Engineering 36
Chp10:
Moment of Interia
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Moments of Inertia
 The Previously Studied “Area Moment
of Inertia” does Not Actually have True
Inertial Properties
• The Area Version is More precisely Stated
as the SECOND Moment of Area
 Objects with Real mass DO have inertia
• i.e., an inertial Body will Resist Rotation by
An Applied Torque Thru an F=ma Analog
T  Iα
I  Mass Momentof Inertia
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Moment of Inertia
The Moment
of Inertia is
the Resistance
to Spinning
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Linear-Rotational Parallels
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Moment of Inertia
 The Angular acceleration, , about the axis
AA’ of the small mass m due to the
application of a couple is
proportional to r2m.
• r2m  moment of inertia of the mass m
with respect to the axis AA’
 For a body of mass m the resistance to
rotation about the axis AA’ is
I  r12 m  r22 m  r32 m     r 2 m
  r 2 dm  m ass m om entof inertia
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Radius of Gyration
 Imagine the entire Body Mass
Concentrated into a single Point
 Now place this mass a distance k from
the rotation axis so as to create the
same resistance to rotation as the
original body
• This Condition Defines, Physically, the
Mass Radius of Gyration, k
 Mathematically
I
I   r dm  k m or k 
m
2
Engineering-36: Engineering Mechanics - Statics
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2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Ix, Iy, Iz
 Similarly, for the
moment of inertia with
respect to the x and z
axes
2
2


I z   x 2  y 2 dm
 Units Summary
I   r 2 dm  kg  m 2 
I x   y  z dm
 Mass Moment of inertia
with respect to the
y coordinate axis  r is
the ┴ distance to y-axis


I y   r 2 dm   z 2  x 2 dm
Engineering-36: Engineering Mechanics - Statics
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• SI
• US Customary Units
 lb  s 2 2 
2
I  slug  ft  
ft   lb  ft  s 2
 ft






Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx

Parallel Axis Theorem
x  x ' x
 The Translation
Relationships
y  y ' y
 Then Write Ix
z  z ' z

 

  y  2  z  2 dm  2 y  y dm  2 z  z dm  y 2  z 2  dm
I x   y 2  z 2 dm    y   y 2   z   z 2 dm
0
0
 In a Manner Similar to
 Consider CENTRIODAL
the Area Calculation
Axes (x’,y’,z’) Which are
• Two Middle Integrals are
Translated Relative to
1st-Moments Relative to
the Original CoOrd
the CG → 0
Systems (x,y,z)
• The Last Integral is
the Total Mass
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
m
Parallel Axis Theorem cont.
 Similarly for the Other
two Axes

 mx
I y  I y  m z  x
I z  I z
 So Ix




I x   y2  z2 dm  0  0  y 2  z 2 m

 I x  I x'  m y  z
 so
2
2


I x  I x  m y  z
2
Engineering-36: Engineering Mechanics - Statics
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2

2
2
2
 y2


 In General for any axis
AA’ that is parallel to a
centroidal axis BB’
I  I  md
 Also the Radius of
Gyration
2
k  k d
2
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
2
Thin Plate Moment of Inertia
 For a thin plate of uniform thickness t and
homogeneous material of density , the mass
moment of inertia with respect to axis AA’
contained in the plate
I AA   r 2 dm   r 2 tdA   t  r 2 dA   t I AA,area
 Similarly, for perpendicular axis BB’ which is also
contained in the plate
I BB  t I BB,area
 For the axis CC’ which is PERPENDICULAR to the
plate note that This is a POLAR Geometry
I CC   t J C ,area   t I AA,area  I BB,area 
 I AA  I BB
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Polar Moment of Inertia
 The polar moment of inertia is an important parameter in
problems involving torsion of cylindrical shafts, Torsion
in Welded Joints, and the rotation of slabs
 In Torsion Problems, Define a Moment of Inertia
Relative to the Pivot-Point, or “Pole”, at O
J O   r dA
2
 Relate JO to Ix & Iy Using The
Pythagorean Theorem


J O   r 2 dA   x 2  y 2 dA   x 2 dA   y 2 dA
 Iy  Ix
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Thin Plate Examples
 For the principal centroidal axes on a
rectangular plate
I AA  t I AA,area  t 121 a3b  121 taba2  121 ma2
I BB  t I BB,area  t

