1. Statics
Chapter Two
Force Vectors By
Laith Batarseh

2. Force Vectors Statics Definition Constant Variable Velocity Velocity
Engineering mechanics
Deformable body mechanics
Rigid body mechanics
Dynamics
Statics
Fluid mechanics
Constant
Velocity
Variable
Velocity

3. Force Vectors Statics Definition
Rigid body is the body that has the same volume parameters before and after applying the load
Deformable body is the body changes its volume parameters when the load is applied on it.

4. Velocity is changeable Acceleration or Deceleration
Force Vectors
Statics
Definition
Static Cases:
Dynamic Case:
Velocity = 0 P B A P
Velocity = Constant B A P
Velocity is changeable
Acceleration or Deceleration

5. Force Vectors Statics Dynamics
Definition
Start
End P
At rest
Acceleration
Constant velocity
Deceleration
Statics
Dynamics

6. z y y x x y-z plane x-y plane Force Vectors x-z plane x-y plane
Statics
Cartesian coordinate system z
y-z plane y
x-y plane
x-z plane y
x-y plane x x

7. Force Vectors
Statics
Definition y
P(x,y) y x

8. Newtown's Laws of Motion
Force Vectors
Statics
Newtown's Laws of Motion
Newtown's Laws of Motion
First Law: a particle at rest or moves in constant velocity will remain on its state
unless it is subjected to unbalance force. F1 F4 F3 F2 F
Second Law: a particle subjected to unbalance force will move at acceleration has the same direction of the force.
Third Law: each acting force has a reaction equal in magnitude and opposite in direction A
Action: Force of A acting on B B
Reaction: Force of B acting on A

9. Force Vectors Statics Newtown's Law of Gravity
Newtown proved his most famous law at all in the 17th centaury and called it Law of Universal Gravitation.
Statement: any two objects have a masses (M1 and M2 )and far away from each other by a distance (R) will have attraction force (gravitational) related proportionally with the objects masses product (M1 M2 ) and inversely with the square of the distance between them (R2). Mathematically:
Where: G = x m3/(kg.s2)
Weight: according to Newtown's Law of Gravity, weight is the gravitational force acting between the body has a mass (m) and the earth (of mass Me) and is given as: W=mg g is the gravitational acceleration and equal to GMe/R2.

10. Units of Measurements Force Vectors Statics Units of Measurements
According to Newton's 2nd law, the unit of force is a combination of the other three quantities: mass, length and time. So, the force units N (Newtown) and lbf can be written as:
English ft
lbm
lbf SI m Kg N
Units of Measurements
Length
(m or ft)
Mass
(kg or lbm)
Time
(sec)
Force
(N or lbf)

11. Is any physical quantity can be described fully by a magnitude only.
Force Vectors
Statics
Scalar versus vectors
Quantity
Physical
Scalar
Vector
Is any physical quantity can be described fully by a magnitude only.
Is any physical quantity needs both magnitude and direction to be fully described.
Examples: mass, length and time.
Examples: force, position and moment.

12. Force Vectors Example Statics Scalar versus vectors
5 units
This vector has a magnitude of 5 units and it tilted from the horizontal axis by +30o
Example
Vector
Magnitude (M)
Sense of Direction
Direction (θ)
Fixed axis
Sense of Direction
Vector notations: A
V = M∟θ

13. Multiplication by a scalar Example
Force Vectors
Statics
Vector operations
Multiplication by a scalar
30o
A=5∟30o
2A =10∟30o
0.5A =2.5∟30o
-A =-5∟30o = 5∟180+30o
Example A 2A
0.5A -A

14. Force Vectors Statics Vector operations Vector addition (A+B) A A B A
Vector subtraction (A-B) A A B A -B
R=A+B B A
R=A-B A -B
R=A-B A -B OR
R=A+B B

15. Force Vectors Example: Solution: Statics Vector operations
Assume the following vectors: A = 10∟30o and B = 7∟-20 o
Find:
A+B
A-B
Solution: -B B A
A-B A
A+B

16. Special cases: Force Vectors Collinear vectors: Multiple addition:
Statics
Vector operations
Special cases:
Collinear vectors:
Multiple addition:
A+B+C+…: in such case, the addition can be in successive order or in multiple steps. A B
A + B
B - A
B-A

17. Force Vectors Example: Solution: Statics Vector operations
Assume the following vectors: A = 10∟30o ,B = 7∟-20 o and C = 6∟135o .
Find:
A+B+(A+C)
Solution:
A+B+(A+C)
A+B
A+C C B A
A+C
A+B A

18. Example (cont): Find A+B+(A-C)
Force Vectors
Statics
Vector operations
Example (cont): Find A+B+(A-C)
A+B A
A+B+A
A+B A B
A+B+A-C -C
A+B+A

19. Force Vectors
Examples

20. Force Vectors
Examples

21. Force Vectors
Examples

22. Force Vectors
Examples

23. Coplanar forces Force Vectors
Coplanar forces are the forces that share the same plane.
These forces can be represented by their components on x and y axes which are called the rectangular components.
Cartesian notation
The components can be represented by scalar and Cartesian notations.
Scalar notation y x Fy Fx F θ
Fx = F cos(θ)
Fy= F sin(θ) Fy Fx F x i y j
F = Fx i + Fy j

