1. Force, Moment, Couple and Resultants
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 (kg·m/s2 or kg·m · 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 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

7. FORCE SYSTEMS 2-D Force Systems 3-D Force Systems Moment Moment Couple
Vector (2D&3D)
Basic Concept
2-D Force Systems
3-D Force Systems
Moment
Couple
Resultants
Moment
Couple
Resultants

8. Free Vectors: associated with “Magnitude” and “Direction”
Representation
parallelogram
or V
Magnitude:
or V
Vector :
: Direction
triangle +

9. Vector
Operation Addition #5
Commutative

10. Vector
Operation Addition #6
Associative

11. Operation Scalar Multiplication #2
wrt = with respect to
associative
distributive wrt scalar addition
distributive wrt vector addition

13. Component Resolution of a Vector
A vector may be resolved
into two components.

14. Basic relations of Triangle (C/6, law of cosine, sine)
Law of sine b

15. a c b Hint Given V,  and , find Law of cosine Law of sine 2 1 b bq b c b
Given V,  and , find
Law of cosine
(Law of sine)
Law of sine

16. Vector Component and Projectionb
: 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 x
p-b
minus
(b>90)  b x y
b-q y x b   y x

19. EXAMPLE 2-1 y T  ANS x kN Given the magnitude of the tension
in the cable, T = 9 kN, express T in
terms of unit vector i and j  T x y
3 S.F.
Correct? kN
ANS

20. (a) parallel and perpendicular to arm AB
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
P = 90 N
par
per
Defining
direction
P = 90 N
per
par
arm AB
ANS
arm BC
ANS

21. Vector Component (Algebraic)
2/2 Combine the two forces P and T, which act on the fixed structure at B, into a single equivalent force R
Graphics
P=800 N (8cm) R
T=600 N (6cm)
Geometric P T R
Vector Component (Algebraic)
Correct?
Point of application is B

22. Two forces is not acting
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 y x o
F1y
F1x
F2y
F2x
1. Graphically Ry Rx
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

31. Three Dimensional Coordinate System
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
- cos(x), cos(y), cos(z) : “directional cosines” of
(maybe +/-)
- cos2(x)+cos2(y)+cos2(z) = 1
is a unit vector in the direction of
- 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 x y z A B
Real
directional
cosine B
directionl cosine = -0.92 A

35. x y z A B A B x y z A B B A
Thus,
ANS

36. Directional Cosines by Graphics
cos2(x)+cos2(y)+cos2(z) = 1

37. (a) Two points on the line of action of force is given (F also given).
- 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
2) 2-point construction
0.4 y B A
0.3
1.2 x kN
Ans

39. ANS 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
where
= unit vector from B to A
Thus
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. Fx = Fxy cos() = F cos() cos() Fy = Fxy sin() = F cos() sin()
(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. y x z F
Fxy Fx Fy Fz
65o
50o
Ans

51. y x z T A C B
15o x
TAB 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/ / /110

55. Operation Products Dot Products Cross Products Mixed Triple Products
Vector
Dot Products
Cross Products
Mixed Triple Products

56. scalar product 
(unit vector)
( three orthogonal vector )

57. Application of Dot Operation
Angle between two vectors
Component’izing Vector
line
which direction?

58. y x z F
Fxy Fx Fy Fz
65o
50o
Ans

59. y x z T A C B
15o x
TAB 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+ 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 A C y B
Area = ? O x

66. Mixed Triple Product

67. Why mixed triple product?
Why mixed triple product?
Mathematical Representation of Moments along the axis. O
Volume Calculation
Volume must always +
Volume = ?
Area * Height

68. Operation Product Summary
Vector
Dot Product
Scalar
Cross Product
Vector
Mixed Triple Product
Scalar

72. Homepage URLs
Statics official HP (User: Prince Password: Caspian)
Session 1 HP
(after the end of registration period)

73. FORCE SYSTEMS 2-D Force Systems 3-D Force Systems Moment Moment Couple
Vector
Basic Concept
2-D Force Systems
3-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 (kg·m/s2 or kg·m · 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 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. Force
Cables & Springs
Cable in tension

78. 2/2 Combine the two forces P and T, which act on the fixed structure at B, into a single equivalent force R
P=800 N (8cm)
Graphical R
T=600 N (6cm)
Geometric P T R
Algebraic
Correct?
Point of application is B

79. How to add sliding vectors (forces)?
Principle of
Transmissibility
is applied at point A A
Point of application
Not OK. !
Still OK.
Point of Application
is wrong A A

80. Special case: Addition of Parallel Sliding Force
Point of application
line of
action R2 R R1 R2 R1 R 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” 80

81. T VD1 Ty Tx x y 60 Ans Move all forces to that concurrent point
Point of application,
But no physical meaning
Ans
Application Point

82. How to add sliding vectors (forces)?
is applied at point A
Point of application
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.

