the force F2 into components acting along the u and v axes and determine the magnitudes of the components.
9
10
p.31, Problem 2-29The beam is to be hoisted using two chains. If the resultant force is to be 600 N directed along the positive y axis, determine the magnitudes of forces FA and FB acting on each chain and the angle of FB so that the magnitude of FB is a minimum. FA acts at 30
from the y axis as shown.
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12
Addition of a System of Coplanar Forces
Cartesian VectorF = Fx i + Fy j
Resultants
j )(F i )(F
j )FF F ( i )FF (F
F
RyRx
3y2y1y
3x2x1x
R
+=
+++++=++= 321 FFF
13
Addition of a System of Coplanar Forces
yRy
xxR
F FFF
=
=
Rx
Ry
FF1
Ry2
RxR
tan
FFF 2
=
+=
14
p.41, Problem 2-50The three forces are applied to the bracket. Determine the range of values for the magnitude of force P so that the resultant of the three forces does not exceed 2400 N.
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16
Cartesian Vectors
Right-Handed Coordinate SystemRectangular Components of a Vector
A=Ax+Ay+Az
Unit VectorCartesian Unit Vector i , j , k
uA = = AA
AA
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Cartesian Vectors
Cartesian Vector Representation
A = Ax i + Ay j + Az k
Magnitude
222zyx AAA ++
22'zAA +|A| = =
Axy
18
Cartesian Vectors
Direction
(unit vector)A
A = cos A
A=cos
AA=cos
cosine) (direction ,,
zyx
k A+ j A+ i A =k Acos j Acos i Acos =
uA = A1=cos+cos +cos
k cos j cos i cos=
kA
A+ jA
A+i
AA =
AA =
zyx
A
222
zyxA
++
++
u
19
Cartesian Vectors
Addition and Subtraction of Cartesian VectorsR = A+B = (Ax+Bx)i + (Ay+By)j
+ (Az+Bz)kR= A-B = (Ax - Bx)i + (Ay - By)j
+ (Az - Bz)k
Concurrent Force SystemskFjFiFF zyx ++==RF
20
p.55, Problem 2-84Determine the coordinate direction angles of F1 and FR .
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22
Position Vectors
x, y, z Coordinates ()
(6,-1,4) (0,2,0)
(4,2,-6)
23
Position Vectors
Position Vector
Relative Position Vector
O
r = xi + yj +zk
24
Force Vector Directed along a Line
)ABAB( F = uF = F
r-r =AB
F
AB
25
p.62, Ex.2-15,
Determine the resultant force acting at A.
( )( )( )
mACAC
mABAB
CBA
0.6)4,2,4(
657.5)4,0,4(
0,2,40,0,44,0,0
==
==
L
L
N ) 80- , 40 , 80 (=
(4,2,-4)6
120 =
ACAC F = F
N ) 70.71- , 0 , 70.71 (=
(4,0,-4)5.657100=
ABAB
F =F
ACAC
ABAB
N 217 =(-150.71)40150.71=R
N 150.71)- , 40 , (150.71 = F + F = R 222
ACAB
++
26
p.65, Problem 2-93The chandelier is supported by three chains which are concurrent at point O. If the force in each chain has a magnitude of 300 N, express each force as a Cartesian vector and determine the magnitude and coordinate direction angles of the resultant force.
27
28
Dot Product
Cartesian Vector Formulation
0
01
=
==
ik
ijii
A
B = AB cos , oo 1800
ex: Work = F
r
0
1
0
=
=
=
jk
jj
ji
1
00
=
==
kk
kjki
)()( kBjBiBkAjAiABA ZyxZyx ++++=
zzyyxx BABABA ++=
29
Dot Product
Applications
ABBA
0=BA BA
1. angle = cos-1( )
when = 90o , cos = 0 , 2. components of a vector parallel and perpendicular to a line
sin
)(coscos
||
||
||
AAAAA
uuAuAAuAAA
=
=
==
==
30
p.72, Ex. 2-17, Determine the parallel and perpendicular components of the force to member AB.
7.154)2.110(80)5.73(
257110.222073.5F
N ) 110.2- , 80 , 73.5- ( F-FF
N ) 110.2 , 220 , 73.5 (
(2,6,3)71
76300
)(
)3,6,2(71
362)3,6,2(
ABAB u
) 3 , 6 , 2 B( ) 0 , 0 , 0 A(
222
222AB
AB
222AB
NF
N
uuFF ABABAB
=++=
=++=
===
=
=
=++
==
31
p.76, Problem 2-116Two forces act on the hook. Determine the angle between them.Also, what are the projections of F1 and F2 along the y axis?
32
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p.78, Problem 2-133Two cables exert forces on the pipe. Determine the magnitude of the projected component of F1 along the line of action of F2 .
34
Chap. 2 Force Vectors Chapter OutlineScalar and VectorsVector Operations Vector OperationsVector OperationsVector Operations 8 9 10 11Addition of a System of Coplanar Forces Addition of a System of Coplanar Forces 14 15Cartesian VectorsCartesian VectorsCartesian VectorsCartesian Vectors 20 21Position VectorsPosition VectorsForce Vector Directed along a Line 25 26 27Dot ProductDot Product 30 31 32 33 34 35