Explanation Given: The element has a thickness of dy and a height of y Show the volume of the thin disc differential element as in Figure 1. Express the parabolic value. y 2 b 2 + z 2 a 2 = 1 (I) Here, the length and height of the ellipsoid of evolution are b and a , respectively. Compute the volume of the element. d V = π z 2 d z (II) Rearrange Equation (I) to calculate z 2 . z 2 a 2 = ( 1 − y 2 b 2 ) z 2 = a 2 ( 1 − y 2 b 2 ) Substitute a 2 ( 1 − y 2 b 2 ) for y 2 in Equation (II). d V = π a 2 ( 1 − y 2 b 2 ) d y Compute the centroid of the differential element along the y axis using the formula. y ˜ = y Determine the location of the centre of gravity of the homogeneous rod along the y axis ( y ¯ ) using the relation, y ¯ = ∫ V y ˜ d V ∫ V d V (III) Conclusion: Apply the limits from 0 to b , substitute y for y ˜ and π a 2 ( 1 − y 2 b 2 ) d y for dV in Equation (III). y ¯ = ∫ 0 b y ( π a 2 ( 1 − y 2 b 2 ) d y ) ∫ 0 b π a 2 ( 1 − y 2 b 2 ) d y (IV) Calculate the numerator part from Equation (IV)

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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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 9.1, Problem 46P

To determine

The centroids x¯,y¯, and z¯ of the ellipsoid of revolution.

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Chapter 9.1, Problem 45P

Chapter 9.1, Problem 47P

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Chapter 9 Solutions

Engineering Mechanics: Statics & Dynamics (14th Edition)

Ch. 9.1 - In each case, use the element shown and specify...Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid (x,y) of the shaded area....Ch. 9.1 - Determine the centroid of the shaded area. Prob....Ch. 9.1 - Locate the center of mass x of the straight rod if...Ch. 9.1 - Locate the centroid of the homogeneous solid...Ch. 9.1 - Locate the centroid z of the homogeneous solid...Ch. 9.1 - Locate the center of mass of the homogeneous rod...Ch. 9.1 - Prob. 2P Ch. 9.1 - Locate the center of gravity x of the homogeneous...

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The centroids x ¯ , y ¯ , and z ¯ of the ellipsoid of revolution.

Solution Summary: The author shows the parabolic value of the ellipsoid of evolution.

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The centroids x¯,y¯, and z¯ of the ellipsoid of revolution.

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Chapter 9 Solutions