ME 221 Statics
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Sections 2.2 – 2.3
ME 221
Lecture 3
1
Announcements
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ME 221
Lecture 3
2
Announcements
•HW#1 Due on Wednesday 09/03
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 09/05
ME 221
Lecture 3
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 3
4
Law of Cosines
This will be used often in balancing forces
g
b
a
b
a
c
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Lecture 3
5
Law of Sines
Again, start with the same triangle
g
b
a
b
a
c
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Lecture 3
6
Example
25o
200 lb
ME 221
45o 300 lb
Lecture 3
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 3
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 3
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Forces
•
•
•
•
Review definition
Shear and normal forces
Resultant of coplanar forces
Problems
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Lecture 3
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 3
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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Shear and Oblique
• Shear internal force has line of action
contained in cutting plane
P
Intenal
shear
forces
P
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Oblique Internal Forces
• Oblique cutting planes have both normal
and shear components
N
P
S
Where N + S = P
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Lecture 3
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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 3
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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 3
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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 in the future
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Lecture 3
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 3
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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 3
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 3
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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Sample Problem
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