Concept explainers
9.75 through 9.78 Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Fig. P9.75
9.82 Determine the moments of inertia and the product of inertia of the area of Prob. 9.75 with respect to new centroidal axes obtained by rotating the x and y axes 45° clockwise.
Find the moment of inertia and product of inertia with respect new centroid axes obtained x and y axes
Answer to Problem 9.82P
The moment of inertia with respect new centroid axes obtained x axes
The moment of inertia with respect new centroid axes obtained y axes
The product of inertia with respect new centroid axes obtained x axes
Explanation of Solution
Calculation:
Refer to problem 9.75.
The product of inertia of the area with respect to x and y axes by using direct parallel axis theorem is
Sketch the rectangular section as shown in Figure 1.
Find the moment of inertia
Here,
Substitute
Find the moment of inertia
Here,
Substitute
Find the total moment of inertia
Substitute
Find the moment of inertia
Substitute
Find the moment of inertia
Substitute
Find the total moment of inertia
Substitute
Find the value of
Find the value of
Find the moment of inertia with respect new centroid axes obtained x axes
Refer to Equation 9.18 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia with respect new centroid axes obtained x axes
Find the moment of inertia with respect new centroid axes obtained y axes
Refer to Equation 9.19 in section 9.3B in the textbook.
Substitute
Thus, the moment of inertia with respect new centroid axes obtained y axes
Find the product of inertia with respect new centroid axes obtained x and y axes
Substitute
Thus, the product of inertia with respect new centroid axes obtained x axes
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Chapter 9 Solutions
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