PROBLEM 9.111 Using Mohr’s circle, prove that the expression 2xyxyIII′′′′−is independent of the orientation of the x′and y′axes, where xI′, yI′, and x yIrepresent 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 Aand origin Oof a rectangular coordinate system, the values of aveIand Rare the same for all orientations of the coordinate axes. Shown below is a Mohr’s circle, with the moments of inertia, xI′and ,yI′and the product of inertia. ,x yIhaving been computed for an arbitrary orientation of the
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