PROBLEM 9.180 Given a homogeneous body of mass mand of arbitrary shape and three rectangular axes x, y, andzwith origin at O, prove that the sum xyzIII++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 xis the axis of revolution, its moment of inertia yIabout a transverse axis y cannot be smaller than2310ma, where ais the radius of the sphere of the same mass and the same material. SOLUTION (i) Using Equation (9.30), we have ()()()222222xyzIIIyzdmzxdmxydm++=+++++∫∫∫()2222xyzdm=++∫22r dm=∫where ris the distance from the origin
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