ch.02 force vectors
DESCRIPTION
Engineering Mechanics - StaticsTRANSCRIPT
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2/13/2013
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02. Force Vectors
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.01 Force Vectors
Chapter Objectives
To show how to add forces and resolve them into components using the Parallelogram Law
To express force and position in Cartesian vector 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
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.02 Force Vectors
1. Scalars and Vectors
- Scalar: a scalar is any positive or negative physical quantity
that can be completely specified by its magnitude
Examples: length, mass, and time
- Vector: a vector is any physical quantity that requires both a
magnitude and a direction for its complete description
Examples: force, position, and moment
A vector is shown graphically by an arrow
the length of the arrow: the magnitude of the vector
the angle between the vector and a fixed axis: direction of its line of action
the head or tip of the arrow: the sense of direction of the vector
- Notation: , | |
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.03 Force Vectors
2. Vector Operations
- Multiplication and division of a vector by a scalar =
- Vector addition = +
Parallelogram law
Triangle rule
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.04 Force Vectors
2. Vector Operations
- Vector subtraction = = + ()
Parallelogram law
Triangle construction
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.05 Force Vectors
3. Vector Addition Forces
- Experimental evidence has shown that a force is a vector
quantity since it has a specified magnitude, direction, and
sense and it adds according to the parallelogram law
- Finding a resultant force
The two component forces 1 and 2 acting on the pin can be
added together to form the resultant force = 1 + 2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.06 Force Vectors
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3. Vector Addition Forces
- Finding the components of a force
To resolve a force into two components in order to study its
pulling or pushing effect in two specific directions
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.07 Force Vectors
3. Vector Addition Forces
- Addition of several forces
To resolve a force into two components in order to study its
pulling or pushing effect in two specific directions
= 1 + 2 + 3 = ( 1 + 2 12
) + 3
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.08 Force Vectors
3. Vector Addition Forces
- Example 2.1 The screw eye is subjected to two forces, 1 and
2. Determine the magnitude and direction of the resultant force
Solution
= 1002 +1502 2100 150 1150 = 212.6 150
=
212.6
1150 = 39.80, = 39.80 + 15.00 = 54.80
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.09 Force Vectors
3. Vector Addition Forces
- Example 2.2 Resolve the horizontal 600 force into components acting along the and axes and determine the magnitudes of these components
Solution
1200
=600
300 = 600
1200
300= 1039
300
=600
300 = 600
300
300= 600
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.10 Force Vectors
3. Vector Addition Forces
- Example 2.3 Determine the magnitude of the component
force and the magnitude of the resultant force if is directed along the positive axis
Solution
600=
200
450 = 200
600
450= 245
750
=200
450 = 200
750
450= 273
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.11 Force Vectors
3. Vector Addition Forces
- Example 2.4 It is required that the resultant force acting on
the eyebolt be directed along the positive axis and that
2 have a minimum magnitude. Determine this magnitude, the angle , and the corresponding resultant force
Solution
Magnitude of 2 is a minimum its line of action is perpendicular to the line of action
of , that is, when = 900
and = 800 600 = 400
2 = 800 600 = 693
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.12 Force Vectors
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Fundamental Problems
- F2.1: Determine the magnitude of the resultant force acting on the
screw eye and its direction measured clockwise from the axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.13 Force Vectors
Fundamental Problems
- F2.2: Two forces act on the hook. Determine the magnitude of
the resultant force
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.14 Force Vectors
Fundamental Problems
- F2.3: Determine the magnitude of the resultant force and its
direction measured counterclockwise from the positive axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.15 Force Vectors
Fundamental Problems
- F2.4: Resolve the 30 force into components along the and axes, and determine the magnitude of each of these
components
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.16 Force Vectors
Fundamental Problems
- F2.5: The force acts on the frame. Resolve this force into
components acting along members and , and determine the magnitude of each component
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.17 Force Vectors
Fundamental Problems
- F2.6: If force is to have a component along the axis of = 6, determine the magnitude of and the magnitude of its component along the axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.18 Force Vectors
