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

Statics and Mechanics of Materials (5th Edition)

5th Edition

ISBN: 9780134382593

Author: Russell C. Hibbeler

Publisher: PEARSON

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Center of Gravity and Centroid

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 18P

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Find the centroid x¯ 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 x ¯ of the parabolic shaded area.

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Chapter 6 Solutions