1
12

ab3  121 tabb2  121 mb2

ICC  I AA,mass  I BB,mass  121 m a2  b2
 For centroidal axes on a
circular plate



I AA  I BB  t I AA,area  t 14  r 4  14 mr 2
I CC  I AA  I BB  12 mr 2
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
3D Mass Moments by Integration
• The Moment of inertia of a
homogeneous body is obtained from
double or triple integrations of the form
I    r 2 dV
• For bodies with two planes of symmetry,
the moment of inertia may be obtained
from a single integration by choosing thin
slabs perpendicular to the planes of
symmetry for dm.
• The moment of inertia with respect to a
particular axis for a COMPOSITE body
may be obtained by ADDING the
moments of inertia with respect to the
same axis of the components.
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Common Geometric Shapes
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Example 1
 SOLUTION PLAN
 Determine the moments
of inertia of the steel
forging with respect to
the xyz coordinate
axes, knowing that the
specific weight of steel
is 490 lb/ft3 (0.284 lb/in3)
Engineering-36: Engineering Mechanics - Statics
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• With the forging divided
into a Square-Bar and
two Cylinders, compute
the mass and moments
of inertia of each
component with respect
to the xyz axes using the
parallel axis theorem.
• Add the moments of
inertia from the
components to determine
the total moments of
inertia for the forging.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Example 1 cont.
 Referring to the
Geometric-Shape Table
for the Cylinders
•
•
•
•
 Then the Axial (x)
Moment of Inertia
 For The Symmetrically
Located Cylinders
m
V
g

490lb/ft  1  3in
1728in ft 32.2 ft s 
3
3
2
3
m  0.0829lb  s 2 ft
Engineering-36: Engineering Mechanics - Statics
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a = 1” (the radius)
L = 3”
xcentriod = 2.5”
ycentriod = 2”
3
2
I x  I x  my 2  12 m a2  my 2

1
2
0.0829121 2  0.0829122 2
 2.59103 lb  ft  s 2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Example 1 cont.2
 Now the Transverse
(y & z) Moments of Inertia


I y  I y  mx 2  121 m 3a 2  L2  mx 2


 121 0.0829 3121   123   0.0829 212.5 
2
2
2
dz
 4.17 103 lb  ft  s 2
d z2

 
I z  121 m 3a 2  L2  m x 2  y 2




 121 0.0829 3121   123   0.0829  212.5   122 
2
2
2
2

 6.48103 lb  ft  s 2
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Example 1 cont.3
 Referring to the
Geometric-Shape Table
for the Block
• a = 2”
• b = 6”
• c = 2”
 Then the Transverse (x
& z ) Moments of Inertia
 For The Sq-Bar
m
V
g
490lb/ft 2  2  6in

1728in ft 32.2 ft s 
3
3
3
3
m  0.211lb  s 2 ft
Engineering-36: Engineering Mechanics - Statics
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2

I x  I z  121 m b 2  c 2

1
12

0.211   122 2 
6 2
12
 4.88103 lb  ft  s 2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Example 1 cont.4
 Add the moments of
inertia from the
components to
determine the total
moment of inertia.

I x  4.88 103  2 2.59 103



1
12

I y  9.32  103 lb  ft  s 2

0.211     
2 2
12
I x  10.06 103 lb  ft  s2
I y  0.977  103  2 4.17  103
 And the Axial (y)
Moment of Inertia
I y  121 m c 2  a 2

2 2
12

I z  4.88 103  2 6.48 103

I z  17.84 103 lb  ft  s2
 0.977103 lb  ft  s 2
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
T = Iα
 When you take ME104 (Dynamics) at
UCBerkeley you will learn that the Rotational
Behavior of the CrankShaft depends on its
Mass Moment of inertia
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
WhiteBoard Work
Some Other
Mass
Moments
 For the
Thick Ring
2
2
Router
 Rinner
Iz  m
2
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
WhiteBoard Work
Find MASS
Moment of
Inertia
for Prism
 About the y-axis in this case
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Engineering 36
Appendix
dy
µx µs
 sinh

dx
T0 T0
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
WhiteBoard Work
Find MASS
Moment of
Inertia
for Roller
 About axis AA’ in this case
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx
Mass Moment of Inertia
 Last time we discussed the “Area
Moment of Intertia”
• Since Areas do NOT have Inertial
properties, the Areal Moment is more
properly called the “2nd Moment of Area”
 Massive Objects DO physically have
Inertial Properties
• Finding the true “Moment of Inertia” is very
analogous to determination of the 2nd
Moment of Area
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx