Calculus: Analytic Geometry and Calculus, with Vectors

4.8 Moments and centroids in Es and Es 273

In case p = 0, the pth moment is the mass of the body. We shall not

comment upon second moments of solid bodies about planes, but brief
comments about first moments may be appropriate. As was the case for
laminas, our body B determines a point in E3 which is called the centroid
of the body. With reference to the particular coordinatesystem which
we have chosen, the x coordinate of the centroid is the number x for which
Mme = 0 when E = x. Thus

(4.88) ffb (x- 9),4 (x) dx = 0

and it follows that

(4.881) MR = S

rb

x4(x) dx, x = a

S f

b

x.4(x) dx

Ja S f 'A (x) dx ,

where M, the denominator in the second formula, is the mass of the body

B. Analogous formulas serve to determine the other coordinates y and z

of the centroid. As was the case for centroids of laminas, the centroid

of a body B in E3 has an important physical property. An ordinary

wheel mounted on an axle through its center balances in the gravitational

field of the earth which is (so far as an ordinary wheel near the surface is

concerned) nearly a uniform parallel force field. Similarly, the body B,

when mounted on an axis through its centroid, must balance in a uniform

parallel force field. If a plane it is a plane of symmetry of the body B,

then the centroid of B is a point in ir. If a line L is a line of symmetry of

B, then the centroid of B is a point on L. If a point P is a center of sym-

metry of B, then the centroid of B is P.

All through this section, the moments that have appeared have been

"moments of mass," that is, moments of lamina or solid bodies that possess

mass. Our methods and formulas are easily modified to produce numbers

that are "moments of area," that is, moments of sets in B2 that possess

positive area, and "moments of volumes," that is, moments of sets in E3

that possess positive volumes. The moments and the centroid of a set

S in E2 which possesses positive area are, by definition, the same as those

of the lamina of unit areal density (unit mass per unit area) which coin-

cides with the set. Similarly, the moments and the centroid of a set S

in E3 possessing positive volume are defined to be the moments and the

centroid of a body of unit density (unit mass per unit volume) which

coincides with S. Thus formulas for moment and centroids of "geo-

metrical" sets are obtained by putting S = 1 in formulas for "moments

and centroids of mass." These concepts are introduced because they

are useful. For example, calculations involving forces which bend a beam

depend upon a number I which is the moment of inertia of a cross section

of the beam about a line through the centroid of the cross section.