ME 221 Statics
Lecture #4
Sections 2.4 – 2.5
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Homework Problems
• Due Today:
– 1.1, 1.3, 1.4, 1.6, 1.7
– 2.1, 2.2, 2.11, 2.15, 2.21
• Due Wednesday, September 9:
– Chapt 2: 22, 23, 25, 27, 29, 32, 37, 45, 47 & 50
– On 2.50:
• Solve with hand calculations first
• Then use MathCAD, MatLab, Excel, etc. to solve
• Quiz #1 – Friday, 9/5
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TA Hours
• Help Sessions – ME Help Room – 1522EB - Cubicle #2
• TA’s: – Jimmy Issa, Nanda Methil-Sudhakaran & Steve Rundell
Mondays & Wednesdays – 10:15am to 5:00pm
Tuesdays & Thursdays – 8:00am to 5:00pm
Fridays – 8:00am to 11:00am
• Grader – Jagadish Gattu
2415EB – Weds: 10:00am to 12:00am
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Last Lecture
• Scalar Multiplication of Vectors
•
Perpendicular Vectors
• Vector Components
•
Example 2.3
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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
• Example problem
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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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3-D Direction Cosines
The angle between the vector and coordinate
axis measured in the plane of the two
y
qy
A
qx
O
qz
z
x
Where: x2+y2+z2=1
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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|
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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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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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Position Vectors in Space
• Points A and B in space are referred to in
terms of their position vectors.
y
rA
O
• Relative position defined
by the difference
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x
z
rB
rB/A
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Vectors in Matrix Form
• When using MathCAD or setting up a
system of equations, we will write vectors
in a matrix form:
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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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