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. Refer Figure 1, Express the parabolic values, y 1 and y 2 . y 1 = h − h a x y 2 = h − h a n x n Compute the area of the element. d A = ( y 2 − y 1 ) d x Here, the area of element is dA . Substitute h − h a x for y 1 and h − h a n x n for y 2 . d A = ( h − h a n x n − ( h − h a x ) ) d x = ( h a x − h a n x n ) d x Compute the centroid of the differential element along the x- axis using the formula. x ˜ = x 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 (1) Conclusion: Apply the limits from 0 to a , and then substitute x for x ˜ and ( h a x − h a n x n ) d x for dA in Equation (1). x ¯ = ∫ 0 a x ( h a x − h a n x n ) d x ∫ 0 a ( h a x − h a n x n ) d x (2) Calculate the numerator part from Equation (2). ∫ 0 a x ( h a x − h a n x n ) d x = ∫ 0 a ( − h a n x n ⋅ x + h a x ⋅ x ) d x = − ∫ 0 a − h a n x n x + h a x x d x = [ − h a n ⋅ x n + 2 n + 2 + h a ⋅ x 3 3 ] 0 a = h ( n − 1 ) a 2 3 ( n + 2 ) − ( − h ⋅ 0 n + 2 a − n n + 2 ) = h ( n − 1 ) a 2 3 ( n + 2 ) Calculate the denominator part from 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 29P

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Find the centroid x¯ of the 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 shaded area.

Solution Summary: The author calculates the area of the differential element as shown in Figure 1.

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Find the centroid x¯ of the shaded area.

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Chapter 6 Solutions