Explanation Refer Fig. P5.131 and Fig. 1. An element of area dA of the shell is obtained from the shell by cutting the shell. The planes must be parallel to the xy plane. Write the equation for r from the element. r = R sin θ (I) Here, r is the radius of the element and R is the radius of the shell. Write the expression for the area of the shell using equation (I). d A = ( π r ) ( R d θ ) = π R 2 sin θ d θ (II) Write the expression for y ¯ of the area. y ¯ A = ∫ y ¯ E L d A y ¯ = ∫ y ¯ E L d A A (III) 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 = − 2 r π = − 2 R π sin θ Conclusion: Calculate the area using equation (II). A = ∫ 0 π / 2 d A = ∫ 0 π / 2 π R 2 sin θ d θ = π R 2 [ − cos θ ] 0 π / 2 = π R 2 (IV) Calculate ∫ y ¯ E L d A for the area obtained by rotating the shaded area

Loose Leaf for Vector Mechanics for Engineers: Statics and Dynamics

12th Edition

ISBN: 9781259977206

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

Publisher: McGraw-Hill Education

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Chapter 5.4, Problem 5.131P

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The location of the centroid of one-half of a thin, uniform hemispherical shell of radius R.

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Chapter 5.4, Problem 5.130P

Chapter 5.4, Problem 5.132P

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The location of the centroid of one-half of a thin, uniform hemispherical shell of radius R .

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The location of the centroid of one-half of a thin, uniform hemispherical shell of radius R.

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