April 27, 2009
Statics Final Grades
Statics Final Exam
Your Final grade will be computed as follows:
•Homework: 10%, (the lowest one dropped)
•Quizzes: 10%, (the lowest one dropped)
Friday, May 8th
at 8:00 am - 9:45 am
•3 exams: 45%
•Binder is 5%;
•Final Exam: 30%
http://www.utpa.edu/academic_calendar/final_exams.htm
Final Exam
Ten short problems covering each one of
the student learning outcomes in the
syllabus, from material in chapters 1-6, 9,
and 10 in the textbook. To find the
syllabus go to:
Final Exam
Additional problems:
1- A problem about chapters 2,3,4,5,
and 9: centroids, unit vectors, moments
of a force with respect to a point,
equilibrium of particles and rigid
bodies.
http://www.engr.panam.edu/~hvasquez/Statics/Statics_syllabus_FALL_2008.pdf
1
Final Exam
2- A problem about a truss or a
frame like in chapter 6.
Moments of Inertia of Areas
The moments of inertia of areas about the x and y axes
and about the origin of the coordinate system O depend
only on the geometry of the area.
I x = ∫ y 2 dA
I y = ∫ x 2 dA
Final Exam
3- A problem about a machine
like in chapter 6.
Moments of Inertia of Areas
Moments of inertia of
area of some common
shapes are found
inside the front cover
of the textbook.
JC =
π 4
r
2
Polar moment of inertia or
product of inertia about O:
J o = I xy = ∫ r 2 dA = I x + I y
2
Parallel-Axis Theorem
Parallel-Axis Theorem
The moment of inertia about two parallel axes, one
of them passing through the centroid, are related by
the parallel-axis theorem.
Axes y and y1 are parallel and axis y passes through the
centroid of the area.
_
Axes x and x1 are parallel, and axis x passes through
the centroid.
y
3
ba
12
_
I x1 = I x + Ad 12
ba 3
2
I x1 =
+ bad 1
12
Iy =
_
Ix =
d2
_
x
C
d1
a
b
Determine the moments of inertia of the shaded area with
respect to axes that pass through the centroid and are
parallel to the x and y axes.
1
C1
C
C2
x
C
b
Location of Centroid:
y
1
_
Centroid of 2
C1
C
_
yc
2
C2
x
Centroid of 1
Centroidal axis
_
a
Example of Moments of Inertia
xc = 0
y
2
2
ab3
2
I y1 =
+ abd2
12
x1
y1
y
I y1 = I y + Ad 2
Example of Moments of Inertia
yc
ab 3
12
_
_
yc =
∑yA
∑A
i
i
=
i
x
( 35 + 20 )(1400 ) + 10 (1000 )
1400 + 1000
_
y c = 36 . 25 mm
3
Example of Moments of Inertia
y
d 1 = 55 − 36 . 25 = 18 . 75 mm
1
d 2 = 36 . 25 − 10 = 26 . 25 mm
C1
C2
2
_
d2
_
_
_
_
_
_
Iy =
_
70 ( 20 )
20 ( 50 )
+
12
12
6
I y = 0 . 255 x 10 mm
3
3
3
4
Moments of Inertia of Masses
y
1
x
3
20 ( 70 )
50 ( 20 )
2
2
+ 20 ( 70 )(18 . 75 ) +
+ 50 ( 20 )( 26 . 25 )
12
12
_
2
I y = I1y + I 2 y
_
_
I x = 1 . 79 x10 6 mm
Example of Moments of Inertia
Moment of Inertia about
the centroidal vertical axis:
x
_
Ix =
I x = I 1 + A 1 d 12 + I 2 + A 2 d 22
d1
d2
_
I x = I 1 x + A 1 d 12 + I 2 x + A 2 d 22
y c = 36 . 25 mm
_
1
yc 2
x
_
y
Moment of Inertia about the
centroidal horizontal axis:
d1
C
_
yc
Example of Moments of Inertia
The moments of
inertia of mass of
some common
bodies are
presented in tables,
like in tables in the
textbook.
4
4
Moments of Inertia of Masses
, z’
Moments of Inertia of Masses
The parallel axis theorem also applies for
the moment of inertia of masses.
Axes x and x’ are parallel separated a distance “d”
and axis x passes through the centroid.
_
I x ' = I x + md 2
Moments of Inertia of Masses
Determine the moment of inertia of mass
about axis x’ knowing the moment of
inertia about axis x.
_
I x ' = I x + m (l / 2 )
ml 2 ml 2
+
12
4
2
ml
=
3
I x' =
I x'
2
Moments of Inertia of Masses
Determine the moment of inertia of mass
about z’ knowing the moment of inertia
about z.
_
d
I z ' = I z + md
I z ' = md
z’
2
2
5
Example
Determine the moment of inertia with respect to the
x and y axes. The three rectangular sections are
identical.
y
20 mm
100 mm
x
C
6