Calculus: Analytic Geometry and Calculus, with Vectors

4.8 Moments and centroids in E2 and E3 269

This answer can be greatly improved if we notice that methods very similar to
those which we haveused enable us to show that the integral in (3) is the total
mass M of theball B. Thus we can put (3) in the form
GmM.
(4) F = b2 t.

This proves that the force exerted upon m by a radially homogeneous ball is
the same as the force exerted upon m by a single particle, at the center of the
ball, whose mass is the total mass of the ball. Thus, when computing forces

upon particles outsidethe ball, we may "consider the mass of the ball to be con-
centrated at its center," the assertion in quotation marks being rather weird
because mass is a number and we do not ordinarily squeeze numbers.
19 Use the method of Problem 18 to show that if S is a radially homogeneous
spherical shell having inner and outer radii r, and r2 for which rl < r2, then
F = 0 when F is the gravitational force which the shell exerts upon a particle
inside the shell.
20 With the aid of arguments involving continuity, the final formulas of
preceding problems for gravitational forces upon particles exerted by solid
spherical balls and solid unconcentrated spherical shells can be proved to be
correct when the particles lie on boundaries of the balls and shells. Using this
fact, show how it is possible to split a given radially homogeneous solid ball into
an inner solid ball and an outer spherical shell to calculate the force which the
given ball exerts upon a particle m* of mass m concentrated at an inner point
of the given ball.

4.8 Moments and centroids in E2 and E3 Section 4.7 introduced us
to moments, about a point on a line, of material concentrated upon the

line. This section introduces us to two similar ideas. In the first place,
we consider moments, about a line, of material concentrated in a plane

containing that line. In the second place we consider moments, about a

plane, of material in E3.

To begin, let R be a bounded region in the xy plane which lies between

the lines having the equation x = a and x = b. It is supposed that when

a 5 x < b, the line having the equation x = x intersectsR in an

interval (or collection of intervals) having length (or total length) f(x*).

It is not necessary that f be continuous, but we do assume that R has

area IRI and that JRI =f ab f (x) dx. If the region R is, as in Figure 4.81,

Y=f2(x)

* -xk

Figure 4.81

b x