STATICS-Chap2 Force Vectors 2.42.6

8/10/2019 STATICS-Chap2 Force Vectors 2.42.6

1/14

EGB 2042/ Ch 2 1

CHAPTER 2: FORCE VECTORS

Ainul Akmar bt. Mokhtar

Mechanical EngineeringJuly Semester 2004


2/14

EGB 2042/ Ch 2 2

CHAPTER 2: FORCE VECTOR

2.1 Scalars and Vectors

2.2 Vector Operations

2.3 Vector Addition of Forces2.4 Addition of a System of Coplanar Forces

2.5 Cartesian Vectors

2.6 Addition and Subtraction of Cartesian Vectors

2.7 Position Vectors

2.8 Force Vector Directed Along a Line

2.9 Dot Product


3/14

EGB 2042/ Ch 2 3

Chapter Objectives

To show how to add forces and resolve them

into components using the parallelogram law.

To express force and position in Cartesianvector form and explain how to determine the

vectors magnitude and direction.

To introduce the dot product in order to

determine the angle between two vectors or

the projection of one vector onto another.


4/14

EGB 2042/ Ch 2 4

2.4 Addition of a System of

Coplanar Forces

2 ways to obtain the resultant force of more than

2 forces:

Parallelogram law (discussed in Sec 2.3)tedious Resolve each force into its rectangular components FXand

FY

For analytical work, we must establish a notation

to represent the directional sense of the

rectangular components Scalar notation

Cartesian vector notation


5/14

EGB 2042/ Ch 2 5


Coplanar Forces (cont)

Scalar Notation

Magnitude and directional sense of the rectangular

components of a force can be expressed in terms ofalgebraic scalars

To be used ONLY for computational purposes, not for

graphical representationsy

x

Fy

Fx

F


6/14

EGB 2042/ Ch 2 6



Cartesian Vector Notation

Cartesian unit vectors

idesignate thedirections ofxaxis

jdesignate thedirections of yaxis

Fx=Fxi

Fy=Fyj

F

i

j

y

x

jiF yx FF

where Fx=Fcos Fy=Fsin


7/14

EGB 2042/ Ch 2 7



Coplanar Force Resultants

Using Scalar Notation

(+) FRx= F1x- F2x+ F3x

(+ ) FRy= F1y+ F2y- F3y

Magnitude of FR=

Direction angle, =

In general,

22

RyRx FF

x

y

R

R1

F

Ftan

FRx= Fx FRy= Fy

F2F1

F3

y

x


8/14

EGB 2042/ Ch 2 8



Coplanar Force Resultants

Using Cartesian Vector Notation

F1= F

1xi + F

1yj

F2= - F2xi + F2yj

F3= F3xi - F3yj

FR = F1+ F2+ F3

= F1xi + F1yj -F2xi + F2yj +F3xi F3yj

`` =(F1xF2x+ F3y) i + (F1y+F2x- F3y)j= (FRx )i +(FRy)j

22

YXR FFF

Rx

Ry

F

F1

tan

F2F1

F3

y


9/14

EGB 2042/ Ch 2 9

2.5 Cartesian Vectors

Used to solve 3D problems

Will present a general method to

represent a vector in Cartesian form

Right-handed Coordinate System


10/14

EGB 2042/ Ch 2 10

2.5 Cartesian Vectors (cont)

Rectangular Components of a Vector

A= Ax+ Ay+ Az

Unit Vector

Magnitude of 1

Dimensionless

A=AuAWhereA defines the magnitude of A

uAdefines the direction and sense of A

Cartesian Unit Vectors

idesignate the directions ofxaxis

jdesignate the directions of yaxis

kdesignate the directions of zaxis

A

Au A


11/14

EGB 2042/ Ch 2 11


Cartesian Vector Representation

A= Axi+ Ayj+ Azk

Magnitude of a Cartesian Vector

Direction of a Cartesian Vector

A

AxcosA

A ycos

A

Az

cos

2

z

2

y

2

x AAAA

cos 2+ cos 2 + cos 2 = 1


12/14

EGB 2042/ Ch 2 12


A

AxcosA

A ycos

A

Az

cos

cos 2+ cos 2 + cos 2 = 1


13/14

EGB 2042/ Ch 2 13


A

Au A

kjikjiuAA

AA

AA

AAAA zyxzyx

A

kjiu coscoscosA

kAz

jiA

kjiuA A

yx AA

cosAcosAcosAA

AA

uA1

Thus,

A=Aua


14/14

EGB 2042/ Ch 2 14

2.6 Addition and Subtraction of

Cartesian Vectors

To find the resultant of a concurrent force system, express eachforce as a Cartesian vector and add the i, j ,k components of allthe force system

FR= F =Fxi +Fyj+ FzkLet

A = Ax+ Ay+ Az

B= Bx+ By+ BzThus

R= A+ B= (Ax + Bx)i + (Ay + By)j + (Az + Bz)k

R= A- B

= (Ax - Bx)i + (Ay - By)j + (Az - Bz)k

STATICS-Chap2 Force Vectors 2.42.6

Documents

Transcript of STATICS-Chap2 Force Vectors 2.42.6