Explanation Refer Fig. P5.130 and Fig. 1. A horizontal slice element is chosen with a thickness dy . Write the equation for area of the base of the pyramid. A base = k b 2 (I) Here, k = k ( N ) and b is the length of the base. Write the equation from the triangle in figure 1. s b = h − y h s = b h ( h − y ) (II) Here, s is the length of the element in the slice, y is the distance from the base of the triangle to the element and h is the height of the section. Write the expression for the volume of the slice using equation (I). d V = A slice d y = k s 2 d y = k b 2 h 2 ( h − y ) 2 d y (III) Write the expression for y ¯ of the volume. y ¯ V = ∫ y ¯ E L d V y ¯ = ∫ y ¯ E L d V V (IV) Here, y ¯ is the y coordinate of the centroid and y ¯ E L is the y coordinate of the centroid of the element. Write the expression for y ¯ of the element. y ¯ E L = y Conclusion: Calculate the volume using equation (III). V = ∫ 0 h d V = ∫ 0 h k b 2 h 2 ( h − y ) 2 d y = k b 2 h 2 [ − 1 3 ( h − y ) 3 ] 0 h = 1 3 k b 2 h (V) Calculate ∫ y ¯ 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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Chapter 5.4, Problem 5.130P

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To show that for a regular pyramid of height h, the centroid of the volume of the pyramid is located at a distance h/4 above the base.

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Chapter 5.4, Problem 5.129P

Chapter 5.4, Problem 5.131P

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Vector Mechanics for Engineers: Statics and Dynamics

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To show that for a regular pyramid of height h , the centroid of the volume of the pyramid is located at a distance h / 4 above the base.

Solution Summary: The author illustrates how the centroid of the volume of a regular pyramid is located at h/4 above the base.

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To show that for a regular pyramid of height h, the centroid of the volume of the pyramid is located at a distance h/4 above the base.

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