   Chapter 15.8, Problem 33E

Chapter
Section
Textbook Problem

Use spherical coordinates.33. (a) Find the centroid of a solid homogeneous hemisphere of radius a.(b) Find the moment of inertia of the in part (a) about a diameter of its base.

(a)

To determine

To find: The centroid of the solid in example 4 by using spherical coordinates.

Explanation

Given:

The solid is the hemisphere of radius a.

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)

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 . Then, by the equation (1), the mass of the solid is,

m=02π0π20aKdzdydx=02π0π20aK(ρ2sinϕ)dρdϕdθ=K02π0π20a(ρ2sinϕ)dρdϕdθ

Use the equation (2) to separate the integrals and integrate it.

m=K02πdθ0π2sinϕdϕ0aρ2dρ=K[θ]02π[cosϕ]0π2[ρ33]0a=K[2π0][cosπ2+cos(0)][a33033]=2Kπ(0+1)(a33)

= 23Kπa3

Thus, the mass of the given solid is 23Kπa3

(b)

To determine

To find: The moment of inertia about the diameter of its base for the region in part (a).

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