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257_Problem CHAPTER 9 - PROBLEM 9.177 For the homogeneous...

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PROBLEM 9.177 For the homogeneous circular cylinder shown, of radius a and length L , determine the value of the ration a / L for which the ellipsoid of inertia of the cylinder is a sphere when computed ( a ) at the centroid of the cylinder, ( b ) at point A . SOLUTION ( a ) From Figure 9.28 ( ) 2 2 2 1 1 3 2 12 x y z I ma I I m a L = = = + Now observe that symmetry implies 0 xy yz zx I I I = = = Using Equation (9.48), the equation of the ellipsoid of inertia is then ( ) ( ) 2 2 2 2 2 2 2 2 2 2 1 1 1 1: 3 3 1 2 12 12 x y z I x I y I z ma x m a L y m a L + + = + + + + = For the ellipsoid to be a sphere, the coefficients must be equal. Therefore,
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