   Chapter 15.6, Problem 46E

Chapter
Section
Textbook Problem

Assume that the solid has constant density k.46. Find the moment of inertia about the z-axis of the solid cone x 2 + y 2   ≤   z   ≤   h .

To determine

To find: The momentum of inertia of the solid about the z-axis if the solid has constant density k.

Explanation

Given:

The cone is x2+y2zh.

Calculation:

Initially, calculate the moment of inertia over the z-axis. Therefore, the integration will be over (x2+y2)ρ(x,y,z) but here the density is constant.

The moment of inertia over the z-axis is, Iz=E(x2+y2)ρ(x,y,z)dV

From the given x2+y2zh is in a polar form, let x=rcosθ and r=rsinθ.

r varies from 0 to z and θ varies from 0 to 2π and z varies from 0 to h.

Iz=kE(r2)rdrdzdθ

Integrate the above integral with respect to r and apply the limit of it.

Iz=k02π0h[r44]0zdzdθ=k02π0h[z44044]dzdθ=k402π0h

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