
Concept explainers
Calculate the moment of intertia with respect to the X-X centroidal axes for the areas shown.
a.

Moment of inertia with respect to X-X centroidal axes.
Answer to Problem 8.1P
Explanation of Solution
Given:
Concept Used: Since the shape is symmetrical about X and Y axis, so the centroidal axis lies in the center of the cross-section. The moment of inertia of the ring will be equal to the moment of inertia of the outer rectangle minus the moment of inertia on the inner circle.
Calculation: Moment of inertia of the Square about the centroid is
Moment of inertia of the Circle about the centroid is
Total moment of inertia of the shape is
Conclusion: Moment of inertia with respect to X-X centroidal axes
b.

Moment of inertia with respect to X-X centroidal axes.
Answer to Problem 8.1P
Explanation of Solution
Given:
Concept Used: Since the shape is symmetrical about X and Y axis, so the centroidal axis lies in the center of the cross-section. The moment of inertia of the ring will be equal to the moment of inertia of the outer circle
Calculation: Total moment of inertia of the circular ring is
Conclusion:
Moment of inertia of the circular ring is
c.

Moment of inertia with respect to X-X centroidal axes.
Answer to Problem 8.1P
Explanation of Solution
Given:
Concept Used: Since the shape is symmetrical about X and Y axis so the centroidal axis lies in the center of the cross-section. The moment of inertia of the ring will be equal to the moment of inertia of the outer square minus the moment of inertia on the inner rectangle.
Calculation: Moment of inertia of the Outer square is
Moment of inertia of the Inner rectangle is
Total moment of inertia of the cross-section is
Conclusion: Moment of inertia with respect to X-X centroidal axes
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