   Chapter 15, Problem 43RE

Chapter
Section
Textbook Problem

(a) Find the centroid of a solid right circular cone with height hand base radius a. (Place the cone so that its base is in the xy-plane with center the origin and its axis along the positive z-axis.)(b) If the cone has density function ρ ( x , y , z ) = x 2 + y 2 , find the moment of inertia of the cone about its axis (the z-axis).

(a)

To determine

To find: The centroid of the given region E.

Explanation

Given:

The region B is the right circular cone with radius a and height h.

Calculation:

The base of the right circular cone lies in the xy-plane. So, its axis is z-axis.

Here, it is more appropriate to use the cylindrical coordinates because the given solid is a cone. It is given that the density is constant.

By the symmetric of the given function, it is observed that x and y coordinates of the centroid is 0. Thus, the third coordinate is given by,

z¯=0a02π0h(1ra)z(r)dzdθdrm=0a02π0h(1ra)zrdzdθdrm

First find the mass of the lamina,

m=DdA=volume of the given region=13πr2h=13πa2h

Integrate with respect to z and apply the limit of it.

0a02π0h(1ra)zrdzdθdr=0a02πr[z22]0h(1ra)dθdr=0a02πr((hhra)22(0)22)dθdr=120a02πr(h2+h2r2a22h2ra0)dθdr=120a02π(h2r+h2r3a22h2r2a)dθdr

Integrate with respect to θ and apply the limit of it

(b)

To determine

To find: The moment of inertia about its axis for the given region by using cylindrical coordinates.

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