Explanation Given: The density of the material is ρ . Show the intersection of body at the arbitrary point ( x , y , z ) as in Figure (1). Conclusion: From Figure (1), Express the radius of the disk element. r = y (I) Here, the centroid of differential element along the y -axis is y . Compute the centroid of the differential element along the y -axis. z = h a 2 ( a 2 − y 2 ) y = a 1 − z / h Substitute a 1 − z / h for y in Equation (I). r = y = a 1 − z / h Express the mass of the disk element. d m = ρ d V (II) Here, the density of the material is ρ , and the volume of the body is V . Substitute π r 2 d z for d V in Equation (II). d m = ρ π r 2 d z (III) Substitute a 1 − z / h for r in Equation (III). d m = ρ π ( a 1 − z / h ) 2 d z = ρ π a 2 ( 1 − z / h ) d z Calculate the mass moment of inertia of this element about the z- axis. d I z = 1 2 d m r 2 (IV) Substitute ρ π r 2 d z for d m in Equation (IV). d I z = 1 2 ( ρ π r 2 d z ) r 2 = 1 2 ρ π r 4 d z (V) Substitute a 1 − z / h for r in Equation (V). d I z = 1 2 ρ π [ a 1 − z / h ] 4 d z = ρ π a 4 2 ( 1 − z / h ) 2 d z Express the mass of the solid. m = ∫ 0 h / 2 d m (VI) Substitute ρ π a 2 ( 1 − z / h ) d z for d m in Equation (VI)

13th Edition

ISBN: 9780132915540

Author: Russell C. Hibbeler

Publisher: Prentice Hall

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Chapter 10.8, Problem 90P

To determine

The mass moment of inertia Iz of the solid formed by revolving the shaded area around the z-axis in terms of the mass m of the solid.

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Chapter 10.8, Problem 89P

Chapter 10.8, Problem 91P

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Chapter 10 Solutions

Engineering Mechanics: Statics

Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Determine the moment of inertia of the shaded area...Ch. 10.3 - Prob. 1P Ch. 10.3 - Prob. 2P Ch. 10.3 - Prob. 3P Ch. 10.3 - Prob. 4P Ch. 10.3 - Prob. 5P Ch. 10.3 - Prob. 6P

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The mass moment of inertia I z of the solid formed by revolving the shaded area around the z -axis in terms of the mass m of the solid.

The mass moment of inertia Iz of the solid formed by revolving the shaded area around the z-axis in terms of the mass m of the solid.

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Chapter 10 Solutions