Explanation Given: The width of the cone is h . Half of the height of the cone is r . The constant density of cone is ρ . Show the intersection of the cone at the arbitrary point ( x , y ) as in Figure (1). Calculation: From Figure 1, Calculate the differential mass of the cone. d m = ρ d V (I) Here, the density of the material is ρ , and the volume of the disc is V . Substitute π y 2 d x for d V in Equation (I). d m = ρ ( π y 2 d x ) Here, the centroid of the differential element along the y - axis is y 2 . Compute the moment of inertia of the differential element parallel to the y -axis and about the x -axis. d I x = 1 2 d m y 2 (II) Substitute ρ ( π y 2 d x ) for d m in Equation (II). d I x = 1 2 ( ρ π y 2 d x ) y 2 = 1 2 ρ π y 4 d x (III) Substitute r h x for y in Equation (III). d I x = 1 2 ρ π [ r h x ] 4 d x = 1 2 ρ π r 4 h 4 x 4 d x Conclusion: Express the moment of inertia of the cone about the x -axis. I x = ∫ 0 h d I x (IV) Substitute 1 2 ρ π r 4 h 4 x 4 d x for d I x in Equation (IV)

INTERNATIONAL EDITION---Engineering Mechanics: Statics, 14th edition (SI unit)

14th Edition

ISBN: 9780133918922

Author: Russell C. Hibbeler

Publisher: PEARSON

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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 93P

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The moment of inertia about the x -axis, in terms of the total mass m of the cone.

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Chapter 10.8, Problem 92P

Chapter 10.8, Problem 94P

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INTERNATIONAL EDITION---Engineering Mechanics: Statics, 14th edition (SI unit)

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The moment of inertia about the x - axis, in terms of the total mass m of the cone.

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The moment of inertia about the x -axis, in terms of the total mass m of the cone.

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