I.Note that the solid is symetric about the x axis. This means that the center of mass will lie on the x axis. II. The center of mass is determined by summing the mass x distance from an arbitrary point, divided by the total mass. Here, pick the origin. III. Since everything is symetric about the x-axis, we form an integral taking succesive slice from our half sphere. The slice will have the area of a circle with the radius equal to the height, which is h = √(r^2-x^2). According, our integral will be: =0-r of ∫ h^2 * x dx = 0-r of ∫ (r^2- x^2) * x dx = 0-r of ∫ (r^2x- x^3) dx = 0-r [ (r^2x^2/2- x^4/4)] = (r^4/2 - r^4/4) = r^4/4 IV The volume of a sphere is 4/3 r^3, which means a half sphere is 2/3 r^3, but lets do an integral doing the same basic method. The integral will be: =0-r of ∫ h^2 dx .... Note this is the same as above except the multiplication by x is missing = 0-r of∫ (r^2-x^2) dx = 0-r of [ r^2x - x^3/3] = (r^3 - r^3/3) = 2/3 r^3 V. Divide III by IV cm = ( r^4/4) / (2/3 r^3) cm = 3/8 r
