Explanation Given information: The length of the shaded area is 1 m. The height of the shaded area is 1 m. Calculation: Show the area of the differential element as in Figure 1. Using Figure 1, Express the parabolic value. y = x 3 x = y 1 / 3 Compute the area of the element. d A = x d y Substitute y 1 / 3 for x . d A = y 1 / 3 d y Compute the centroid of the differential element along the x- axis using the formula. x ˜ = x 2 (1) Substitute y 1 / 3 for x in Equation (1). x ˜ = y 1 / 3 2 Determine the location of the centre of gravity of the homogeneous rod along the x -axis ( x ¯ ) using the relation. x ¯ = ∫ A x ˜ d A ∫ A d A (2) Conclusion: Apply the limits from 0 to 1 m, substitute y 1 / 3 2 for x ˜ and y 1 / 3 d y for dA in Equation (2). x ¯ = ∫ 0 1 m ( y 1 / 3 2 ) y 1 / 3 d y ∫ 0 1 m y 1 / 3 d y = 1 2 ∫ 0 1 m y 2 / 3 d y ∫ 0 1 m y 1 / 3 d y (3) Calculate the numerator part from Equation (3). 1 2 ∫ 0 1 m y 2 / 3 d y = 1 2 [ y 2 3 + 1 2 3 + 1 ] = 1 2 [ y 5 / 3 5 / 3 ] 0 1 m = 3 10 [ y 5 / 3 ] 0 1 m = 3 10 m 5 / 3 Calculate the denominator part from Equation (3). ∫ 0 1 m y 1 / 3 d y = y 1 3 + 1 ( 1 3 + 1 ) = 3 4 [ y 4 3 ] 0 1 m = 3 4 m 4 / 3 Substitute 3 4 m 4 / 3 for ∫ 0 1 m y 1 / 3 d y and 3 10 m 5 / 3 for 1 2 ∫ 0 1 m y 2 / 3 d y in Equation (3)

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 1FP

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Find the centroid (x¯, y¯) of the parabolic-shaded area.

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Chapter 6.1, Problem 1PP

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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 ¯ , y ¯ ) of the parabolic-shaded area.

Solution Summary: The author calculates the area of the differential element using the formula.

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Find the centroid (x¯, y¯) of the parabolic-shaded area.

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Chapter 6 Solutions