Explanation Given: The length of the shaded area is b . The height of the shaded area is h . Show the area of the differential element as in Figure (1). Using Figure 1, Express the parabolic equation. y = h b 3 x 3 (I) Compute the area of the element. d A = ( x ) d y The product of inertia of this element with respect to the x and y axes is determined using the parallel-axis theorem. d I x y = d I ¯ x ′ y ′ + d A x ˜ y ˜ (II) Here, the centroid of the element is x ˜ and y ˜ axes and the set of axes passing through the centroid of the area is x ′ and y ′ , respectively. Compute the value of x ˜ and y ˜ . x ˜ = x y ˜ = y 2 Due to symmetry, d I ¯ x ′ y ′ = 0 . Substitute 0 for d I ¯ x ′ y ′ , x for x ˜ , y / 2 for y ˜ , and ( x ) d y for dA in Equation (II). d I x y = 0 + x d y ( x ) y 2 = x 2 y 2 d y Determine the product of inertia for the shaded area with respect to x and y axes. I x y = ∫ d I x y (III) Conclusion: Apply the limits from 0 to h and substitute x 2 y 2 d y 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 55P

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 54P

Chapter 10.7, Problem 56P

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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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