   Chapter 15.8, Problem 34E

Chapter
Section
Textbook Problem

Use spherical coordinates.34. Find the mass and center of mass of a solid hemisphere of radius a if the density at any point is proportional to its distance from the base.

To determine

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

Explanation

Given:

The solid H is the hemisphere of radius a.

The density function is proportional to its distance from the base say Kz.

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 given solid is,

m=02π0π20aKzdzdydx=02π0π20aK(ρcosϕ)(ρ2sinϕ)dρdϕdθ=K02π0π20a(ρ3cosϕsinϕ)dρdϕdθ

Use the equation (2) to separate the integrals and integrate it. For that, substitute t=cosϕ,dt=sinϕdϕ

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