7. Find the centroid of the region bounded by the surfaces r2 + y2 2-2az , 0 and

Question

7. Find the centroid of the region bounded by the surfaces r2 + y2 2-2az , 0 and 3r2 +3y2-12-0, lying above the zy plane. (Consider the inner region). 8. The stem of a mushroom is a right circular cylinder of diameter and length 2 and its cap is a hermisphere of radius "a", f 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, find "a" Hint: Let the center of mass be at the origin 9. Find the moment of inertia about the z-axis for the solid region bounded by the cone z-v 3 zrty from the z-axis and the sphere r2 + y2 +22 = a2 if the density is inversely proportional to the distance

Explanation / Answer

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 isdsms=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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