259_Problem CHAPTER 9 - PROBLEM 9.179 Consider a cube of...

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PROBLEM 9.179 Consider a cube of mass m and side a . ( a ) Show that the ellipsoid of inertia at the center of the cube is a sphere, and use this property to determine the moment of inertia of the cube with respect to one of its diagonals. ( b ) Show that the ellipsoid of inertia at one of the corners of the cube is an ellipsoid of revolution, and determine the principal moments of inertia of the cube at that point. SOLUTION ( a ) At the center of the cube have (using Figure 9.28) ( ) 2 2 2 1 1 12 6 x y z I I I m a a ma = = = + = Now observe that symmetry implies 0 xy yz zx I I I = = = Using Equation (9.48), the equation of the ellipsoid of inertia is 2 2 2 2 2 2 1 1 1 1 6 6 6 ma x ma y ma z + + = or ( ) 2 2 2 2 2 6 x y z R ma + + = = W which is the equation of a sphere. Since the ellipsoid of inertia is a sphere, the moment of inertia with respect to any axis OL through the center O of the cube must always
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