of 101
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FTUI
Force System I
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2/101
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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FTUI EM 2008- 5R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia
Operations by Scalars
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6/101
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Vector Addition
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FTUI EM 2008- 8R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia
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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FTUI EM 2008- 10R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia
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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FTUI
Vector Addition of Forces
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FTUI EM 2008- 14R a h m a t S a p t o n o - Department of Metallurgy and MaterialsFaculty of Engineering University of Indonesia
8/10/2019 P1 - 2014.pdf
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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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FTUIEM 2008
- 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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- 31R 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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- 32R 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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- 33R 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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- 35R 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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- 36R 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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- 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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FTUIEM 2008
- 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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- 45R 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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- 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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FTUIEM 2008
- 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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50/101
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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51/101
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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FTUIEM 2008
- 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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FTUIEM 2008
- 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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FTUIEM 2008
- 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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FTUIEM 2008
- 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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FTUIEM 2008
- 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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67/101
FTUIEM 2008
- 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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68/101
FTUIEM 2008
- 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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69/101
FTUI
Addition and Subtraction of
Cartesian Vectors
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FTUIEM 2008
- 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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- 71
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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72/101
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FTUI
Position Vectors
x, y, z coordinates
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FTUIEM 2008
- 74
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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FTUIEM 2008
- 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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FTUIEM 2008
- 76
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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- 77
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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- 79
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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- 80
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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FTUI EM 2008- 85
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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FTUI EM 2008- 86
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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FTUI EM 2008- 87
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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FTUI EM 2008- 88
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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FTUI EM 2008- 89
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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FTUI EM 2008- 93
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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FTUI EM 2008- 94
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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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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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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