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.

11

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.

15

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

17

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 .

21

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

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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

33

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 .

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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