Explanation Given: The length of the cone is h . The radius of the cone is a . Show the intersection of cone at the arbitrary point ( x , y ) as in Figure (1). Conclusion: Express the mass of the differential element. d m = ρ d V (I) Here, density of material is ρ , and the volume of the cone is V . Substitute π y 2 for d V in Equation (I). d m = ρ ( π y 2 ) d x (II) Substitute a h x for y in Equation (II). d m = ρ π a 2 h 2 x 2 d x Express the mass moment of inertia of the cone with respect to y axis. d I y = 1 4 d m y 2 + d m x 2 (III) Substitute ρ π a 2 h 2 x 2 d x for d m and a h x for y in Equation (III). d I y = 1 4 [ ρ π a 2 h 2 x 2 d x ] [ a h x ] 2 + [ ρ π a 2 h 2 x 2 d x ] x 2 = ρ π a 2 4 h 2 ( 4 h 2 + a 2 ) x 4 d x Express the moment of inertia of the cone about the y axis. I y = ∫ d I y (IV) Substitute ρ π a 2 4 h 2 ( 4 h 2 + a 2 ) x 4 d x for d I y in Equation (IV). I y = ∫ 0 h ρ π a 2 4 h 2 ( 4 h 2 + a 2 ) x 4 d x = ρ π a 2 4 h 2 ( 4 h 2 + a 2 ) ∫ 0 h x 4 d x = [ ρ π a 2 4 h 2 ( 4 h 2 + a 2 ) ] [ x 5 5 ] 0 h = [ ρ π a 2 4 h 2 ( 4 h 2 + a 2 ) ] h 5 5 = ρ π a 2 h 20 ( 4 h 2 + a 2 ) (V) Calculate the entire mass of the cone. m = ∫ 0 h d m (VI) Substitute ρ π a 2 h 2 x 2 d x for d m in Equation (VI)

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ISBN: 9780137616909

Author: HIBBELER

Publisher: VST

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Chapter 21, Problem 2P

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The moment of inertia of the cone with respect to a vertical y¯ axis and the moment of inertia about a parallel y' axis:

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The moment of inertia of the cone with respect to a vertical y ¯ axis and the moment of inertia about a parallel y ' axis:

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The moment of inertia of the cone with respect to a vertical y¯ axis and the moment of inertia about a parallel y' axis:

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