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4. Addition of a System of Coplanar Forces
- Scalar Notation
= +
=
=
- Cartesian Vector Notation
= +
= +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.19 Force Vectors
=
=
4. Addition of a System of Coplanar Forces
- Coplanar Force Resultants
= 1 + 2 + 3
= 1 + 1 + (2 + 2 ) + (3 3 )
= 1 2 + 3 + (1 + 2 3)
+ =
+ =
= 2 +
2 , = 1
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.20 Force Vectors
4. Addition of a System of Coplanar Forces
- Example 2.5 Determine the and components of 1 and
2 acting on the boom. Express each force as a Cartesian vector
Solution
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.21 Force Vectors
4. Addition of a System of Coplanar Forces
Scalar notation
1 = 200300 = 100 = 100
1 = 200300 = 173 = 173
2 = (12/13) 260 = 240 = 240N
2 = (5/13) 260 = 100 = 100
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.22 Force Vectors
4. Addition of a System of Coplanar Forces
Cartesian vector notation
1 = 100 + 173
2 = 240 100
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.23 Force Vectors
4. Addition of a System of Coplanar Forces
- Example 2.6 The link is subjected to two forces 1 and 2. Determine the magnitude and direction of the resultant force
Solution 1
Scalar Notation
+ = : = 600300 400450 = 236.8
+ = : = 600300 + 400450 = 582.8
= (236.8)2+(582.8)2= 629
= 1(582.8/236.8) = 67.90 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.24 Force Vectors
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4. Addition of a System of Coplanar Forces
Solution 2
Cartesian vector notation
1 = 600300 + 600300
2 = 400450 + 400450
= 1 + 2
= (600300 400450) +(600300 + 400450)
= 236.8 + 582.8
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.25 Force Vectors
4. Addition of a System of Coplanar Forces
- Example 2.7 The end of the boom is subjected to three concurrent and coplanar forces. Determine the magnitude and
direction of the resultant force
Solution
+ = : = 400 + 250450 200
4
5= 383.2
+ = : = 250450 + 200
3
5= 296.8
= (383.2)2+(296.8)2= 485
= 1(296.8/383.2) = 37.80 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.26 Force Vectors
Fundamental Problems
- F2.7: Resolve each force acting on the post into its and components
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.27 Force Vectors
Fundamental Problems
- F2.8: Determine the magnitude and direction of the resultant force
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.28 Force Vectors
Fundamental Problems
- F2.9: Determine the magnitude of the resultant force acting
on the corbel and its direction measured counterclockwise from the axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.29 Force Vectors
Fundamental Problems
- F2.10: If the resultant force acting on the bracket is to be 750 directed along the positive axis, determine the magnitude of and its direction
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.30 Force Vectors
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Fundamental Problems
- F2.11: If the magnitude of the resultant force acting on the
bracket is to be 80 directed along the axis, determine the magnitude of and its direction
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.31 Force Vectors
Fundamental Problems
- F2.12: Determine the magnitude of the resultant force and its
direction measured counterclockwise from the positive axis
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.32 Force Vectors
5. Cartesian Vectors
- Right-Handed Coordinate System
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.33 Force Vectors
5. Cartesian Vectors
- Rectangular Components of a Vector
= + +
- Cartesian Unit Vectors
, ,
- Cartesian Vector Representation
= + +
- Magnitude of a Cartesian Vector
= 2 +
2 + 2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.34 Force Vectors
5. Cartesian Vectors
- Direction of a Cartesian Vector: , ,
=
, =
, =
Unit vector
=
+
+
= + +
Relation between the direction cosines
2 + 2 + 2 = 1
and
= = + + = + +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.35 Force Vectors
5. Cartesian Vectors
- The direction of can be specified using two angles, and
= + +
=
=
=
= + +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.36 Force Vectors
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6. Addition of Cartesian Vectors
= + +
= + +
= +
= ( +) + ( +) +( +)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.37 Force Vectors
6. Addition of Cartesian Vectors
- Example 2.8 Express the force as a Cartesian vector
Solution
2 + 2 + 2 = 1
2 + 2600 + 2450 = 1
= 1 0.52 0.7072
= 0.5
= 600
1200
must be in the + direction: = 600
= + +
= 200600 + 200600 + 200450
= 100.0 + 100.0 + 141.4
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.39 Force Vectors
6. Addition of Cartesian Vectors
- Example 2.9 Determine the magnitude and the coordinate
direction angles of the resultant
force acting on the ring
Solution
= 1 + 2
= 50 100 + 100+ 60 + 80
= 50 40 + 180
The magnitude of the sum vector
= 502 + 40 2 +1802
= 191.0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.39 Force Vectors
6. Addition of Cartesian Vectors
