ME 221 Statics
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Sections 2.2 – 2.5
ME 221
Lecture 2
1
Announcements
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• Quiz & exam grading & help room
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ME 221
Lecture 2
2
Announcements
• HW#1 Due on Friday, May 21
Chapter 1 - 1.1, 1.3, 1.4, 1.6, 1.7
Chapter 2 – 2.1, 2.2, 2.11, 2.15, 2.21
• Quiz #1 on Friday, May 21
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Lecture 2
3
Last Lecture
• Chapter 1: Basics
• Vectors, vectors, vectors
• Law of Cosines
• Law of Sines
• Drawing vector diagrams
• Example 1: Addition of Vectors
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Lecture 2
4
Law of Cosines
This will be used often in balancing forces
g
b
a
b
a
c
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Lecture 2
5
Law of Sines
Again, start with the same triangle
g
b
a
b
a
c
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Lecture 2
6
Example
25o
200 lb
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45o 300 lb
Lecture 2
Note: resultant of two
forces is the vectorial
sum of the two vectors
7
155o
200 lb
110o
25o
45o
300 lb
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200 lb
R
25o
a
 = 90o+25o-a
R
300 lb
Lecture 2
8
Scalar Multiplication of Vectors
Multiplication of a vector by a scalar simply
scales the magnitude with the direction
unchanged
Line of action
stays the same
Line of action
0.5 x A
A
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Lecture 2
9
Forces
• Review definition
• Shear and normal forces
• Resultant of coplanar forces
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Lecture 2
10
Characteristics of a Force
• Its magnitude
– denoted by |F|
• Its direction
• Its point of application
– important when we discuss moments later
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Lecture 2
11
Further Categorizing Forces
• Internal or external
– external forces applied outside body
P
P
Cut plane through body
• A section of the body exposes internal body
P
Internal
tension
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Lecture 2
12
Shear and Oblique
• Shear internal force has line of action
contained in cutting plane
P
Intenal
shear
forces
P
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Lecture 2
13
Oblique Internal Forces
• Oblique cutting planes have both normal
and shear components
N
P
S
Where N + S = P
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Lecture 2
14
Transmissibility
• A force can be replaced by a force of equal
magnitude provided it has the same line of
action and does not disturb equilibrium
B
A
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Lecture 2
15
Weight is a Force
• Weight is the force due to gravity
– W = mg
• where m is mass and g is gravity constant
• g = 32.2 ft/s2 = 9.81 m/s2
• English and metric
– Weight lb or N
– Mass
slugs or kg
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Lecture 2
16
Resultant of Coplanar Forces
A body’s motion depends on the resultant of all the
forces acting on it
In 2-D, we can use the Laws of Sines and Cosines to
determine the resultant force vector
In 3-D, this is not practical and vector components
must be utilized
• more on this later
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Lecture 2
17
Perpendicular Vectors
y
y
A
Ax
A
Ay
y
Ay
y
Ay
x
x
Ax
Ax
x
x
Ax is the component of vector A in the x-direction
Ay is the component of vector A in the y-direction
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Lecture 2
18
Vector Components
Vector components are a powerful way to
represent vectors in terms of coordinates.
y
y
y
A
where
Ax = |A| cos x
Ay
Ay = |A| cos y
= |A| sin x
x
Ax
A=
x
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x
Lecture 2
Ax
Ay
19
Vector Components (continued)
Ax = |A| cos x
Ay = |A| cos y
= |A| sin x
cos x = Ax / |A| = x
cos y = Ay / |A| = y
x and y are called direction cosines
x2 + y2 = 1
Note: To apply this rule the two axes must be orthogonal
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Lecture 2
20
Summary
• External forces give rise to
– tension and compression internal forces
– normal and shear internal forces
• Forces can translate along their line of
action without disturbing equilibrium
• The resultant force on a particle is the vector
sum of the individual applied forces
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3-D Vectors; Base Vectors
• Rectangular Cartesian coordinates (3-D)
• Unit base vectors (2-D and 3-D)
• Arbitrary unit vectors
• Vector component manipulation
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3-D Rectangular Coordinates
• Coordinate axes are defined by Oxyz
y
Coordinates can be rotated
any way we like, but ...
O
z
x
• Coordinate axes must be a right-handed
coordinate system.
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Writing 3-D Components
• Component vectors add to give the vector:
y
y
A
Az
=
O
O
x
z
A=
Ay
z
Ax
x
Ax + Ay + Az
Also,
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24
3-D Direction Cosines
The angle between the vector and coordinate
axis measured in the plane of the two
y
y
A
x
O
z
z
x
Where: x2+y2+z2=1
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25
Unit Base Vectors
Associate with each coordinate, x, y, and z, a unit
vector (“hat”). All component calculations use the
unit base vectors as grouping vectors.
y
Now write vector as follows:
where Ax = |Ax|
O
z
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Ay = |Ay|
x
Az = |Az|
Lecture 2
26
Vector Equality in Components
• Two vectors are equal if they have equal
components when referred to the same
reference frame. That is:
if
Ax = Bx , Ay = By , Az = Bz
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27
Additional Vector Operations
• To add vectors, simply group base vectors
• A scalar times vector A simply scales all the
components
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General Unit Vectors
• Any vector divided by its magnitude forms
a unit vector in the direction of the vector.
– Again we use “hats” to designate unit vector
y
b
O
z
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Lecture 2
x
29
Position Vectors in Space
• Points A and B in space are referred to in
terms of their position vectors.
y
rA A
O
• Relative position defined
by the difference
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Lecture 2
x
z
rB B
rB/A
30
Vectors in Matrix Form
• When using MatLab or setting up a system
of equations, we will write vectors in a
matrix form:
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Lecture 2
31
Summary
• Write vector components in terms of base
vectors
• Know how to generate a 3-D unit vector
from any given vector
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Example Problem
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