Chapter 2 Force

1
2 Force Systems

• Force, Moment, Couple and Resultants

2
Objectives Students must be able to 1

• Course Objective
• Describe the characteristics and properties of
  forces and moments, analyse the force system,
  obtain the resultant and equivalent force
  systems.

• Chapter Objectives
• Use mathematical formulae to manipulate physical
  quantities
• Obtain position vectors with appropriate
  representation.
• Use and manipulate force vectors
• Use and manipulate moment vectors
• Analyse the force system resultants
• Describe and obtain equivalent systems

3
Force Definition

• Force is a vector quantity (why?)

• Force is the action of one body on another.
  Statics

• Force is an action that tends to cause
  acceleration of an object. Dynamics
• The SI unit of force magnitude is the newton (N).
  
• One Newton is equivalent to one kilogram-meter
  per second squared (kgm/s2 or kgm s 2)

Examples of mechanical force include
the thrust of a rocket engine, the impetus that
causes a car to speed up when you step on the
accelerator, and the pull of gravity on your body.

• Force can result from the action of
  electric fields, magnetic fields, and various
  other phenomena.

3
4
FORCE SYSTEMS
Force is a vector
Line of action is a straight line colinear with
the force
Force System
concurrent if the lines of action intersect at a
point
parallel if the lines of action are parallel
y
coplanar if the lines of action lie on the same
plane
x
5
Writing Convention
Hand Print
Scalar
Vector
Unit Vector
Magnitude of Vector
same symbol
In this course, you have to write in this
convention.
Recommended Style
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Vector (2D3D) Basic Concept
3-D Force Systems
2-D Force Systems
Moment Couple Resultants
Moment Couple Resultants
8
Free Vectors associated with Magnitude and
Direction
Representation
parallelogram
triangle

9
Operation Addition 5
Vector
Commutative
10
Operation Addition 6
Vector
Associative
11
Operation Scalar Multiplication 2
wrt with respect to
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Component Resolution of a Vector
Vector
A vector may be resolved into two components.
14
Basic relations of Triangle (C/6, law of cosine,
sine)
Law of cosine
a
Law of sine
c
b
15
2
Hint
a
1
b
b
q
c
b
Given V, ? and ?, find
Law of cosine
Law of sine
16
Vector Component and Projection
b
vector components of (along axis a
and b)
a
projections of
(onto axis a and b)
b
special case projection vectors are
orthogonal to each other
a
orthogonal projections vector components
17
Rectangular Components
vector component vector projection

• Most commonly used

y
q
x
18
Fx? Fy?
y
F
p-b
x
minus (bgt90)
b
x
?
y
b-q
y
x
b
?
?
y
x
19
EXAMPLE 2-1
Given the magnitude of the tension in the cable,
T 9 kN, express T in terms of unit vector i and
j
3 S.F.
Correct?
ANS
kN
20
We are using robot arm to put the cylindrical
part into a hole. Determine the components of the
force which the cylindrical part exerts on the
robot along axes
(a) parallel and perpendicular to arm AB
(b) parallel and perpendicular to arm BC
par
per
Defining direction
per
par
arm AB
ANS
arm BC
ANS
21
2/2 Combine the two forces P and T, which act on
the fixed structure at B, into a single
equivalent force R
P800 N (8cm)
Graphics
Geometric
Vector Component (Algebraic)
Correct?
Point of application is B
22
Example Hibbeler Ex 2-1 1
Determine the magnitude and direction of the
resultant force.
Two forces is not acting at the same point.
Geometric
23
Vector Component (Algebraic)
Geometric
Good? (get full score?)
- more explanation
- mark answer
- 5S.F. Then 3S.F.
24
Good Answer Sheet
Geometric
O
a
25
Point of Application
26
Example Hibbeler Ex 2-6 1
Vector
27
Example Hibbeler Ex 2-6 2
Vector
28

• Reference axis (very very important)
• Many problems do not come with ref. axis.
• Assignment based on convenience/experience

Originally pass through O

• Vector summation (addition)
• Three ways to be mastered

1. Graphically
2. Geometrically
3. Vector component (algebraically)
The calculations do not reveal the point of
application of the resultant force.
In case where forces do not apply at the same
point of application, you have to find it too!
29
Recommended Problem

