   Chapter 15.8, Problem 31E

Chapter
Section
Textbook Problem

Use spherical coordinates.31. (a) Find the centroid of the solid in Example 4. (Assume constant density K.)(b) Find the moment of inertia about the z-axis for this solid.

(a)

To determine

The centroid of the solid in example 4 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)

Calculation:

From example 4, it is known that m=π8K. From the symmetry of the given region, it is observed that the Myz and Mxz are equal to 0. The value of Mxy is given by,

Mxy=02π0π40cosϕKzdzdydx=K02π0π40cosϕρcosϕ(ρ2sinϕ)dρdϕdθ=K02π0π40cosϕρ3cosϕsinϕdρdϕdθ

Integrate with respect to ρ and apply the limit of it

(b)

To determine

The moment of inertia about the z axis.

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