Explanation Given: The element has a thickness of dz and a height of z . Show the volume of the hemispherical differential element as in Figure 1. Express the parabolic equation. y 2 + z 2 = a 2 (I) Compute the volume of the element. d V = π y 2 d z (II) Re-arrange Equation (I). y 2 = a 2 − z 2 Substitute a 2 − z 2 for y 2 in Equation (II). d V = π ( a 2 − z 2 ) d z Compute the centroid of the differential element along the z- axis using the formula. z ˜ = z Determine the location of the centre of gravity of the hemisphere along the z -axis ( z ¯ ) using the relation. z ¯ = ∫ V z ˜ d V ∫ V d V (III) Conclusion: Apply the limits from 0 to a ; substitute z for z ˜ and π ( a 2 − z 2 ) d z for dV in Equation (III)

15th Edition

ISBN: 9780137569830

Author: HIBBELER

Publisher: VST

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Chapter 9, Problem 3RP

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The centroid z¯ of the hemisphere.

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The centroid z ¯ of the hemisphere.

Solution Summary: The author shows the volume of the hemispherical differential element as in Figure 1.

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The centroid z¯ of the hemisphere.

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