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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 x y z I I I + + 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 ( ) ( ) ( ) 2 2 2 2 2 2 x y z I I I y z dm z x dm x y dm + + = + + + + + ( ) 2 2 2 2 x y z dm = + + 2 2 r dm = where r is the distance from the origin
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