Calculus: Analytic Geometry and Calculus, with Vectors

4.7 Mass, linear density, and moments 265

concentrated at the point (c,0,0). Hint:

Assuggested by Figure 4.793, make a
partition P of the interval a S x < b.
Consider the part of the shell between

the planes havingthe equations x = xk_1
and x = Xk to be acircular ring having
its mass Mkconcentrated in the plane
having the equation x =xk. Let AFk

Figure 4.793

be the force exerted uponm by this ring. Because of symmetry, the components
of AFk orthogonal to i are zero. Moreover, the i component of OFk (which is
iFk) is the same as the i component of the force on m produced by a single
particle of mass Mk concentrated at the point (xk,R,0) in E3. Therefore,
Mk(x - c)i
(1) OFk = Gin[(x* - c)2 + R2] '

and we are ready to calculate Mk and get on with the calculus. Ins.:

(2) F = 2a6GmR ('1(a

• c)2 +R2 c)2 +R2) i.
  13 We can claim that if the density S and the radius R of the cylindrical shell
  of Problem 12 are so related that the total mass is M, then the answer to Problem
  12 should be nearly the same as one of the answers to Problem 10 when R is
  near zero. Is it so? Ins.: Yes, unless misprints disrupt the harmony.
  14 A circular disk of radius H has its center on the x axis of an x, y, z coordi-
  nate system and lies in the plane having the equation x = xo. This disk has
  constant areal density (mass per unit area) S. Set up an integral for the gravita-
  tional force F which the disk exerts upon a particle m of mass m which is con-
  centrated at the point (c,0,0) when c ; xo. Hint: Make a partition with the
  aid of which the disk is split into a collection of concentric rings so that a repre-
  sentative ring has radius rk. The hint of Problem 12 provides a formula that
  can be adapted to give the force which the representative ring exerts upon m.
  Ans.:

(1)

H r

F = 2aSGm(xo - c)i Io [r2 + (xo- c)21 dr

(2) F 27r&Gm xo - c (

xo1 c

() ( )L I 1/H2 + (xo - c)2J 1 i.

When M is the total mass of the disk so that M = irH26, the answer can be put

in the form

(3) _

GmM xo - c xo - c 1

F-2 H2 Elxo - cI H2 +(xo -

c)2.1i.

Remark: We really should look at these formulas. For example, (2) gives very

interesting information when H is large and our disk is a huge part of a homo-

geneous plane. One who wishes additional mental elevation should undertake

to realize that we can replace gravitational laws and constants by electrostatic

ones and obtain information about forces on electrons produced by charges on

plates of capacitors.