Explanation Given information: The length of the shaded area is 4 ft. The height of the shaded area is 16 ft. Calculation: Show the area of the differential element as in Figure 1. Using Figure 1, Express the parabolic value of y. y = ( 4 − x 1 / 2 ) 2 Compute the area of the element. d A = y d x Here, the area of element is dA . Substitute ( 4 − x 1 / 2 ) 2 for y , d A = ( 4 − x 1 / 2 ) 2 d x = ( x − 8 x 1 / 2 + 16 ) d x Compute the centroid of the differential element along the y axis using the formula. y ˜ = y 2 Substitute ( 4 − x 1 / 2 ) 2 for y . y ˜ = ( 4 − x 1 / 2 ) 2 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: Apply the limits from 0 ft to 4 ft and substitute ( 4 − x 1 / 2 ) 2 2 for y ˜ and ( x − 8 x 1 / 2 + 16 ) d x for dA in Equation (1). y ¯ = ∫ 0 4 ft ( ( 4 − x 1 / 2 ) 2 2 ) ( x − 8 x 1 / 2 + 16 ) d x ∫ 0 4 ft ( x − 8 x 1 / 2 + 16 ) d x (2) Calculate the numerator part from Equation (2). ∫ 0 4 ft ( ( 4 − x 1 / 2 ) 2 2 ) ( x − 8 x 1 / 2 + 16 ) d x = ∫ 0 4 ft ( 1 2 x 2 − 8 x 3 / 2 + 48 x − 128 x 1 / 2 + 128 ) d x = [

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 16P

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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.

Solution Summary: The author explains how to find the centroid of the parabolic shaded area using the formula.

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