1. CHAPTER TWO
Force Vectors

2. 2.1 Scalars and Vectors Force Vectors
Scalars : A quantity represented be a number (positive or negative)
Ex: Mass, Volume, Length
(in the book scalars are represented by italics)
Vectors : A quantity which has both
A – magnitude (scalar)
B – direction (sense)
Ex: position, force, moment
Line of action
Magnitude
Sense

3. Forces Classification of Forces Contact
1 – Contacting or surface forces (mechanical)
2 – Non-Contacting or body forces (gravitational, weight)
Area
1 – Distributed Force, uniform and non-uniform
2 – Concentrated Force

4. Forces Classification of Forces Force System
1 – Concurrent : all forces pass through a point
2 – Coplanar : in the same plane
3 – Parallel : parallel line of action
4 – Collinear : common line of action
Three Types
1 – Free (direction, magnitude and sense)
2 – Sliding
3 – Fixed A O
Origin

5. 2.2 Basic Vector Operations
Properties of Vectors
1 – Vector Addition
2 – Vector Subtraction
3 – Vector Multiplication

6. Trigonometric Relations of a triangle
Phythagorean theorem is valid only for a right angled triangle. For any triangle (not necessarily right angled)
Sine Law
Cosine Law A B C

7. Example Ex: If the angle between F1 and F2 =60o and F1 = 54N
Find: the Resultant force and the angle b. R F1 F2
120o
60o

8. Example… contd R F1 F2
120o
60o

9. Properties of Parallelogram
Parallelogram Law
Properties of Parallelogram
A – sum of the three angles in a triangle is 180o
B – sum of the interior angles is 360o
C- opposite sides are equal

10. Vector Addition R A B B C
A + B A R

11. Vector Subtraction
Vector Subtraction A
A - B -B B A

12. 2.3 Vector addition of forces
If we consider, only two forces at a time then the result
can be obtained using parallelogram law, and by using
law of sines and cosines of triangles.
Even if we have multiple (say 5) forces, we take two at
a time to resolve the resultant one by one.
Consider another example in Example 2.4 (a) page 24
Given
To find: Resultant
Procedure: Use law of cosines to find the magnitude, law of sines find the angles

13. 2.3 Vector addition of forces (example 2.4)
Given
To find:
Solution:
Draw the vector diagram
Use the sine law to find

14. 2.4 Cartesian Coordinate Systems
A much more logical way to add/subtract/manipulate vectors is to represent the vector in Cartesian coordinate system. Here we need to find the components of the vector in x,y and z directions.
Simplification of Vector Analysis x y A z n en i j k x y z

15. Vector Representation in terms unit vector
Suppose we know the magnitude of a vector in any
arbitrary orientation, how do we represent the vector?
Unit Vector : a vector with a unit magnitude A en ^

16. 2.5 Right-Handed Coordinate Systemx y z
Right-Handed System
If the thumb of the right hand points in the direction of the positive z-axis when the fingers are pointed in the x-direction & curled from the x-axis to the y-axis.

17. Components of a Vector y A Ay Ax Az z x In 3-D y A Ay Ax x In 2-D
Cartesian (Rectangular) Components of a Vector y A Ay Ax Az z x
In 3-D y A Ay Ax x
In 2-D

18. Cartesian Vectors z Az = Azk A = Aen en y Ax = Axi Ay = Ayj x
Cartesian Unit Vectors y
A = Aen
Ay = Ayj
Ax = Axi
Az = Azk z x n i j k en

19. Cartesian Vectors
Magnitude of a Cartesian Vector y A Ay Ax Az z x

20. 2.6. Addition and Subtraction of vectorsy F Fy Fx Fz z x

21. Force Analysis y F Fy Fx Fz z x

22. Force Analysis
Ex: y z x
10ft
6ft
8ft
F = 600

23. Summary of the Force Analysisz F y F x q =
cos - 1 y y F

24. 2.6 Addition/Subtraction of Cartesion vectors
Since any vector in 3-D can be expressed as components in x,y,z directions, we just need to add the corresponding components since the components are scalars.
Then the addition
Then the subtraction

25. Example 2-9, pg 41
Determine the magnitude and the coordinate direction angles of the resultant force on the ring
Solution:

26. 2.6 ---Example 2-9…2 Unit vector and direction cosines
From these components, we can
determine the angles

27. 2.4---Problem 2-27 (pg 35) Solution:
Four concurrent forces act on the plate. Determine the magnitude of the resultant force and its orientation measured counterclockwise from the positive x axis.
Solution:
Draw the free body diagram. Then resolve forces in x and y directions.

28. 2.4---Problem 2-27…..2
Find the magnitude and direction.

29. 2.7. Position Vectors z
(xb, yb, zb) r
( xa, ya, za) y x
Position vectors can be determined using the coordinates of the end and beginning of the vector

30. 2.8---Problem 2-51, page 56 Solution
Determine the length of the connecting rod AB by first formulating a Cartesian position vector form A to B and then determining its magnitude.
Solution

31. Force Vector Along a Line
A force may be represented by a magnitude & a position
Force is oriented along the vector AB (line AB) B z F
Unit vector along
the line AB A y x

32. 2.6---Problem 2-36 (pg 47)
Solution:

33. 2.6---Problem 2-36…..2

34. 2.6---Problem 2-36…..3

35. 2.9. Dot Product y x A
Dot Product (Scalar Product)

36. 2.9---Problem 2-76, pg 65
A force of F = 80 N is applied to the handle of the wrench. Determine the magnitudes of the components of the force acting along the axis AB of the wrench handle and perpendicular to it.
Hand first rotates 45 in the xy plane and 30 in that plane

37. 2.9---Problem 2-76…..2

38. Application of Dot Product
Component of a Vector along a line
A n A

39. Application of dot product
Angle between two vectors A B