Explanation Refer Fig. P5.126 and Fig. 1. Because of the symmetry of the diagram, y ¯ = 0 and z ¯ = 0 . Write the given equation for y . y = k ( X − h ) 2 Find the expression for k when x = 0 and y = a . a = k ( − h ) 2 k = a h 2 Let the element of volume of the disk be of radius r and thickness dx . Write the expression for the volume of the element. d V = π r 2 d x = π ( a h 2 ( x − h ) 2 ) 2 d x = π a 2 h 4 ( x − h ) 4 d x (I) Here, h is the radius of the element, x is the centroid of the element and d x is the thickness of the element. Find the centroid of the element. x E L ¯ = x Here, x E L ¯ is the centroid of the element. Write the expression for the x ¯ for the volume. x ¯ V = ∫ x E L ¯ d V x ¯ = ∫ x E L ¯ d V V (II) Conclusion: Calculate the volume obtained by rotating the shaded area. V = ∫ 0 h d V = ∫ 0 h π a 2 h 4 ( x − h ) 4 d x = π 5 a 2 h 4 [ ( x − h ) 4 ] 0 h = 1 5 π a 2 h Calculate ∫ x E L ¯ d V for the volume obtained by rotating the shaded area

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ISBN: 9781260912814

Author: BEER

Publisher: MCG

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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 5.4, Problem 5.126P

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To locate the centroid of the volume.

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Chapter 5.4, Problem 5.125P

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