Explanation Given information: The height of the shaded area is 2 m . The width of the shaded area is 1 m . Calculation: Show the area of the differential element parallel to the x axis as in Figure 1. From the Figure 1, Compute the area of the differential element parallel to the x axis as shown with shaded area from the Figure 1. d A = x d y Here, the centroid of the differential element along x axis is x and the area of the differential element is d A . Compute the centroid of the differential element along x and y axis using the formula. x ˜ = x 2 Here, the centroid of the differential element along x axis is x ˜ . Compute the moment of inertia of the differential element parallel to the x axis about the y axis. d I y = d I ¯ y ′ + d A x ˜ 2 (1) Here, the first integral of the moment of inertia of the area about the centroid axis is I ¯ y ′ and the centroid of the differential element along x axis is x ˜ . Substitute 1 12 ( d y ) x 3 for d I ¯ y ′ , x d y for d A , and x / 2 for x ˜ in Equation (1). d I y = 1 12 ( d y ) x 3 + x d y ( x / 2 ) 2 = [ x 3 12 + x x 2 4 ] ( d y ) = [ x 3 12 + x 3 4 ] ( d y ) = x 3 3 d y (2) Solve the given equation for centroid of the differential element along y axis. y 2 = 1 − x x = 1 − y 2 Substitute 1 − y 2 for x in Equation (2). d I y = ( 1 − y 2 ) 3 3 d y = 1 3 ( − y 6 + 3 y 4 − 3 y 2 + 1 ) d y Conclusion: Express the moment of inertia for the shaded area about the y axis

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 6.4, Problem 75P

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Find the moment of inertia for the shaded area about the y axis.

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Chapter 6.4, Problem 74P

Chapter 6.4, Problem 76P

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Chapter 6 Solutions

Statics and Mechanics of Materials (5th Edition)

Ch. 6.1 - In each case, use the element shown and specify...Ch. 6.1 - Prob. 1FP Ch. 6.1 - Determine the centroid (x,y) of the area. Prob....Ch. 6.1 - Determine the centroid y of the area. Prob. F63 Ch. 6.1 - Locate the center of gravity x of the straight rod...Ch. 6.1 - Prob. 5FP Ch. 6.1 - Locate the centroid z of the homogeneous solid...Ch. 6.1 - Locate the centroid x of the area. Prob. 61 Ch. 6.1 - Locate the centroid of the area. Prob. 62 Ch. 6.1 - Locate the centroid x of the area. Probs. 63/4

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