• 2/9, H2-17, 2/12, 2/26, H2-28

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Three Dimensional Coordinate System
y
Real-life Coordinate System is 3D.
Introduce rule for defining the 3rd axis -
right-hand rule x-y-z - for consistency
in math calculation (cross vector)
z
x
How does 2D differs from 3D?
y
2D
z
x
32
Rectangular Components (3D)
projection
component
z
y
x
(maybe /-)
- cos2(?x)cos2(?y)cos2(?z) 1
- If you known the magnitude and all directional
cosines, you can write force in the form of
directional cosine Method
33
Example Hibbeler Ex 2-8
Find Cartesian components of F
z
x
y
34
Given the cable tension T 2 kN. Write the
vector expression of
1) directional cosine method
Real directional cosine
directionl cosine -0.92
B
A
35
B
A
36
Directional Cosines by Graphics
cos2(?x)cos2(?y)cos2(?z) 1
37
- Usually, the direction of force is not given
using the directional cosines. Need some
calculation. - Two examples (a) Two points on the
line of action of force is given (F also given).
z
B (x2, y2, z2)
Position vector
Two-Point Method
A (x1, y1, z1)
y
x
38
z
0.5
y
2) 2-point construction
0.4
B
A
0.3
1.2
x
kN
Ans
39
Write vector expression of . Also determine
angle ?x, ?y, ?z, of T with respect to positive
x, y and z axes
Consider T as force of tension acting on the bar
ANS
40
Example Hibbeler Ex 2-9 1
Vector
Determine the magnitude and the coordinate
direction angles of the resultant force acting on
the ring
41
Example Hibbeler Ex 2-9 2
Vector
42
Example Hibbeler Ex 2-11 1
Vector
Specify the coordinate direction angles of F2 so
that the resultant FR acts along the positive y
axis and has a magnitude of 800 N.
43
Example Hibbeler Ex 2-11 2
Vector
44
Example Hibbeler Ex 2-11 3
Vector
45
Example Hibbeler Ex 2-15 1
Force
The roof is supported by cables as shown. If the
cables exert forces FAB 100 N and FAC 120 N
on the wall hook at A as shown, determine the
magnitude of the resultant force acting at A.
46
Example Hibbeler Ex 2-15 2
Force
47
Example Hibbeler Ex 2-15 3
Force
48
Example Hibbeler Ex 2-15 4
Force
49
(b) Two Angles orienting the line of action of
force are given (?, ?)
Othorgonal projection Method
Resolve into two components at a time
z
y
Fz F sin(?) Fxy F cos(?)
?
?
Fx Fxy cos(?) F cos(?) cos(?) Fy Fxy
sin(?) F cos(?) sin(?)
x
50
Ans
51
x
TZ
Ans
52
2/110 A force F is applied to the surface of
the sphere as shown. The 2 angles (zeta, phi)
locate Point P, and point M is the midpoint of
ON. Express F in vector form, using the given
x-,y- z-coordinates.
53
Recommended Problems

• 3D Rectangular Component
• 2/99 2/100 2/107 2/110

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Operation Products
Vector

1. Dot Products
2. Cross Products
3. Mixed Triple Products

56
scalar product
?
(unit vector)
( three orthogonal vector )
57
Application of Dot Operation

• Angle between two vectors

• Componentizing Vector

line
which direction?
58
Ans
59
x
TZ
which direction??
Ans
60
Cross Product of Vectors
right-hand rule (A then B)
line which are perpendicular with both vectors
61
Operation Cross Product
Laws of Operations

• Commutative Law is not valid

• Associative wrt scalar multiplication

• Distributive wrt vector addition

62
x-y-z complies with right-hand rule
y

z
x
63
How to calculate cross product
This term can be written in a determinant form
64
Cross Product
-
-
-



65
Why cross product?

• Mathematical Representation of Moments, Torque

• Perpendicular Direction
• Area Calculation

z
C
A
y
Area ?
B
O
x
66
Mixed Triple Product
67
?
Why mixed triple product?

• Mathematical Representation of Moments along
  the axis.

O

• Volume Calculation

Volume must always
Area Height
Volume ?
68
Operation Product Summary
Vector
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Homepage URLs

• Statics official HP http//www.lecturer.
  eng.chula.ac.th/fmekmn/ (User
  Prince Password Caspian)

• Session 1 HP

http//pioneer.netserv.chula.ac.th/lsawat/course/
statics/
http//blackboard.it.chula.ac.th/ (after the
end of registration period)
73
Vector Basic Concept
3-D Force Systems
2-D Force Systems
Moment Couple Resultants
Moment Couple Resultants
74
Force Definition

• Force is a vector quantity (why?)

