Variable-density solids Find the coordinates of the center ofmass of the following solid with variable density. The solid bounded by the cone z = 9 - r and z = 0 with ρ(r, θ, z) = 1 + z
Variable-density solids Find the coordinates of the center of
mass of the following solid with variable density.
The solid bounded by the cone z = 9 - r and z = 0 with ρ(r, θ, z) = 1 + z
We have to find out the center of mass of the region bounded
by the cone z = 9 - r and z = 0 with
variable density ρ(r, θ, z) = 1 + z
The center of mass lies on the z-axis
because of the symmetry of the solid and the density function;
therefore, .
To obtain the center of mass,
we have to calculate and m ,
We know that, the z-coordinate of the centroid of the region is given by
The cone is described by the equation z = 9 - r . The inner integral in z runs from the plane z=0 (the lower surface) to the cone z = 9 - r (upper surface).
We project D onto the xy-plane to produce the region R, whose boundary is determined by the intersection of the two surfaces. Equating the z-coordinates in the equations of the two surfaces, we have
0 = 9 - r
r = 9.
Therefore, the projection of D on the xy-plane is
which is a disk of radius 9 centered at (0,0).
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