Explanation The figure 1 shows the center of mass of solid hemisphere. Consider elemental discs as shown in the figure. The mass of the elemental disc is, d m = ρ d V (I) Here, d m is the elemental mass of the disc, ρ is the density of the disc, and d V is the change in volume of the elemental disc. The relation between mass and density is, ρ = M V (II) Here, M is the mass of the solid hemisphere and V is the volume of the hemisphere. The volume of the hemisphere is, V = 2 3 π R 3 (III) Here, R is the radius of the hemisphere. The change in volume of the elemental disc is, d V = π r 2 d y (IV) Here, r is the radius of the elemental disc and d y is the thickness of the disc. The radius of the elemental disc is, r = R 2 − y 2 (V) The center of mass of the solid hemisphere is, y c m = ∫ d m ⋅ y ∫ d m (VI) Conclusion: Substitute the equations (II) and (III) in equation (I). d m = M V d V = M 2 3 π R 3 d V Substitute the equations (IV) and (V) in above relation

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ISBN: 9780135272992

Author: Wolfson

Publisher: VST

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Chapter 9, Problem 55P

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Chapter 9, Problem 54P

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