Explanation Given: The cylindrical coordinates of the cone indicated by z = h ( 1 − r r 0 ) shown in the following figure: Formula Used: The mass of the solid is, mass = volume ⋅ density The moments are found by, M y z = ∭ Q x ρ ( x , y , z ) d V M x z = ∭ Q y ρ ( x , y , z ) d V M x y = ∭ Q z ρ ( x , y , z ) d V The volume is V = ∭ Q d V Also use the integration formula ∫ x n d x = x n + 1 n + 1 Calculation: First, the volume of the cone using the cylindrical coordinates V = ∭ E r d z d r d θ Apply limits and use ∫ x n d x = x n + 1 n + 1 V = ∫ 0 2 π ∫ 0 r 0 ∫ 0 h ( 1 − r r 0 ) r d z d r d θ = ∫ 0 2 π ∫ 0 r 0 [ r z ] 0 h ( 1 − r r 0 ) d r d θ = ∫ 0 2 π ∫ 0 r 0 r h ( 1 − r r 0 ) d r d θ = h r 0 ∫ 0 2 π ∫ 0 r 0 ( r r 0 − r 2 ) d r d θ On further simplification, V = h r 0 ∫ 0 2 π [ r 2 2 r 0 − r 3 3 ] 0 r 0 d θ = h r 0 ∫ 0 2 π r 0 3 6 d θ = h r 0 [ r 0 3 6 ] 0 2 π = h r 0 [ r 0 3 6 ⋅ 2 π ] On calculation, V = h r 0 2 π 3 The mass is thus found to be for density ρ M = ρ ⋅ V = ρ h r 0 2 π 3 From the diagram, it can be seen that the centroid is on the positive z -axis, thus x ¯ = 0 , y ¯ = 0

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ISBN: 9781337275576

Author: Larson, Ron, Edwards, Bruce

Publisher: Cengage Learning,

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Chapter 14.7, Problem 24E

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To calculate: The centroid of the given cone, the cone has equation z=h(1−rr0)

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Chapter 14.7, Problem 23E

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The centroid of the given cone, the cone has equation z = h ( 1 − r r 0 )

Solution Summary: The author explains how to calculate the centroid of the given cone using the cylindrical coordinates.

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To calculate: The centroid of the given cone, the cone has equation z=h(1−rr0)

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Chapter 14 Solutions