261_Problem CHAPTER 9

# 261_Problem CHAPTER 9 - PROBLEM 9.180 Given a homogeneous...

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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.

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