The coordinate direction angles
, , are determined from the components of the unit vector
acting in the direction of
=
=50
191
40
191 +
180
191
= 0.2617 0.2094 + 0.9422
So that
= 0.2617 = 74.80
= 0.2094 = 1020
= 0.9422 = 19.60
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.40 Force Vectors
6. Addition of Cartesian Vectors
- Example 2.10 Express the force as a Cartesian vector
Solution
The magnitudes of the components of
= 100600 = 86.6
= 100600 = 50
= 450 = 50 2/2 = 35.4
= 450 = 50 2/2 = 35.4
Realizing that < 0, we have
= 35.4 35.4 + 86.6
= 35.42 + 35.4 2 +86.62
= 100
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.41 Force Vectors
6. Addition of Cartesian Vectors
| | = 100, = 35.4, = 35.4, = 86.6
The coordinate direction angles of can be determined from
the components of the unit vector acting in the direction of
=
=
+
+
=35.4
100
35.4
100 +
86.6
100
= 0.354 0.354 + 0.866
So that
= 0.354 = 69.30
= 0.354 = 1110
= 0.866 = 30.00
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.42 Force Vectors
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6. Addition of Cartesian Vectors
- Example 2.11 Two forces act on the hook. Specify the
magnitude of and its coordinate
direction angles of that the resultant
force acts along the positive axis and has a magnitude of 800
Solution
1 = 11 + 11 + 11
= 300450 + 300600
+3001200
= 212.1 + 150 150
2 = 2 + 2 + 2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.54 Force Vectors
6. Addition of Cartesian Vectors
We require
= 1 + 2
where
1 = 212.1 + 150 150
2 = 2 + 2 + 2
= 800
800 = 212.1 + 2 + 150 + 2 + (150 + 2)
To satisfy this equation the , , components of must be
equal to the corresponding , , components of 1 + 2
0 = 212.1 + 2 2 = 212.1
800 = 150 + 2 2 = 650
0 = 150 + 2 2 = 150
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.44 Force Vectors
6. Addition of Cartesian Vectors
= 800
1 = 212.1 + 150 150
2 = 212.1 + 650 + 150
The magnitude of 2
2 = (212.1)2+6502 +1502= 700
The coordinate direction angles of 2
2 = 212.1/700 2 = 1080
2 = 650/700 2 = 21.80
2 = 150/700 2 = 77.60
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.45 Force Vectors
Fundamental Problems
- F2.13: Determine its coordinate direction angles of the force
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.46 Force Vectors
Fundamental Problems
- F2.14: Express the force as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.47 Force Vectors
Fundamental Problems
- F2.15: Express the force as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.48 Force Vectors
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Fundamental Problems
- F2.16: Express the force as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.49 Force Vectors
Fundamental Problems
- F2.17: Express the force as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.50 Force Vectors
Fundamental Problems
- F2.18: Determine the resultant force acting on the hook
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.51 Force Vectors
7. Position Vectors
- , , Coordinates
A right handed coordinate system
to reference the location of points
in space
- Position Vectors
= + +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.52 Force Vectors
7. Position Vectors
- The position vector may be directed from point to point in space
+ =
=
= + + + +
= + +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.53 Force Vectors
7. Position Vectors
- Example 2.12
An elastic rubber band is attached to points
and . Determine its length and its direction measured from toward
Solution
Establish a position vector from to
= (21) + (2 0) + (3 3)
= 3 + 2 + 6
These components of can also be determined directly by realizing that they
represent the direction and distance one
must travel along each axis in order to move
from to
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.54 Force Vectors
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7. Position Vectors
The length of the rubber band
= (3)2+22 +62= 7
Formulating a unit vector in the direction of
=
=
3
7 +
2
7 +
6
7
The components of this unit vector give the
coordinate direction angles
= 3/7 = 1150
= 2/7 = 73.40
= 6/7 = 31.00
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.55 Force Vectors
8. Force Vector Directed Along A Line
Quite often, the direction of a force is specified by two points
through which its line of action passes
Formulate as a Cartesian vector by realizing that it has the same direction and sense as the position vector directed from point to point on the cord
= =
=
+ + ( )
2 + 2 + ( )2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.56 Force Vectors
8. Force Vector Directed Along A Line
- Example 2.13
The man pulls on the cord with
a force of 70. Represent this force acting on the support as a Cartesian vector and
determine its direction
Solution
Force is shown in the figure
The direction of this vector, , is determined from the position
vector ( {24} {8} {12} )
= 12 8 24
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.57 Force Vectors
8. Force Vector Directed Along A Line
The magnitude of , which represents the length of cord
| | = 122+(8)2+(24)2= 28
Forming the unit vector that
defines the direction and sense
of both and
=
=
12
28
8
28
24
28
