   Chapter 15.8, Problem 32E

Chapter
Section
Textbook Problem

Use spherical coordinates.32. Let H be a solid hemisphere of radius a whose density at any point is proportional to its distance from the center of the base.(a) Find the mass of H.(b) Find the center of mass of H.(c) Find the moment of inertia of H about its axis.

(a)

To determine

To find: The mass of the given solid by using spherical coordinates.

Explanation

Formula used:

If f is a spherical region E given by aρb,αθβ,cϕd , then, Ef(x,y,z)dV=αβabcdf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)ρ2sinϕdϕdρdθ (1)

If g(x) is the function of x and h(y) is the function of y and k(z) is the function of z  then, abcdefg(x)h(y)k(z)dzdydx=abg(x)dxcdh(y)dyefk(z)dz (2)

The spherical coordinates (ρ,θ,ϕ) corresponding to the rectangular coordinates (x,y,z) is,

ρ=x2+y2+z2ϕ=cos1(zρ)θ=cos1(xρsinϕ)

Given:

The solid H is the hemisphere of radius a.

The density function is proportional to its distance from the center of the base say K.

Calculation:

From the given condition, it is observed that ρ varies from 0 to a, θ varies from 0 to 2π and ϕ varies from 0 to π2

(b)

To determine

To find: The centroid of the given region H.

(c)

To determine

To find: The moment of inertia about the z axis.

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