Explanation Given: The radius of the sphere is r . The constant density of the sphere is ρ . Explanation: Show the intersection of the sphere at the arbitrary point ( x , y ) as in Figure (1). Calculation: From Figure 1, Calculate the differential mass of the sphere. d m = ρ d V (I) Here, the density of the material is ρ and the volume of the sphere is V . Substitute π y 2 d x for d V in Equation (I). d m = ρ ( π y 2 d x ) (II) Compute the centroid of the differential element along the y - axis. x 2 + y 2 = r 2 y 2 = r 2 − x 2 Substitute r 2 − x 2 for y 2 in Equation (II). d m = ρ π ( r 2 − x 2 ) d x Compute the moment of inertia of the differential element parallel to the y -axis and about the x -axis. d I x = y 2 2 d m (III) Here, the differential mass is d m , and the centroid of the differential element along the y - axis is y 2 . Substitute ρ π ( r 2 − x 2 ) d x for d m and r 2 − x 2 for y 2 in Equation (III). d I x = ( r 2 − x 2 ) 2 ρ π ( r 2 − x 2 ) d x = 1 2 ρ π ( r 2 − x 2 ) 2 d x Conclusion: Express the moment of inertia of the paraboloid about the x -axis. I x = ∫ − r r d I x (IV) Substitute 1 2 ρ π ( r 2 − x 2 ) 2 d x for d I x in Equation (IV). I x = ∫ − r r [ 1 2 ρ π ( r 2 − x 2 ) 2 d x ] = 1 2 ρ π ∫ − r r ( r 4 + x 4 − 2 r 2 x 2 ) d x = 1 2 ρ π ∫ − r r r 4 d x − ∫ − r r 2 r 2 x 2 d x + ∫ − r r x <

Engineering Mechanics: Statics Plus Mastering Engineering with Pearson eText -- Access Card Package (14th Edition) (Hibbeler, The Engineering Mechanics: Statics & Dynamics Series, 14th Edition)

14th Edition

ISBN: 9780134160689

Author: Russell C. Hibbeler

Publisher: PEARSON

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Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provi…

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Chapter 10.8, Problem 92P

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The moment of inertia Ix of the sphere, in terms of the total mass m of the sphere.

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Chapter 10.8, Problem 91P

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Engineering Mechanics: Statics Plus Mastering Engineering with Pearson eText -- Access Card Package (14th Edition) (Hibbeler, The Engineering Mechanics: Statics & Dynamics Series, 14th Edition)

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The moment of inertia I x of the sphere, in terms of the total mass m of the sphere.

Solution Summary: The author calculates the moment of inertia of the differential element parallel to the y -axis.

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The moment of inertia Ix of the sphere, in terms of the total mass m of the sphere.

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