Mathematical Methods for Physics and Engineering - Matematica.NET

6.3 APPLICATIONS OF MULTIPLE INTEGRALSThe coordinates of the centre of mass of a solid or laminar body may also bewritten as multiple integrals. The centre of mass of a body has coordinates ¯x, ȳ,¯z given by the three equations∫¯x∫∫dM =∫xdMȳ dM = ydM∫ ∫¯z dM = zdM,where again dM is an element of mass as described above, x, y, z are thecoordinates of the centre of mass of the element dM and the integrals are takenover the entire body. Obviously, for any body that lies entirely in, or is symmetricalabout, the xy-plane (say), we immediately have ¯z = 0. For completeness, we notethat the three equations above can be written as the single vector equation (seechapter 7)¯r = 1 ∫r dM,Mwhere ¯r is the position vector of the body’s centre of mass with respect to theorigin, r is the position vector of the centre of mass of the element dM andM = ∫ dM is the total mass of the body. As previously, we may divide the bodyinto the most convenient mass elements for evaluating the necessary integrals,provided each mass element is of constant density.We further note that the coordinates of the centroid of a body are defined asthose of its centre of mass if the body had uniform density.◮Find the centre of mass of the solid hemisphere bounded by the surfaces x 2 + y 2 + z 2 = a 2and the xy-plane, assuming that it has a uniform density ρ.Referring to figure 6.5, we know from symmetry that the centre of mass must lie onthe z-axis. Let us divide the hemisphere into volume elements that are circular slabs ofthickness dz parallel to the xy-plane. For a slab at a height z, the mass of the element isdM = ρdV = ρπ(a 2 − z 2 ) dz. Integrating over z, we find that the z-coordinate of the centreof mass of the hemisphere is given by¯z∫ a0ρπ(a 2 − z 2 ) dz =∫ a0zρπ(a 2 − z 2 ) dz.The integrals are easily evaluated and give ¯z =3a/8. Since the hemisphere is of uniformdensity, this is also the position of its centroid. ◭6.3.3 Pappus’ theoremsThe theorems of Pappus (which are about seventeen centuries old) relate centroidsto volumes of revolution and areas of surfaces, discussed in chapter 2, and may beuseful for finding one quantity given another that can be calculated more easily.195