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. 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 y axis using the formula. y ˜ = y 1 + ( y 2 − y 1 2 ) Substitute h − h a x for y 1 and h − h a n x n for y 2 . y ˜ = h − h a x + ( h − h a n x n − ( h − h a x ) 2 ) = 1 2 ( 2 h − h a n x n − h a x ) Determine the location of the centre 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 to a and substitute 1 2 ( 2 h − h a n x n − h a x ) for y ˜ and ( h a x − h a n x n ) d x for dA in Equation (1). y ¯ = ∫ 0 a 1 2 ( 2 h − h a n x n − h 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)

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 30P

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Find the centroid y¯ of the shaded area.

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Chapter 6.1, Problem 29P

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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 shaded area.

Solution Summary: The author calculates the centroid of the differential element along the y axis using the formula.

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Find the centroid y¯ of the shaded area.

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Chapter 6 Solutions