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 = h − h a 2 y 2 r 2 = a 2 h ( h − x ) Here, the y component is equal to the radius, h is the radius of the paraboloid, 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 h ( h − x ) 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 h / 2 d V = ∫ 0 h / 2 π a 2 h ( h − x ) d x = π ( a 2 h ) [ h x − x 2 2 ] 0 h / 2 = 3 8 π a 2 h Calculate ∫ 1 x E L ¯ d V for component 1. ∫ 1 x E L ¯ d V = ∫ 0 h / 2 x [ π a 2 h ( h − x ) d x ] = π ( a 2 h ) [ h x 2 2 − x 3 3 ] 0 h / 2 = 1 12 π a 2 h 2 Substitute 1 12 π a 2 h 2 for ∫ 1 x E L ¯ d V and 3 8 π a 2 h for V 1 in equation (II). x 1 ¯ = 1 12 π a 2 h 2 3 8 π a 2 h = 2 9 h Thus, the value of x ¯ for the first component is 2 9 h

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Technical drawing has been around since ancient times. As in the ancient Renaissance, complicated drawings were created, including Leonardo da Vinci's drawing. Modern engineering drawing, including its specific orthographic projection with scale conventi…

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Chapter 5.4, Problem 5.124P

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

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