   Chapter 16.7, Problem 42E

Chapter
Section
Textbook Problem

Let S be the part of the sphere x2 + y2 + z2 = 25 that lies above the plane z = 4. If S has constant density k, find (a) the center of mass and (b) the moment of inertia about the z-axis.

(a)

To determine

To find: The center of mass.

Explanation

The center of mass (x¯,y¯,z¯) is (0,0,92)_ .

Formula used:

m=Sρ(x,y,z)dS (1)

rϕ=xϕi+yϕj+zϕk (2)

rθ=xθi+yθj+zθk (3)

z¯=1mSzρ(x,y,z)dS (4)

By using the spherical coordinates to parameterize the sphere, consider r(ϕ,θ)=5sinϕcosθi+5sinϕsinθj+5cosϕk , as the portion of the sphere where z4 then the limits are 0θ2π and 0ϕtan1(34) .

Find rϕ .

Substitute 5sinϕcosθ for x , 5sinϕsinθ for y and 5cosϕ for z in equation (2),

rϕ=ϕ(5sinϕcosθ)i+ϕ(5sinϕsinθ)j+ϕ(5cosϕ)k=(5cosθ)ϕ(sinϕ)i+(5sinθ)ϕ(sinϕ)j+(5)ϕ(cosϕ)k=5cosθcosϕi+5sinθcosϕj5sinϕk

Find rθ .

Substitute 5sinϕcosθ for x , 5sinϕsinθ for y and 5cosϕ for z in equation (3),

rθ=θ(5sinϕcosθ)i+θ(5sinϕsinθ)j+θ(5cosϕ)k=(5sinϕ)θ(cosθ)i+(5sinϕ)θ(sinθ)j+(5cosϕ)θ(1)k=(5sinϕ)(sinθ)i+(5sinϕ)(cosθ)j+(5cosϕ)(0)k=5sinϕsinθi+5sinϕcosθj

Find rϕ×rθ .

rϕ×rθ=(5cosθcosϕi+5sinθcosϕj5sinϕk)×(5sinϕsinθi+5sinϕcosθj)=|ijk5cosθcosϕ5sinθcosϕ5sinϕ5sinϕsinθ5sinϕcosθ0|={(0+25sin2ϕcosθ)i+(25sin2ϕsinθ+0)j+(25cos2θcosϕsinϕ+25sin2θcosϕsinϕ)k}={25sin2ϕcosθi+25sin2ϕsinθj+25cosϕsinϕ(cos2θ+sin2θ)k}={25sin2ϕcosθi+25sin2ϕsinθj+25cosϕsinϕk} {cos2θ+sin2θ=1}

Find |rϕ×rθ|

(b)

To determine

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

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