Consider a right circular solid cone (S) standing on its tip at the origin. The height of the cone is 4 and the radius of the top is 8. Find the centroid of the cone by following the steps. Assume the density of the cone is constant 1.
a. The mass of the cone is m=[1∫∫∫ s 1d(x,y,z)=
b. Let Q(z) be the disk that is the intersection of the cone with the horizontal plane at height z. The radlus of this disk is . So the area of the disk is
c. We can compute mz =∫∫∫S zd(x,y,z)=∫∫ O(z) zd(x,y)dz where a= and b=
d. Hence mz = ∫ z∫∫ Q(z) 1d(x,y)dz=∫ab zArea(Q(z)dz=∫ ab dz=
e. Using the symmetry of the cone, the center of gravity is (0,0,zˉ) where
z
ˉ
=
m
m
e
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