   Chapter 15.6, Problem 48E

Chapter
Section
Textbook Problem

Set up, but do not evaluate, integral expressions for (a) the mass, (b) the center of mass, and (c) the moment of inertia about the z-axis.48. The hemisphere x2 + y2 + z2 ≤ 1, z ≥ 0; ρ (x, y, z) = x 2 + y 2   +   z 2

(a)

To determine

To find: The integral expression for the mass and not to evaluate the value.

Explanation

Given:

The hemisphere is, x2+y2+z21 , z0 and the density is given by ρ(x,y,z)=x2+y2+z2 .

Calculation:

Mass of the solid, mass(m)=density×volume . Here the density is given by ρ(x,y,z)=x2+y2+z2 .

The mass of the solid is, m=liml,m,ni=1lj=1mk=1nρ(xij*,yij*,zij*)ΔA=Eρ(x,y,z)dA .

Here, the density function is given by ρ(x,y,z) and E is the region that is occupied by the solid.

From the given, x2+y2+z21 then the value of z, z=±1x2y2 but take the upper hemisphere z=1x2y2 and z0

(b)

To determine

To find: The integral expression for the center of mass and not to evaluate the value.

(c)

To determine

To find: The integral expression for the moment of inertia about the z-axis and not to evaluate the value.

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