• Force is the action of one body on another.
  Statics

• Force is an action that tends to cause
  acceleration of an object. Dynamics
• The SI unit of force magnitude is the newton (N).
  
• One newton is equivalent to one
  kilogram-meter per second squared (kgm/s2 or
  kgm s 2)

Examples of mechanical force include the
thrust of a rocket engine, the impetus that
causes a car to speed up when you step on the
accelerator, and the pull of gravity on your body.

• Force can result from the action of electric
  fields, magnetic fields, and various other
  phenomena.

75
Force Representation
Force

• Use different colours in diagrams
• Body outline ? blue
• Load ? red
• Miscellaneous ? black
• (dimension, angle, etc.)

• Vector quantity
• Magnitude
• Direction
• Point of application

10 N
76
Type of Forces
Applied force
External force
Reactive force
Force
Stress
Internal force
Strain
Concentrated
Contact force
Force
Force
Distributed
Body force
77
Cables Springs
Force
78
2/2 Combine the two forces P and T, which act on
the fixed structure at B, into a single
equivalent force R
P800 N (8cm)
Graphical
Geometric
Algebraic
Correct?
Point of application is B
79
How to add sliding vectors (forces)?
Principle of Transmissibility
A
Point of application
Not OK. !
Still OK.
Point of Application is wrong
A
A
80
Special case Addition of Parallel Sliding Force
F
F
F
F
F
F1
F
F2
F2
F2
F2
F1
F1
R2
F1
R1
line of action
R2
R
R1
R2
R
R1
R
This graphical method can be used to find Line of
action
The better and efficient way will be discussed
later, when we learn the concept of moment,
couple, and resultant force
81
Move all forces to that concurrent point
Point of application, But no physical meaning
Ans
Application Point
82
How to add sliding vectors (forces)?
There is better way to find the point of
application (or line of action), but you have to
learn the concept of moment and couples first.
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84
Moment
In addition to the tendency to move a body, force
may also tend to rotate a body about an axis
(magnitude) summation
From experience (experiment) magnitude depends
only on F and d
moment axis
Direction
Moment is a vector
85
Moment Definition

• Moment is a vector quantity.
• Magnitude
• Direction
• Axis of Rotation

• The unit of moment is Nm
• The moment-arm d (perpendicular distance)
• The right-hand rule
• determined by vector cross product
• Sign convention 2D k or CCW is positive.
• Moment of a force or torque

86
Mathematical Definition (3D)
d
A
Moment about point A
-Magnitude
a
-Direction
right-hand rule
X
-Point of application point A
d
(Unit newton-meters, N-m)
2D
- 2D, need sign convention and be consistent
e.g. for counter- clockwise and for clockwise
MFd
d
87
can be used with more than 2 components
Varignons Theorem (Principle of Moment)
Same?
The moment of a force about any point is equal to
the sum of the moments of the components of the
force about that point
sum of moment (of each force) moment of sum
(of all force)
Useful with rectangular components
d2
Mo -Fxd2Fyd1
y
O
d1

x
88
Principle of Transmissibility Moment
Principle of Transmissibility is based on the
fact that moving force along the line of action
causes no effect in changing moment
position vector from A to any point on line
of action of the force.
O
convenient
a
X
r
A
Y
d
Z
- direction same
- magnitude
M Fr sin a Fd
Sliding force has the same moment
O
89
Sample 2/5 Calculate the magnitude of the moment
about the base point O of 600N force in five
different ways.
2m
A
400
2m
600N
4m
A
d
400
y
O
x
600N
Solution II 3D Vector Approach
4m
Solution I 2D Scalar Approach
O
CW or CCW?
CW
CW
Correct?
90
F1
B
F1
2m
F2
A
400
F2
600N
2m
A
400
4m
d1
600N
4m
F1
O
C
O
d2
F2
Solution III Varignons theorem
Solution V Transmissibility
Solution IV Transmissibility
91
EXAMPLE 2.8
In raising the flagpole, the tension T in the
cable must supply a Moment about O of 72 kN-m.
Determine T.
15 m
ANS
92
Example Hibbeler Ex 4-7 1
Moment
Determine the moment of the force about point O.
Correct?
93
Example Hibbeler Ex 4-7 2
Moment
3D Vector Approach
Scalar Approach (Varignons theorem)
94
Couple
- Couple is a summed moment produced by two force
of equal magnitude but opposite in direction.
M F(ad) Fa Fd
magnitude does not depend on distance a (point
O), i.e. any point on the body has the same
magnitude.
Effect of Pure Rotation
- tendency to rotate the whole object.
- no effect on moving object as translation.
2D representations (Couples)
C
C
C
couple is a free vector
95
Moment Couple Definition 2
Moment
Moment of a Couple
B
A
O

• A couple moment is a free vector
• It can act at any point since M depends only upon
  the position vector r directed between the forces.

