Explanation Refer Fig. P5.21 and Fig. 1. 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. x 2 + y 2 = a 2 r 2 = a 2 − x 2 Here, the y component is equal to the radius, a is the radius of the hemisphere, x is the centroid of the element and d x is the thickness of the element. Write the expression for the volume of the element. d V = π r 2 d x = π ( a 2 − x 2 ) d x (I) 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: Refer Fig.2 Calculate the volume of component 1. V 1 = ∫ 0 a / 2 d V = ∫ 0 a / 2 π ( a 2 − x 2 ) d x = π [ a 2 x − x 3 3 ] 0 a / 2 = 11 24 π a 3 Calculate ∫ 1 x E L ¯ d V for component 1. ∫ 1 x E L ¯ d V = ∫ 0 a / 2 x [ π ( a 2 − x 2 ) d x ] = π [ a 2 x 2 2 − x 4 4 ] 0 a / 2 = 7 64 π a 4 Substitute 7 64 π a 4 for ∫ 1 x E L ¯ d V and 11 24 π a 3 for V 1 in equation (II). x 1 ¯ = 7 64 π a 4 11 24 π a 3 = 21 88 a Thus, the value of x ¯ for the first component is 21 88 a

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

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

Publisher: McGraw-Hill Education

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Chapter 5.4, Problem 5.122P

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To determine the value of x¯ for the two volumes.

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