Calculus: Analytic Geometry and Calculus, with Vectors

424 Curves, lengths, and curvatures

the curve C to be the ordered set of points P(t) = P(u,v,w) for which u, v, w

are functions of t having continuous derivatives and

(1) OP = u(i)i + v(t)j + w(t)k (a 5 t 5 b),

and suppose that the linear density (mass per unit length) of C at P(t) is S(t),

where S is a continuous function of t. Letting Q be a partition of the interval

a < t 5 b with partition points tk and supposing that tk_i S It S tk, we take the

number S(tk*) JPk_1PkI as an approximation to the mass of an element of the

wire. The force t1Fk which this element produces upon the particle P* of mass

m at P(x,y,z) should have approximate magnitude

Gm3(tk*) IPk_lPx1

(2) I PP(tk) I2

Introducing vectors and coordinates gives the approximation

uk^2 AVk 2 11 wk 2

(3) AF, = GmS(tk)

(L1Ot)

+(Ot) +(fit

(

[u(tk) - x]i + [e(tk) - y]j + [w(tk) - z)]k

l[u(tk) - x]2 + [a(tk) - y]2 + [w(tk) - R.]2}3fe

At.

The limit of the sum of these things should be F, and we are led to the definition

(4) F = Gm Lb S(t) [u'(t)]2 + [e'(t)]2 + [w'(t)]2

[u(t) - x]i + [v(t) - y]j + [w(t) - z]k

{[u(t) - x]2 + [9(t) -' y]2 + [w(t) - z]2} dt.

This is our result. In case the wire coincides with a circle of radius a in the yz

plane having its center at the origin, we can set u(t) = 0, o(t) = a cos t, w(t) _

a sin t and put (4) in the form

2* -xi+[acost-y]j+[asint-z]k

(5) F =Gma fo S(t) {x2± [a cos t - y]2 + [a sin t - z]2}3dt.

In caseP* is on the axis of the wire so that y = z = 0, this takes the much simpler

form

(6) F =Gma fo2*S(t) -xi + a cos tj + a sin tk

(x2 + a2)H A

In case the wire is a uniform wire so that, for some constant So we have S(t) = So

for each t, the coefficients of j and k are zero because

(7) o

2ar

cos t dt =

02r

sin t dt = 0

(or, as is usually said, "because of symmetry") and (6) reduces to

(8)

F - - GmMx

(x2 + a2)

where M = 22rabo, the total mass of the wire. Remark: With slight differences

in notation, (8) was used extensively in some of the Problems 4.79.