135_Problem CHAPTER 9

135_Problem CHAPTER 9 - PROBLEM 9.111 Using Mohr's circle,...

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PROBLEM 9.111 Using Mohr’s circle, prove that the expression 2 x yx y I II ′′ is independent of the orientation of the x and y axes, where x I , y I , and x y I represent the moments and product of inertia, respectively, of a given area with respect to a pair of rectangular axes x and y through a given point O . Also show that the given expression is equal to the square of the length of the tangent drawn from the origin of the coordinate system to Mohr’s circle. SOLUTION First observe that for a given area A and origin O of a rectangular coordinate system, the values of ave I and R are the same for all orientations of the coordinate axes. Shown below is a Mohr’s circle, with the moments of inertia, x I and , y I and the product of inertia. , x y I having been computed for an arbitrary orientation of the
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