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PROBLEM 9.180
Given a homogeneous body of mass
m
and of arbitrary shape and three
rectangular axes
x
,
y
, and
z
with origin at
O
, prove that the sum
xyz
I
II
++
of the moments of inertia of the body cannot be smaller than the similar sum
computed for a sphere of the same mass and the same material centered at
O
. Further, using the results of Prob. 9.178, prove that if the body is a solid
of revolution, where
x
is the axis of revolution, its moment of inertia
y
I
about a transverse axis
y
cannot be smaller than
2
3
10
ma
, where
a
is the radius
of the sphere of the same mass and the same material.
SOLUTION
(i)
Using Equation (9.30), we have
( ) (
)(
)
22
I
I
I
y
z
dm
z
x
dm
x
y
dm
++=
+
+
+
+
+
∫∫∫
( )
222
2
x
yz
d
m
=+
+
∫
2
2
rdm
=
∫
where
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This note was uploaded on 07/11/2011 for the course EGM 2511 taught by Professor Jenkins during the Spring '08 term at University of Florida.
 Spring '08
 Jenkins
 Statics

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