how to calculate centre of mass of solid hemisphere

If the mass of the sphere is uniformly distributed,  i.e it has the same amount of mass in every unit area, then its centre of mass lies on its geometrical centre (Centre of mass=centre of sphere).

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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

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