Explanation Given: The height of the shaded area is b . The width of the shaded area is a . Show the area of the differential element parallel to the y -axis as in Figure (1). Conclusion: From Figure 1, Compute the area of the differential element parallel to the y -axis as shown with shaded area in Figure 1. d A = y d x (I) Here, the centroid of the differential element along the y - axis is y . Substitute b a n x n for y in Equation (I). d A = b a n x n d x Compute the coordinate of the centroid of the differential element along the x -axis. x ˜ = x Compute the coordinate of the centroid of the differential element along the y -axis. y ˜ = y 2 Compute the product of inertia of the differential element with respect to the x and y axes. d I x y = d I ¯ x ′ y ′ + d A x ˜ y ˜ (II) Here, first integral of the moment of inertia of the area about the centroidal x and y axes axis is I ¯ x ′ y ′ . Substitute 0 for d I ¯ x ′ y ′ , b a n x n d x for d A , x for x ˜ , and y 2 for y ˜ in Equation (II). d I x y = 0 + x [ y 2 ] [ b a n x n d x ] = x [ y 2 ] [ b a n x n d x ] Compute the product of inertia of the shaded area with respect to the x and y axes. I x y = ∫ 0 a d I x y (III) Substitute x [ y 2 ] [ b a n x n d x ] for d I x y in Equation (III)

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Question

Chapter 10.7, Problem 57P

To determine

The product of inertia for the shaded area with respect to the x and y axes.

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Chapter 10.7, Problem 56P

Chapter 10.7, Problem 58P

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Chapter 10 Solutions

Engineering Mechanics: Statics

Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Prob. 1P Ch. 10.3 - Prob. 2P Ch. 10.3 - Prob. 3P Ch. 10.3 - Prob. 4P Ch. 10.3 - Prob. 5P Ch. 10.3 - Prob. 6P

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