They are both integral problems. Thanks in advance! The stem of a mushroom Ls a

Question

They are both integral problems. Thanks in advance!

The stem of a mushroom Ls a right circular cylinder of diameter 1 and length 2 and its cap is a hemisphere of radius "a". If the mushroom is a homogeneous solid with axial symmetry and if its center of mass lies in the plane where the stem joins the cap, End "a". Find the moment of inertia about the z-axis for the solid region bounded by the cone x = square root 2(x^2 + y^2) and the sphere x^3 + y^2 + z^3 = a^2 if the density is inversely proportional to the distance horn the z-axis.

Explanation / Answer

(8)

Let's the mass density is .

Then the mass of the stem ms will be Vs, where Vs is the volume of the stem.

Likewise for the cap, mc=Vc.

So, ms=r^2h=2

and mc=(2/3)r^3=(2/3)a^3.

If we take the plane joining them to be z=0, positive up, then the center of mass of the stem is dsms=2(1)=2, where ds is the displacement of the center of mass from the plane.

This means that dcmc=+2.

The center of mass of a hemispherical solid of radius a lies 3/8ths of the way up from the base.

Hence,

(3a/8)(2a^3)/3=2a=(8)^(1/4)1.682.

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