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Force System I

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FTUI

Force Vectors

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FTUI EM 2008- 3R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

Problem 1

Determine the magnitude of the resultant force of F 1, F 2and its direction measured counter clockwise from the positive axis

FR = 867 Nf = 108 o

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FTUI EM 2008- 4R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

Graphical Representation ofa Vector A

Line of Action

Head

O

P20 o

Tail

1

A

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Operations by Scalars

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Vector Addition

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Addition of Collinear Vector

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FTUI EM 2008- 9R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

IMPORTANT POINTS

A scalar is a positive or negative number. A vector is a quantity that has magnitude ,

direction , and sense .

Multiplication or division of a vector by ascalar will change the magnitude of the vector.The sense of the vector will change if the scalaris negative .

As a special case, if the vectors are collinear ,the resultant is formed by an algebraic or scalar addition.

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Vector Subtraction

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FTUI EM 2008- 12R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

Resolution of a Vector

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Vector Addition of Forces

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FTUI EM 2008- 15R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

Problem 1

Determine the magnitude of the resultant force of F 1, F 2and its direction measured counter clockwise from the positive axis

FR = 867 N

f = 108 o

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FTUI EM 2008- 16R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

R F

105

75cos221

2

2

2

1 F F F F F R

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FTUI EM 2008- 17R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

R F

75

75cos221

2

2

2

1 F F F F F R

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FTUI EM 2008- 18R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

R F

75cos221

2

2

2

1 F F F F F R

75

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FTUI EM 2008- 19R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia

N8679751533

259.09600001000000

75cos960000640000360000

75cos8006002800600 22

R

R

R

R

R

F F

F

F

F

75cos2 212

2

2

1 F F F F F R

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FTUIEM 2008

- 20R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

...arcsin

...sin

75sin867

800sin

sinsin75sin12

F F F R

sinsin75sin12 F F F R

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FTUIEM 2008

- 21R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Problem 2

Determine the magnitude of the resultant force of F 1, F 2, F 3and its direction measured counter clockwise from the positive axis

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FTUIEM 2008

- 22R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

12 R F

12 3 R F

93cos231

2

3

2

12 F F F F F R R R

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FTUIEM 2008

- 23R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

N998

51.997995026

93cos4508672450867 22

R

R

R

F F

F

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FTUIEM 2008

- 24R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

sinsin93sin112 F F F R R

93sin998

867sin

sinsin93sin

312

F F F R R

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FTUI

Procedure of Analysis

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FTUIEM 2008

- 26R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Parallelogram Law

Make a sketch showing the vector additionusing the parallelogram law.

Two component forces add according tothe parallelogram law, yielding a resultantforce that forms the diagonal of theparallelogram.

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- 27R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

If a force is to be resolved into

components along two axes directed fromthe tail of the force, then start at the headof force and construct lines parallel to theaxes, thereby forming the parallelogram.

The sides of the parallelogram representthe components. Label all the known and unknown force

magnitudes and the angles on sketch andidentify the two unknowns.

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FTUIEM 2008

- 28R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Trigonometry

Redraw a half portion of the parallelogramto illustrate the triangular head-to-tailaddition of the components.

The magnitude of the resultant force canbe determined from the law of cosines,and its direction is determined from thelaw of sine.

The magnitude of two force componentsare determined from the law of sine.

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FTUIEM 2008

- 29R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

c

C

b

B

a

Asinsinsin

Sine Law

c AB B AC cos222

Cosine Law

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FTUIEM 2008

- 30R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

The screw eye in Figure is

subjected to two forces, F 1 andF2. Determine the magnitudeand direction of the resultantforce.

Example 1

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Faculty of Engineering University of Indonesia

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Faculty of Engineering University of Indonesia

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Faculty of Engineering University of Indonesia

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Faculty of Engineering University of Indonesia

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Faculty of Engineering University of Indonesia

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- 37R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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- 38R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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- 39R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

The force F acting on the frame shown inFigure has a magnitude of 500 N and is to

be resolved into two components actingalong members AB and AC. Determine

the angle , measured below thehorizontal, so that the component F AC is

directed from A toward C and has amagnitude of 400 N.