24. Force Vectors 1 2 Scalar notation cases Fx = F cos(θ) Fx /F= a/cy x Fy Fx c
Fx /F= a/c
Fy /F= b/c a b F y x Fy Fx F θ
Fx = F cos(θ)
Fy= F sin(θ) 1

25. Cartesian notation cases
Force Vectors
Scalar notation cases
Cartesian notation cases
F2,y F2
F2,x
F1,y F1
F1,x
F4,x F4
F4,y x y
F3,x
F3,y F3
F 2 = -F2,x i + F2,y j Fy Fx F x i y j
F 1= F1,x i + F1,y j
F 3= -F3,x i - F3,y j
F 4= F4,x i - F4,y j

26. And the direction is found by:
Force Vectors
Force summation
For both cases of notations, the magnitude of the resultant force is found by:
And the direction is found by:
FR,x =∑Fx
+ve
FR,y
FR,x FR θ
FR,y =∑Fy
+ve

27. Force Vectors
Examples

28. Force Vectors
Examples

29. Force Vectors
Examples

30. Force Vectors
Examples

31. Cartesian Vectors
Forces
One dimension
Two dimensions
Three dimensions

32. Cartesian Vectors
Note that
The projection of vector on x-y plane represent a new vector called A’
In three dimensions system, new component appeared (Az).
The new dimension direction is represented by unit vector (k)
The position of the vector has to be located by three angles one from each axis A Az z y x Ax Ay A’

33. Cartesian Vectors Right hand coordinate system
This method is used to describe the rectangular coordinate system.
The system is said to be right handed if the thumb points to the positive z-axis and the fingers are curled about this axis and points from the positive x-axis to the positive y-axis.

34. Cartesian Vectors Notation
To find the components of a vector oriented in three dimensions, two successive applications of the parallel-ogram must be done.
One of the parallelogram applications is to resolve A to A’ and Az and the other is used to resolve A’ into Ax and Ay.
A = A’ + Az = Ax + Ay + Az A Az z y x Ax
A y A’

35. Cartesian Vectors Angles α is the angle between the vector and x-axis
β is the angle between the vector and y-axis
γ is the angle between the vector and z-axis α β γ

36. Cartesian Vectors Cartesian unit vectors k A = Ax i + Ay j + Az k j
Ax = Acos(α)
Ay = Acos(β)
Az = Acos(γ)

37. Cartesian Vectors
Magnitude and Direction
Magnitude
Direction

38. Force Vectors
Examples

39. Force Vectors
Examples

40. Force Vectors

41. Force Vectors

42. Force Vectors

43. Force Vectors

44. Force Vectors

45. uA = cos(α) i + cos(β) j + cos(γ) k
Cartesian Vectors
Other direction definition
A = Ax i + Ay j + Az k
uA = cos(α) i + cos(β) j + cos(γ) k
A = |A| uA
= |A| cos(α) i + |A| cos(β) j + |A| cos(γ) k
= Ax i + Ay j + Az k

46. Cartesian Vectors Other direction definition cont Az = A cos(ϕ)θ ϕ
Az = A cos(ϕ)
A’ = A sin(ϕ)
Ax = A’ cos(θ) = A sin(ϕ) cos(θ)
Ay = A’ sin(θ) = A sin(ϕ) sin(θ)

47. FR = ∑F = ∑Fx i + ∑Fy j + ∑Fz k
Cartesian Vectors
Cartesian Vectors Addition
A = Ax i + Ay j + Az k
B = Bx i + By j + Bz k
R = A+ B
= (Ax + Bx ) i + (Ay + By) j +(Az + Bz ) k
General rule:
FR = ∑F = ∑Fx i + ∑Fy j + ∑Fz k

48. Cartesian Vectors Example [1]: Question: assume the following forces:
F1 = 5 i + 6j -4k
F2 = -3i +3j +3k
F3=7i-12j+2k
Find the resultant force F = F1 + F2 +F3 and represent it in both Cartesian and scalar notations
Solution:
Given: F1, F2 and F3
Required: find resultant force F and represent it in both Cartesian and scalar notation

49. Cartesian Vectors Example [1]: Solution:
F = (5-3+7)i + (6+3-2)j +(-4+3+2)k = 9i +7j +10k (Cartesian notation)
Scalar notation:
Magnitude:
Unit vector u:

50. Cartesian Vectors Position vector
in this case, vector r is a position vector relates point P to point O.
Assume: r = ai + bj + ck
At this point, r is called position vector.
Position vector is a vector locates a point in space with respect to other point.

51. Cartesian Vectors Position vector
r = (xB – xA)i + (yB – yA)j + (zB – zA)k
Assume vectors rA and rB are used to locate points A and B from the origin (ie. Point 0,0,0).
Define a position vector r to relate point A to point B.
rB = r + rA → r = rB – rA

52. Explain figure b) 52

53. 53

54. 54

55. 55

56. 56

57. 57

58. 58

59. 59

60. Force Vectors
End of Chapter Two