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 Directionx y z O
Moment is a vector quantity.
Magnitude
Direction
Axis of Rotation
The unit of moment is N·m
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)
from A to point of application of the force
Moment about point A :
-Magnitude: a
-Direction:
right-hand rule X r
-Point of application: point A A d
(Unit: newton-meters, N-m) 2D
- 2D, need sign convention and be consistent; e.g. + for counter- clockwise and – for clockwise
M=Fd d +

87. sum of moment (of each force) = moment of sum (of all force)
Varignon’s Theorem (Principle of Moment)
can be used with
more than
2 components
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 = -Fxd2+Fyd1 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 d Y 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 4m
600N A d
400 y x O
600N
Solution II: 3D Vector Approach 4m
Solution I: 2D Scalar Approach O
CW or CCW? CW
Correct? CW

90. Solution III: Varignon’s theoremF1 B 2m F1 F2 A
400 F2
600N 2m A
400 4m d1
600N 4m F1 O C O d2 F2
Solution III: Varignon’s 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. o
12 m
30 m d
15 m
ANS

92. Example Hibbeler Ex 4-7 #1 Correct?
Moment
Determine the moment of the force about point O.
Correct?

93. Scalar Approach (Varignon’s theorem)
Example Hibbeler Ex 4-7 #2
Moment
3D Vector Approach
Scalar Approach (Varignon’s theorem)

94. Couple
- Couple is a summed moment produced by two force of equal magnitude but opposite in direction. a d
M = F(a+d) – 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 + O
- 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 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 A B A B B   d
Principle of transmissibility 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. F A B B F C A C A B F B A F F F A B A B from new location (B)
to old location (A) C A C A B F B A F F
No Moment:
Principle of Transmissibility F A B 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 
Principle of transmissibility 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 A B B  D D A

100. Be careful of the direction of momentP
Vector Diagram F
12m P
Ans

101. 2/11 Replace the force F by an equivalent force-couple system at point O.x y
50 kN
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)
couples occuring when moving forces. x y
sum of moments? +
(CW)
ANS
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. b cos20
exactly cancelled b
300N
20o
60 N-m D
Ans

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 y x q
Point of application
-Add two at a time  get line of action of
-Add many  do not get line of action of

108. Easier way to get a resultant + its location
F1d1
2) Replace each force with a force at point O + a couple
F2d2
F3d3
3) Add forces and moments O
Mo=(Fidi) O
arbitrary d1 d2 d3
1) Pick a point (easy to find moment arms)
(forces + couples : same procedures)
resultant
d=Mo/R 2D
any forces + couples system O
 single-force system (no-couple)
Mo=Rd
or single-couple system
4) Replace force-couple system with a single force 3D
any forces + couples system
 single-force + special single-couple (wrench)

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 M
(force-couple system) O R
Note: M depends on the location where we move the force to +
M = -2.4*0.2cos *0.12cos20
-3.6*0.3cos20 kN-m
Note: R is the same regardless with the location point we move the force to
R = ( 2.4cos sin cos20 ) i
+( -2.4sin cos cos20 ) j kN
move the R to point X where
Resultant Moment is zero
find the point X where the Resultant Moment is zero R O X M + N
= 0
But we want to find the line of action of the “pure” resultant force
(the one which has no additional couple)

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

111. Manually Canceling Couples
At point O (0,0) + M O R P
M = kN-m d
Manually Canceling Couples
How to locate Point P
How to find line of action ?
O (0,0) d O P d or P

112. Equivalent System Definition R P 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 driver’s 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. x y G

114. ANS For line of action of resultant y x y b x G G
Sys II
Sys I
Couple Cancellation
At y = 0; x = m.
ANS
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.G

116. ANS y y x x For line of action of resultant O O If y = 0; x = 7.42 m.
x = 0; y = m.
ANS