Explanation Given information: The length of the shaded area is 8 in. The height of the shaded area is 4 in. Calculation: Show the area of the differential element as in Figure 1. Using Figure 1, Express the parabolic value of y . y = x 2 / 3 Compute the area of the element. d A = y d x Here, the area of element is dA . Substitute x 2 / 3 for y . d A = x 2 / 3 d x Compute the centroid of the differential element along the y axis using the formula. y ˜ = y 2 Substitute x 2 / 3 for y . y ˜ = x 2 / 3 2 Determine the location of the center of gravity of the homogeneous rod along the y axis ( y ¯ ) using the relation, y ¯ = ∫ A y ˜ d A ∫ A d A (1) Conclusion: Substitute x 2 / 3 2 for y ˜ and x 2 / 3 d x for dA in Equation (1). y ¯ = ∫ A ( x 2 / 3 2 ) ( x 2 / 3 d x ) ∫ A ( x 2 / 3 ) d x (2) Apply the limits from 0 to 8 in. in Equation (2) and calculate the numerator part. ∫ 0 8 in . ( x 2 / 3 2 ) ( x 2 / 3 ) d x = 1 2 ⋅ ∫ 0 8 in . x 2 / 3 x 2 / 3 d x = 1 2 ⋅ ∫ 0 8 x 4 / 3 d x = 1 2 ⋅ x 4 3 + 1 ( 4 3 + 1 ) = [ 3 14 x 7 / 3 ] 0 8 in

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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The term center of gravity indicates the position or location of a point about which the whole weight of a body or system is supposed to be concentrated, which means the whole weight of the body or system works through this point. It is closely related t…

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Chapter 6.1, Problem 11P

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Find the centroid y¯ of the parabolic shaded area.

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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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Find the centroid y ¯ of the parabolic shaded area.

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