Explanation Given: The height of the shaded area is 200 mm . The width of the shaded area is 200 mm . 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 from Figure 1. d A = y d x (I) Here, the area of the differential element is d A . Compute the centroid of the differential element along the y -axis. y 2 = x y = x 1 / 2 Substitute x 1 / 2 for y in Equation (I). d A = x 1 / 2 d x Compute the coordinate of the centroid of the differential element along the x -axis. x ˜ c = x Compute the coordinate of the centroid of the differential element along the y -axis. y ˜ c = y 2 (II) Substitute x 1 / 2 for y in Equation (II). y ˜ c = x 1 / 2 2 = 1 2 x 1 / 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 ˜ c y ˜ c (III) Here, the first integral of the moment of inertia of the area about the centroidal x and y axes is I ¯ x ′ y ′ . Substitute 0 for d I ¯ x ′ y ′ , x 1 / 2 d x for d A , x for x ˜ c , and 1 2 x 1 / 2 for y ˜ c in Equation (III). d I x y = 0 + ( x 1 / 2 d x ) ( x ) ( 1 2 x 1 / 2 ) = 1 2 x 2 d x Compute the product of inertia of the shaded area with respect to x and y axes. I x y = ∫ 0 4 m d I x y (IV) Substitute 1 2 x 2 d x for d I x y in Equation (IV). I x y = ∫ 0 4 m 1 2 x 2 d x = 1 2 ∫ 0 4 m x 2 d x = [ 1 2 x 2 + 1 2 + 1 ] | 0 4 m = [ x 3 6 ] | 0 4 m = ( 4 ) 3 6 − ( 0 ) 3 6 = 10

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provi…

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Chapter 10.7, Problem 57P

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The product of inertia of the shaded area with respect to the x and y axes and then find the product of inertia of the area with respect to the centroidal x′ and y′ axes.

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Chapter 10.7, Problem 56P

Chapter 10.7, Problem 58P

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Engineering Mechanics: Statics & Dynamics (14th Edition)

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The product of inertia of the shaded area with respect to the x and y axes and then find the product of inertia of the area with respect to the centroidal x ′ and y ′ axes.

Solution Summary: The author shows the area of the differential element parallel to the y -axis as shown in Figure 1.

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The product of inertia of the shaded area with respect to the x and y axes and then find the product of inertia of the area with respect to the centroidal x′ and y′ axes.

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Chapter 10 Solutions