Example 3

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- 40R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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- 43R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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FTUIEM 2008

- 44R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

The ring shown in Figure issubjected to two forces F 1 and

F2. If it is required that theresultant force have a magnitudeof 1 kN and be directed verticallydownward, determine:

(a) the magnitudes of F 1 and F 2

provided = 30o, and(b) the magnitudes of F 1 and F 2 if F2

is to be a minimum.

Example 4

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Faculty of Engineering University of Indonesia

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- 46R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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- 47R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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- 48R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

ForcesSystem

Co-PlanarVectors

CartesianVectors

2D Problem

3D Problem

ScalarNotation

CartesianVector

Notation

AlgebraicScalar

CartesianUnit

Vector

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FTUI

Addition of a System of

Coplanar Forces

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FTUIEM 2008

- 50R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Problem 2

Determine the magnitude of the resultant force of F 1, F 2, F 3and its direction measured counter clockwise from the positive axis

F1x

F 1y

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FTUIEM 2008

- 51R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

j F i F F

j F i F F

j F i F F

y x

y x

y x

333

222

111

y x

y x

y x

F F F F F F

F F F

333

222

111

y y y y

x x x x

F F F F

F F F F

321

321

j F F F F i F F F F

y y y y

x x x x

321

321

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FTUIEM 2008

- 52R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Scalar Notation

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- 53R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Cartesian Vector Notation

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- 54R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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- 55R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Coplanar Force Resultants

y Ry

x Rx

F F F F

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FTUIEM 2008

- 56R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

22 Ry Rx R F F F

Rx

Ry

F

F 1tan

R F

Rx F

Ry F

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- 58R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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FTUI

Cartesian Vectors

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FTUIEM 2008

- 60R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Three forces act on the hook. If the resultant force F R has amagnitude and direction as shown, determine the magnitude

and the coordinate direction angles of F 3.

Problem 3

F3 = 166 N

= o= 63.7 o

g = 27.5 o

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zC i

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FTUIEM 2008

- 62R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

CartesianVectorsRight-handed

CoordinateSystem

RectangularComponents

of a Vector

x

y

x

y

z

A

Ax

Ay

Az UnitVector

A AuA

CartesianUnit

Vectors

Unit vector have a magnitudeof 1 and same direction as A.

0 Au

j

k

Cartesian VectorRepresentation

k jiA z y x A A A

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FTUIEM 2008

- 63R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Rectangular Components of a Vector

Representation

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FTUIEM 2008

- 64R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

p

Magnitude

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- 65R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

g

Direction

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- 66R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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- 67R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Direction

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- 68R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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FTUI

Addition and Subtraction of

Cartesian Vectors

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- 70R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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FTUI

Position Vectors

x, y, z coordinates

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Position Vector

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- 75

R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Position Vector

k jir z y x

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

BA rrr

k jik jir-rr AB A A A B B B z y x z y x

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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FTUI

Force Vector

Directed Along a Line

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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FTUI

Dot Product

P bl 4

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FTUI EM 2008- 83

R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Problem 4

Determine the projection of the force F along the pole

Proj. F= 0.667 kN

D P d

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FTUI EM 2008- 84

R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Dot Product

z z y y x x B A B A B A BA

cos ABBA

L f O i

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Laws of Operating

A li i

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Applications

The angle formedbetween two vectorsor intersecting lines.

The component of avector parallel and

perpendicular to aline.

1800

cos 1

AB

BA

uA.uuAA.u

cos

cos

A

A A

22 A A A

Th A l

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

The Angle

h

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

The Components

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

The scalar projection of A along a line is determinedfrom dot product of A and the unit vector u which

defines the direction of the line.

E l 1

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FTUI EM 2008- 90

R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

Example 1

The frame shown in Figure is subjected to a horizontal force F ={300j} N. Determine the magnitude of the components of this

force parallel and perpendicular to member AB.

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

E l 2

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R a h m a t S a p t o n o - Department of Metallurgy and Materials

Faculty of Engineering University of Indonesia

The pipe in Figure is subjected to the force of F = 80 lb.Determine the angle between F and the pipe segment BA, and the

magnitudes of the components of F, which are parallel andperpendicular to BA.

Example 2

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