Find the center of mass of the solid E bounded by the planes x=0,x=4,y=0,y=3,z=0 and z=3. Assume the density is \rho (x,y,z)=z.

Question

Find the center of mass of the solid E bounded by the planes x=0,x=4,y=0,y=3,z=0 and

z=3. Assume the density is \rho (x,y,z)=z.

Answer 1

The center of mass is (2, 1.5, 2). This is because the mass distribution inside the solid is symmetrical in the x- and y-directions, so those two coordinates of the center of mass are just the averages of the coordinates of the edges of the solid, while the z-coordinate is found by integrating the moment of the mass about the xy-plane and dividing that by the total mass itself. When we do that, we get ∫ z² dz ÷ ∫ z dz = (z³/3) / (z²/2) = 2z/3. Since z ranges from 0 to 3 for this problem, 2z/3 = 2, which is the z-coordinate of the center of mass.