Mathematical Methods for Physics and Engineering : A Comprehensive Guide

6.3 APPLICATIONS OF MULTIPLE INTEGRALS

The coordinates of the centre of mass of a solid or laminar body may also be

written as multiple integrals. The centre of mass of a body has coordinatesx ̄, ̄y,

̄zgiven by the three equations

̄x

∫

dM=

∫

xdM

̄y

∫

dM=

∫

ydM

̄z

∫

dM=

∫

zdM,

where againdMis an element of mass as described above,x,y,zare the

coordinates of the centre of mass of the elementdMand the integrals are taken

over the entire body. Obviously, for any body that lies entirely in, or is symmetrical

about, thexy-plane (say), we immediately have ̄z= 0. For completeness, we note

that the three equations above can be written as the single vector equation (see

chapter 7)

̄r=

1

M

∫

rdM,

where ̄ris the position vector of the body’s centre of mass with respect to the

origin,ris the position vector of the centre of mass of the elementdMand

M=

∫

dMis the total mass of the body. As previously, we may divide the body

into 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 thecentroidof a body are defined as

those of its centre of mass if the body had uniform density.

Find the centre of mass of the solid hemisphere bounded by the surfacesx^2 +y^2 +z^2 =a^2

and thexy-plane, assuming that it has a uniform densityρ.

Referring to figure 6.5, we know from symmetry that the centre of mass must lie on
thez-axis. Let us divide the hemisphere into volume elements that are circular slabs of
thicknessdzparallel to thexy-plane. For a slab at a heightz, the mass of the element is
dM=ρdV=ρπ(a^2 −z^2 )dz. Integrating overz, we find that thez-coordinate of the centre
of mass of the hemisphere is given by

̄z

∫a

0

ρπ(a^2 −z^2 )dz=

∫a

0

zρπ(a^2 −z^2 )dz.

The integrals are easily evaluated and give ̄z=3a/8. Since the hemisphere is of uniform
density, this is also the position of its centroid.

6.3.3 Pappus’ theorems

The theorems of Pappus (which are about seventeen centuries old) relate centroids

to volumes of revolution and areas of surfaces, discussed in chapter 2, and may be

useful for finding one quantity given another that can be calculated more easily.