96
Force-couple systems
- Line of action of a force on a body may be
changed if a couple is added to compensated for
the change in the tendency to rotate of that body.
No changes in net external effect
B
B
?
?
d
A
Force-couple system
The direction and magnitude of Force can not be
changed, only line of action (i.e. only change
to other pararell line)
Procedure may be reversed to combine a force with
a couple
97
B
F
from new location (B) to old location (A)
C
A
B
A
F
F
No Moment Principle of Transmissibility
A
Principle of Transmissibility is based on the
fact that moving force along the line of action
causes no effect in changing moment
B
98
Why using equivalent system?
B
B
?
A
Force-couple system
All force systems are equal.
real (physical) system
In the viewpoint of Mechanics, Result of force to
these systems are equal
?
?
equivalent system
equivalent system
99
Understanding Force-Couple system
Moment about point B of force F tendency
of force F to rotate the object at point B
? couple occurs when moving Force F from A to
B
(? couple occurs when moving Force F parallel
to its line of
action to the point B)
Equivalent System
B
B
?
D
D
A
100
Be careful of the direction of moment
70m
P
P
Ans
101
2/11 Replace the force F by an equivalent
force-couple system at point O.
0.25 m
0.1m
50 kN
M
Couple occurred when moving F to O Moment of F
about O
CCW
Correct?
Ans
102
Engine number 3 fails. Determine the
force-couple system on the body about point o.
Moving all 3 forces to point O
(direction left)
sum of moments?

ANS
(CW)
Sum of couples
Got the meaning?
103
Example Hibbeler Ex 4-14 1
Resultant
Replace the current system by an equivalent
resultant force and couple moment acting A.
104
Example Hibbeler Ex 4-14 2
Resultant
105
Ans
106
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107
2/6 Simplest Resultant

• Resultant of many forces-couple is the simplest
  force-couple combination which can replace the
  original forces/couples without changing the
  external effects on the body they act on

108
Easier way to get a resultant its location
1) Pick a point (easy to find moment arms)
(forces couples same procedures)
any forces couples system
dMo/R
2D
O
? single-force system (no-couple)
or single-couple system
4) Replace force-couple system with a single force
any forces couples system ?
single-force special single-couple (wrench)
3D
109
2/87 Determine the resultant and its line of
action of the following three loads.
why?
Move 3 forces to point O, Sums their force and
couples
Note M depends on the location where we move the
force to

M -2.40.2cos20 -1.50.12cos20
-3.60.3cos20 kN-m
Note R is the same regardless with the location
point we move the force to
R ( 2.4cos20 -1.5sin20 -3.6cos20 ) i
( -2.4sin20 -1.5cos20 3.6cos20 ) j kN
But we want to find the line of action of the
pure resultant force (the one which has no
additional couple)
110

At point O (0,0)
Sys 1
M -1.635 kN-m
At point X (x,y)
couples cancelled
Correct?
Two equivalent systems Moment at any point must
be the same on both system
Sys 2
Pick Point O
O
( line of action )
111
At point O (0,0)

O
M -1.635 kN-m
R
Manually Canceling Couples
How to locate Point P
How to find line of action ?
O (0,0)
d
O
d
or
112
Equivalent System Definition
Equivalent System
Sys 2
Sys 1
M
O
O
?
R
R

• Two force-couple systems are equivalent

113
A car stuck in the snow. Three students attempt
to free the car by exert forces on the car at
point A, B and C while the drivers actions
result in a forward thrust of 200 N as shown in
picture.
Determine 1) the equivalent force-couple
system at the car center of mass G 2) locate
the point on x-axis where the resultant passes.
114
For line of action of resultant
y
b
G
x
Sys II
Sys I
Couple Cancellation
ANS
At y 0 x 1.218 m.
Two equivalent systems Moment at point
G must be the same on both system
If you want to find only b (not line of action
itself)
Two equivalent systems (2D)
or - , you have to find out manually
115
Determine the resultant (vector) and the
point on x and y axes which must pass.
116
For line of action of resultant
If y 0 x 7.42 m.
x 0 y -23.4 m.
ANS
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