Explanation Refer Fig. P5.128 and Fig. 1. Because of the symmetry of the diagram, y ¯ = 0 and z ¯ = 0 . Let the element of volume of the disk be of radius r and thickness dx . Find the equation for radius from the equation of the generating curve. y = b sin π x 2 a r = b sin π x 2 a Here, h is the semi major axis, a is the semi minor axis and y is the distance along y axis. Write the expression for the volume of the shaded region. d V = π r 2 d x = π b 2 sin 2 π x 2 a d x (I) Write the expression for x ¯ of the volume. x ¯ V = ∫ x ¯ E L d V x ¯ = ∫ x ¯ E L d V V (II) Here, x ¯ is the x coordinate of the centroid and x ¯ E L is the x coordinate of the centroid of the element. Conclusion: Calculate the volume obtained by rotating the shaded area using equation (I). V = ∫ a 2 a d V = ∫ a 2 a π b 2 sin 2 π x 2 a d x = π b 2 [ x 2 − sin π x 2 a 2 π a ] a 2 a = π b 2 [ ( 2 a 2 ) − ( a 2 ) ] = 1 2 π a b 2 When d V = sin 2 π x 2 a , the volume is V = x 2 − sin π x 2 a 2 π a . Calculate ∫ x ¯ E L d V for the volume obtained by rotating the shaded area

Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259638091

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

Publisher: McGraw-Hill Education

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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 5.128P

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To locate the centroid of the volume obtained by rotating the shaded area.

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Chapter 5.4, Problem 5.127P

Chapter 5.4, Problem 5.129P

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To locate the centroid of the volume obtained by rotating the shaded area.

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