ME 201 Engineering Mechanics: Statics

1
ME 201Engineering Mechanics Statics

• Chapter 2 Part A
• Vectors
• Resultants

2
Scalars Vectors

• Scalars
• Positive or negative number
• Follows elementary arithmetic/algebra rules
• Examples mass, volume, length

3
Scalars Vectors

• Vectors
• Quantity with both magnitude direction
• Does NOT follow elementary arithmetic/algebra
  rules
• Examples position, force, moment, velocities,
  acceleration

Magnitude
Head
Direction/Angle
Tail
Line of Action
4
Vector Operations

• Multiplication Division of Vector (A) by Scalar
  (a)
• a A aA

2A
A
2 A 2A
A
-.5A
-.5 A -.5A
5
Vector Addition

• Two vectors, A and B, when added form a resultant
  vector
• R A B
• Not simply their magnitudes, except in special
  case

6
Vector Addition

• Two Methods
• Parallelogram Law
• Triangle Construction

7
Parallelogram Law

• The resultant of two forces can be obtained by
• Joining the vectors at their tails

A
R

• Constructing a
• parallelogram

B

• The resultant is the diagonal of the
  parallelogram

8
Triangle Construction

• The resultant of two forces can be obtained by
• Joining the vectors in tip-to-tail fashion

A
B
R

• The resultant extends
• from the tail of A to
• the head of the B

9
Vector Addition

• Does
• A B B A ?

A
B
R
R
A
B
YES! - commutative
10
Vector Subtraction

• A - B A (-B)

A
-B
B
-B
R
A
11
Vector Subtraction

• Does
• A B B - A ?

-B
B
R
-R
A
-A
NO! opposite sense
12
Resultant of Several Concurrent Forces

• How would you find the resultant of more than 2
  forces (or vectors)?
• Repeated application of parallelogram law or
  triangle construction
• How about graphical solutions?

13
Resolution of a Force into Components

• Given R, resolve into 2 components

R

• Can resolve R many ways
• Need to know lines of action of the components

R
A
B
14
Solving Problems

• Use parallelogram law or triangle construction to
  construct force triangle
• Use trigonometry to find unknowns
• Law of sines
• Law of cosines

b
C
A
c
a
B


15
Example Problem

• Given
• F1 100 N
• F2 150 N
• Find
• R

F2
10º
F1
15º
16
Example Problem Solution
Solution
Ø
F1
15º
17
Resolution of a Force into Rectangular Components

• Given F, resolve into 2 components
• Choosing rectangle over general parallelogram
  simplifies
• Typically Fx and Fy are chosen along horizontal
  and vertical axes

F
y
F
Fy
Fx
?
x
O
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Resolution of a Force into Rectangular Components

• Then in vector notation
• F Fx Fy
• Focusing on the magnitudes we have
• Fx F cos ?
• Fy F sin ?
• or

y
F
Fy
Fx
?
x
O
F
Fy
Fx
?
19
Cartesian Vector Notation

• Another way to represent components of a force is
  by using Cartesian Unit Vectors
• Unit vectors i and j will designate x and y
  directions, respectively
• Magnitude will be represented by Fx and Fy
• Then we have in vector notation
• F Fx i Fy j

Using vector notation will greatly simplify
3-dimensional problems
20
Solving ProblemsDetermining the Resultant of
Several Coplanar Forces

• Resolve each force into x and y components
• Add respective components using scalar algebra
• Use trigonometry to find resultant and angle

21
Example Problem

• Given
• F1 400 N
• F2 250 N
• F3 200 N
• Find
• R, ?

F2
F3
5
3
50º
4
F1
22
Example Problem Solution 1

• Solution

F2
F3
5
3
50º
4
F1
23
Example Problem Solution 2
F2

• Solution

F3
5
3
50º
4
F1
24
Questions for Thought

• Must the x and y axes of rectangular components
  always be chosen horizontal and vertical,
  respectively?