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254_Problem CHAPTER 9

# 254_Problem CHAPTER 9 - PROBLEM 9.175 For the rectangular...

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PROBLEM 9.175 For the rectangular prism shown, determine the values of the ratios b / a and c / a so that the ellipsoid of inertia of the prism is a sphere when computed ( a ) at point A , ( b ) at point B . SOLUTION ( a ) Using Figure 9.28 and the parallel-axis theorem have at point A .. ( ) 2 2 1 12 x I m b c = + ( ) ( ) 2 2 2 2 2 1 1 4 12 2 12 y a I m a c m m a c = + + = + ( ) ( ) 2 2 2 2 2 1 1 4 12 2 12 z a I m a b m m a b = + + = + Now observe that symmetry implies 0 x y y z z x I I I ′ ′ ′ ′ ′ ′ = = = Using Equation (9.48), the equation of the ellipsoid of inertia is then 2 2 2 1 x y z I x I y I z + + = or ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 1 1 1 4 4 1 12 12 12 m b c x m a c y m a b z + + + + + = For the ellipsoid to be a sphere, the coefficients must be equal.
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