Calculus: Analytic Geometry and Calculus, with Vectors

266 Integrals

15 A cylindrical solid of radius R has its axis on the x axis of an x, y, z coordi-
nate system and has its ends in the planes having the equations x = a and x = b.
This solid has uniform density (mass per unit volume) S. Find the gravitational
force F which this solid exerts upon a particle m of mass m located at the point
(c,0,0) of E3, it being assumed that c < a. Hint: Make a partition of the interval
a < x b. Consider the part of the cylinder between the planes having equa-
tions x = x7,_1 and x = Xk to be a circular disk in the plane having the equation
x = xk. Let AFk be the force exerted upon the particle m by this disk. A
formula of Problem 14 can then be applied. .1ns.:

F=2rGmSiI [1- xc ]dx

a 1/(x - --C)2+ R2

F = 2irGmS[b - a - (V-(b - c)2 + R2 - \/ (a - c)2 + R2)] J.

This can be put in the form

F - 2 GmM r

1 -

-\/(b - c)2 + R2 - 1/(a - c)2 + R2]

R2 b-a

where M = sR2(b - a)S, the total mass of the cylindrical solid.

.16 Let S be a thin spherical shell which is assumed to be concentrated on a

sphere (surface, not ball) of radius a having its center at the origin. The shell

has constant areal density (mass per unit area) S. Let m* be a particle of mass in

which is concentrated at a point (-b,0,0) which lies at the origin or at distance

b from the origin on the negative x axis. Thus b >= 0, and we suppose that

b , a so m* does not lie on the sphere. The gravitational force F exerted upon

m* by the shell depends upon the location of m*. If 0 5 b < a so that m* is

inside the sphere, then F = 0. If b > a so that m* is outside the sphere, then

(1) F = G

MM

i,

where M is the total mass of the shell. Thus when m* lies outside the shell,

the force on it exerted by the shell is the same as the force exerted on it by a

particle at the center of the shell whose mass is the total mass of the shell. From

our present point of view, proofs of these famous and important results (which are

discussed in more general terms in Section 13.8) can be comprehended more

easily than they can be originated. To start our proof, we slice the spherical

shell into ribbons to which we can apply a basic result given in Problem 12.

The spherical shell is obtained by

rotating the semicircle of Figure 4.794

about the x axis. We make a partition

m* f , P of the interval 0 S 0 5 ir. With the

aid of the basic formula

Figure 4.794 (2) Angle =length of arc

radius

we see that the lines making angle 8,,_, and Ok with the positive x axis have

between them an arc of the circle of length a(9k- Bk_1), or a ABk. When this

arc is rotated about the x axis, it produces a part of the spherical shell which can