Explanation Refer Figure 1. Write an expression for the curve. y = h ( 1 + x L − 2 x 2 L 2 ) (I) Here, h is the height of the curve and k is a constant. Consider an element in the area as shown in figure 1. Write an expression to determine the x component of the centroid of the element. x ¯ E L = x (II) Here, x ¯ E L is the x component of the centroid of the element. Write an expression to determine the y component of the centroid of the element. y ¯ E L = 1 2 y (III) Here, y ¯ E L is the y component of the centroid of the element. Write an expression to calculate the area of the element. d A = y d x (IV) Here, d A is the area of the element. Write an expression to calculate the area of the entire section. x ¯ = ∫ x E L d A A (V) Here, x ¯ is the x component of the centroid of the section. Write an expression to calculate the area of the entire section. y ¯ = ∫ y E L d A A (VI) Here, y ¯ is the y component of the centroid of the section. Write an expression to calculate the area of the section. A = ∫ d A (VII) Here, A is the area of the section. Conclusion: Substitute equation (I), (II), (IV) and (VII) in equation (V) to find x ¯ . x ¯ = ∫ x ( y d x ) ∫ d A = ∫ 0 L x ( y d x ) ∫ 0 L y d x = ∫ 0 L x ( h ( 1 + x L − 2 x 2 L 2 ) ) d x ∫ 0 L ( h ( 1 + x L − 2 x 2 L 2 ) ) d x = h ( x 2 2 + 1 3 x 3 L − 2 4 x 4 L 2 ) | 0 L h ( x + x 2 2 L − 2 3 x 3 L 3 ) | 0 L = ( 1 3 ) h L 2 ( 5 6 ) h L = 2 5 L Substitute equation (I), (III), (IV) and (VII) in equation (V) to find y ¯

Connect 1 Semester Access Card for Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9781259639272

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 5, Problem 5.141RP

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The centroid of the area.

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Chapter 5, Problem 5.140RP

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Connect 1 Semester Access Card for Vector Mechanics for Engineers: Statics and Dynamics

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The centroid of the area.

Solution Summary: The author illustrates how to calculate the area of an element. Write an expression to determine the x component of the centroid.

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