Intro 440:221 to Engineering Mechanics: Statics
Alberto Cuitino, Po Ting Lin
Instructors
Arturo Villegas
Assistants
Fall 2011
Based on Textbook Material: Engineering Mechanics Statics, 12th Edition, R.C. Hibbeler, Pearson 2010.
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440:221 Lectures
Topics to be covered
What is a moment of a force ABOUT A SPECIFIED AXIS?
What is it useful for?
How do we compute a such moment?
Mixed (cross and dot) product, what it is, what is good for?
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Moment of a force about an axis
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Lets think
With the force P, a person is creating a moment MA.
Does all of MA act to turn the socket?
How would you calculate an answer to this question?
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Another question
Sleeve A of this bracket can provide a maximum resisting moment of
125 Nm about the x-axis.
How would you determine the maximum magnitude of F before
turning about the x axis occurs?
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Scalar Analysis
Recall that the moment of a scalar force about any point O is
MO= F dO
where dO is the perpendicular (or shortest) distance from the
point to the forces line of action.
This concept can be extended to find the moment of a force
about an axis.
Finding the moment of a force about an axis can help
answer the types of questions we just considered.
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440:221 Lectures
Scalar Analysis
x
The moment about the y-axis would be My= Fz (dx) = F (r cos ).
In general, Maxis = F axis daxis
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Vector Analysis
However, unless the force can easily be broken into components
and the d found quickly, such calculations are not always trivial
and vector analysis may be much easier (and less likely to
produce errors).
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Vector Analysis
Our goal is to find the moment of F (the
tendency to rotate the body) about the axis a.
First compute the moment of F about any
arbitrary point O that lies on the a axis using
the cross product.
MO = r F
Now, find the component of
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MO along the axis a using the dot product.
Ma = ua MO
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Vector Analysis
Ma can also be obtained as
The above equation is also called the
mixed or triple scalar product.
In the this equation,
ua represents the unit vector along the axis a axis,
r
is the position vector from any point on the a axis to any point A on the
line of Engineering, Fall force
School of action of the 2011 , and F is the force vector.
440:221 Lectures
Example
Given: A force is applied to the tool
to open a gas valve.
Find: The magnitude of of the
moment this force about the z
axis of the value.
A
B
1)
Use Mz = u (r F).
2)
Plan:
Note that u = 1 k.
3) The vector r is the position vector from A to B.
4) Force F is already given in Cartesian vector form.
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Example (cont.)
A
A
u = 1k
B
rAB = {0.25 sin 30 i + 0.25 cos30 j} m
B
= {0.125 i + 0.2165 j} m
F = {60 i + 20 j + 15 k} N
Now find Mz = u (rAB F)
Mz = 1{ 0
0.125 0
(20) 1
0.2165 (60) } Nm
0.125 0.2165 0
-60
20
15
= 15.5 Nm
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Questions
1. The vector operation (P Q) R equals
A) P (Q R).
B) R (P Q).
C) (P R) (Q R).
D) (P R) (Q R ).
B) R (P Q)
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Questions
2. The force F is acting along DC.
Using the triple product to
determine the moment of F
about the bar BA, you could
use any of the following
position vectors except
______.
A) rBC
B) rAD
C) rAC
D) rDB
E) rBD
D) rDB
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Example 2
Given: Sleeve A can provide a
maximum resisting moment of
125 Nm about the x-axis.
Find: The maximum magnitude of F before slipping
occurs at A (the sleeve rotating around the
x-axis).
Plan:
1) We need to use MX = u (rAB F)
2) Find rAB
3) Find F in Cartesian vector form.
4) Complete the triple scalar product & solve for F !
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Example 2 (cont.)
rAB = {( 0.15 0) i + (0.30 0) j + (0.10 0) k} m
F = F {( cos 60 i + cos 60 j + cos 45 k)} N
= { 0.5 F i + 0.5 F j + 0.707 F k} N
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Example 2 (cont.)
Now find the triple product, Mx = u (rAB F)
1
0
0
-0.15
0.3
0.1
-0.5F
MX =
0.5F
0.707F
Nm
Mx = 1 {0.3 (0.707F) 0.1 (0.5F)} + 0 + 0 = 0.162 F Nm
Mx = 125 Nm = maximum moment along x-axis
125 = 0.162 F
FMAX = 771 N
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Attention Quiz
1. For finding the moment of the force F
about the x-axis, the position vector in
the triple scalar product should be ___ .
A) rAC
B) rBA
C) rAB
D) rBC
C)
C) rAB
2. If r = {1 i + 2 j} m and F = {10 i + 20 j + 30 k} N, then the moment of F
about the y-axis is ____ Nm.
A) 10
B) -30
C) -40
D) None of the above.
Hint:
MA = (ry FZ - rZ Fy) i - (rx Fz - rz Fx ) j + (rx Fy - ry Fx ) k
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B) -30
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Take-home message
The moment of a force about an axis measures the tendency
to rotate about that axis
The moment of a force about an axis can be computed using
mixed or triple product
You need the determine a vector position from any point on the axis to
any point in the line of action of the force
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