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12/3/2013
1
Mass Moment of InertiaEngineering Mechanics- Distributed Forces
• Encounter in the engineering problems involving rotational motion of rigid bodies in dynamics
Moment of Inertia of a Mass• Angular acceleration about the axis AA’ of the
small mass Dm due to the application of a couple is related to r2Dm.
r2Dm = moment of inertia of the mass Dm with respect to the axis AA’
• For a body of mass m the resistance to rotation about the axis AA’ is
inertiaofmomentmassdmr
mrmrmrI
2
23
22
21
• The radius of gyration for a concentrated mass with equivalent mass moment of inertia is
mIkmkI 2
Engineering Mechanics- Distributed Forces
12/3/2013
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Moment of Inertia of a Mass
• Moment of inertia with respect to the ycoordinate axis is
dmxzdmrI y222
• Similarly, for the moment of inertia with respect to the x and z axes,
dmyxI
dmzyI
z
x22
22
Engineering Mechanics- Distributed Forces
Parallel Axis Theorem• For the rectangular axes with origin at O and parallel
centroidal axes,
dmzydmzzdmyydmzy
dmzzyydmzyIx2222
2222
22
22 zymII xx
22
22
yxmII
xzmII
zz
yy
• Generalizing for any axis AA’ and a parallel centroidal axis,
2mdII
Engineering Mechanics- Distributed Forces
12/3/2013
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Moments of Inertia of Thin Plates• For a thin plate of uniform thickness t and homogeneous
material of density r, the mass moment of inertia with respect to axis AA’ contained in the plate is
IA A r 2dm t r 2 dA t IA A ,area
• Similarly, for perpendicular axis BB’ which is also contained in the plate,
IB B t IB B ,area
• For the axis CC’ which is perpendicular to the plate,IC C t JC ,area t IA A ,area IB B ,area
IA A IB B
Engineering Mechanics- Distributed Forces
Moments of Inertia of Thin Plates• For the principal centroidal axes on a rectangular plate,
21213
121 mabatItI area,AAAA
21213
121 mbabtItI area,BBBB
22121 bamIII mass,BBmass,AACC
• For centroidal axes on a circular plate,
2414
41 mrrtItII area,AABBAA
221 mrIII BBAACC
Engineering Mechanics- Distributed Forces
12/3/2013
4
Mass Moment of Inertia of a Slendor Rod
2zI r dm
2
0
3 2 2
( )
/
( )3 3 3
L
zI x mdx
mm mass unit lengthL
mL mL L mL
2 2
2 12z zcg
L mLI I m
Engineering Mechanics- Distributed Forces
Moments of Inertia of a 3D Body by Integration
• Moment of inertia of a homogeneous body is obtained from double or triple integrations of the form
dVrI 2
• 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.
Engineering Mechanics- Distributed Forces
12/3/2013
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Mass Moment of Inertia of a Rectangular PrismEngineering Mechanics- Distributed Forces
Find mass moment of inertia about Z axis
Engineering Mechanics- Distributed Forces
Mass Moment of Inertia of a Sphere
12/3/2013
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Moments of Inertia of Common Geometric ShapesEngineering Mechanics- Distributed Forces
Sample Problem: Composite Section
Determine the moments of inertia of the steel forging with respect to the xyz coordinate axes, knowing that the specific weight of steel is 7896 kg/m3.
SOLUTION:
• With the forging divided into a prism 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.
Engineering Mechanics- Distributed Forces
12/3/2013
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Sample Problem: Composite Section
kg0744.01031
kg/m7896
:cylindereach
362
3
mm
Vm
2
222121
222121
kgm5394.0
5.20744.03130744.0
3
xmLamI y
2 2 2 2112
2 2 2 2112
3
0 0744 3 1 3 0 0744 2 5 2
0 837 2kgm
zI m a L m x y
. . .
.
2
2221
2221
kgm5394.0
20744.010744.0
ymmaI x
cylinders :cm2cm,5.2,cm3,cm1 yxLaSOLUTION:• Compute the moments of inertia
of each component with respect to the xyz axes.
Engineering Mechanics- Distributed Forces
Sample Problem: Composite Section
kg1895.0
m10622kg/m7896
:prism36–3
mVm
prism (a = 2 cm, b = 6 cm, c = 2 cm):
2
2212122
121
gmk 6320
2618950
.
.cbmII zx
2
2212122
121
gmk 1260
2218950
.
.acmI y
• Add the moments of inertia from the components to determine the total moments of inertia.
372.02632.0 xI2kgm376.1xI
5394.02126.0 yI2kgm2048.1yI
837.02632.0 zI2kgm306.2zI
Engineering Mechanics- Distributed Forces
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