Since , has a magnitude of 70 and a direction specified by
= = 7012
28
8
28
24
28
= 30 20 60
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.58 Force Vectors
8. Force Vector Directed Along A Line
=
=
12
28
8
28
24
28
From the components of the unit
vector
= 12/28 = 64.60
= 8/28 = 1070
= 24/28 = 1490
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.59 Force Vectors
8. Force Vector Directed Along A Line
- Example 2.14 Express the force acts on the hook as a Cartesian
vector
Solution
The coordinates for points and
(2, 0, 2)
(2,3.464,3)
To go from to : {4} {3.464} {1} , thus
=
=4 +3.464 +
(4)2+3.4642+0.18572=0.7428 +0.6433 +0.1857
Force expressed as a Cartesian vector
= = 750(0.7428 + 0.6433 + 0.1857)
= 577 + 482 + 139
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.60 Force Vectors
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8. Force Vector Directed Along A Line
- Example 2.15
The cables exert forces = 100, = 120 on the wall hook at , determine the resultant force acting at as a Cartesian vector
Solution = +
= 4 4
= 42 + 42 = 5.66
= ( /)
= 70.7 70.7
= 4 + 2 4
= 42+22+(4)2=6
= ( /)
= 80 + 40 80
= 150.7 + 40 150.7 HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.61 Force Vectors
Fundamental Problems
- F2.19: Express tile position vector in Cartesian vector form, then determine its magnitude and coordinate direction angles
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.62 Force Vectors
Fundamental Problems
- F2.20: Determine the length of the rod and the position vector
directed from to . What is the angle ?
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.63 Force Vectors
Fundamental Problems
- F2.21: Express the force as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.64 Force Vectors
Fundamental Problems
- F2.22: Express the force as a Cartesian vector
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.65 Force Vectors
Fundamental Problems
- F2.23: Determine the resultant force at
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.66 Force Vectors
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Fundamental Problems
- F2.24: Determine the magnitude of the resultant force at
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.67 Force Vectors
9. Dot Product
- dot product = scalar product of vectors
=
- Laws of Operation
Commutative law: =
Multiplication by a scalar: = = ()
Distributive law: + = ( ) + ( )
- Cartesian Vector Formulation
= + +
= + +
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.68 Force Vectors
= + +
9. Dot Product
- Applications
The angle formed between two vectors or intersecting lines
= 1
, 00 1800
The components of a vector parallel and perpendicular to a line
= =
= 2 2
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.69 Force Vectors
9. Dot Product
- Example 2.16 Determine the magnitudes of the projection of
the force onto the and axes
Solution
The projection forces
()= 100450 = 70.7
()= 100150 = 96.6
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.70 Force Vectors
9. Dot Product
- Example 2.17 The frame is subjected to a horizontal force
= 300 . Determine the magnitude of the components of
this force parallel and
perpendicular to member
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.71 Force Vectors
9. Dot Product
Solution
=
=2 + 6 + 3
22 + 62 + 32
= 0.286 + 0.857 + 0.429
=
= (300 )(0.286 +0.857 +0.429 )
= 257.1
=
= (257.1) 0.286 +0.857 +0.429
= 73.5 + 220 + 110
=
= 73.5 + 80 110
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.72 Force Vectors
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9. Dot Product
- Example 2.18 The pipe is subjected to the force of = 80. Determine the angle
between and the pipe segment and the projection
of along this segment
Solution
Angle
= 2 2 + , = 3
= 3 + , = 10
=
=2 0 + 2 3 +(1)(1)
3 10=0.7379
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.73 Force Vectors
= 42.50
9. Dot Product
Components of
The component of along is shown in the figure
=
=2 2 +
(2)2+(2)2+12
= 2
3
2
3 +
1
3
= 80
= 803 +
10= 75.89 +25.30
=
= (75.89 +25.30 ) 2
3
2
3 +
1
3
= 59.0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.74 Force Vectors
Fundamental Problems
- F2.25: Determine the angle between the force and the line
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.75 Force Vectors
Fundamental Problems
- F2.26: Determine the angle between the force and the line
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.76 Force Vectors
Fundamental Problems
- F2.27: Determine the angle between the force and the line
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.77 Force Vectors
Fundamental Problems
- F2.28: Determine the component of projection of the force along the line
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.78 Force Vectors
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Fundamental Problems
- F2.29: Find the magnitude of the projected component of the force along the pipe
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.79 Force Vectors
Fundamental Problems
- F2.30: Determine the components of the force acting parallel and perpendicular to the axis of the pole
HCM City Univ. of Technology, Faculty of Mechanical Engineering Nguyen Tan Tien
Engineering Mechanics Statics 2.80 Force Vectors