COSMOS Chapter 9 Solution 110 | Michigan State University - Edubirdie

University:
Michigan State University
Course:
ERG 160 | Success in Science, Technology, Engineering, and Mathematics
Academic year:

2024
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1
Author:

imophulebf10

COSMOS: Complete Online Solutions Manual Organization System Chapter 9, Solution 110. Consider the regular pentagon shown, with centroidal axes x and y. Because the y axis is an axis of symmetry, it follows that I xy = 0. Since I xy = 0, the x and y axes must be principal axes. Assuming I x = I max and I y = I min , the Mohr’s circle is then drawn as shown. Now rotate the coordinate axes through an angle α as shown; the resulting moments of inertia, I x′ and I y′ , and product of inertia, I x′y′ , are indicated on the Mohr’s circle. However, the x′ axis is an axis of symmetry, which implies I x′y′ = 0. For this to be possible on the Mohr’s circle, the radius R must be equal to zero (thus, the circle degenerates into a point). With R = 0, it immediately follows that (a) I x = I y = I x′ = I y′ = I ave (for all moments of inertia with respect to an axis through C) (b) I xy = I x′y′ = 0 (for all products of inertia with respect to all pairs of rectangular axes with origin at C)

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