Calculus: Analytic Geometry and Calculus, with Vectors

712 Iterated and multiple integrals

It is supposed that the sphere has density 5(r,4,9) at the point P having spherical
coordinates (r,4),0). A particle of mass m is supposed to be concentrated at a
point fl which lies outside the ball and on the negative z axis, the rectangular
coordinates of 11 being (0,0, - D), where D > a. We are required to determine
and learn something about the gravitational force F upon the particle of mass m
that is produced by the ball. We start with a basic idea of Newton that lies at
the foundation of classical science. If particles of masses m and AM are located
at points A and P, then each particle pulls the othertoward it with a force of
magnitude Gm AM/17P-12, where G is a universal gravitational constant whose
numerical value depends only upon the units of force and distance that are used.
The actual force upon the particle of mass m is obtained by multiplying this
magnitude by SIP/API, the unit vector which has its tail at .4 and is pointed
toward P. Letting OF denote this force, we have

(1) OF = Gm

14P

p13011.

We need a useful formula for the vector A'P. The rectangular coordinates

x, y, z, the cylindrical coordinates p, gyp, z, andthe spherical coordinates r, 4,, B

of the point P are related by the formulas

x=pcos¢=rsin0cos4,

y=psin4,=rsin0sin4,

z=rcos0

which can be found in this and other books and can be derived from Figure

13.891. Since .l has rectangular coordinates 0, 0, -D, we find that

(2) f1P = r sin 0 cos ¢1 + r sin 0 sin 4)j + (D + r cos 8)k

and hence that

(3) I t1PI = (D2 + 2Dr cos 0 +

Using spherical coordinates, we employ the right member of the formula

(4) AM = 5(r,4),B)r2 sin 0 Or A¢ AO

to approximate the mass AM of a subset (or element) of the ball containing the

point P. Substituting in (1) gives the formula

(5) OF = Gm(D2

S(2D rcos s0+ r2)36

AP Or AO 09

in which the right side is an approximation to the force upon the particle of

mass m produced by one subset of the ball. In (5) and some of the following

formulas, AP is written instead of the right member of (2) to save time and paper.

Supposing that the density function 6 is a reasonably decent function, we employ