The Chemistry Maths Book, Second Edition

460 Chapter 16Vectors

The general case

If the force is notconstant along the path r

1

to r

2

then

the work has the form of a line integral (Section 9.8).

Consider a body moving from point A at r

1

to point

B at r

2

along the curve C under the influence of a

force Fwhose value varies from point to point on C

(Figure 16.22). Let F(r) be the force at point ron the

curve. The work done on the body between positions r

andr 1 + 1 ∆ris∆W 1 ≈ 1 F(r) 1

·

1 ∆rand, in the limit|∆r| 1 → 10

on the curve, the element of work isdW 1 = 1 F(r) 1 ·

1 dr. The

total work done from A to B on C is then

(16.38)

and the integral is a line integral. In terms of components, F 1 = 1 (F

x

, F

y

, F

z

)and

dr 1 = 1 (dx, dy, dz),F 1 ·

1 dr 1 = 1 F

x

dx 1 + 1 F

y

dy 1 + 1 F

z

dz, and

(16.39)

This is a generalization of equation (9.48) for a curve in three dimensions, and the

discussion of Section 9.8 applies with only minor changes. In particular, if the force is

a conservative force then, by a generalization of the discussion of conservative forces

in Sections 5.7 and 9.8, the components of the force can be expressed as (partial)

derivatives of a potential-energy function V(see equation (5.57)),

(16.40)

Then

(16.41)

and dVis the total differential of the potential-energy functionV(r) 1 = 1 V(x,y,z). It

follows that, for a conservative force, the line integral (16.38) is independent of the

path, and is equal to the difference in potential energy between A and B:

(16.42)

0 Exercise 29

WdVVV

AB

A

B

AB

=−Z = −

Fdx Fdy Fdz

V

x

dx

V

y

dy

V

z

dz

xyz

++=−

∂













=−−dV

F

V

x

F

V

y

F

V

z

xyz

=−

∂

=−

∂

=−

∂

W F dx F dy F dz

AB xyz

C

=++













Z

Wd

AB

C

=Z Fr r() 11

·

... ... .... ... .... ... .... ... .... ... .... .... ... ... .... .... ... ..

a

o

•

..... .. .. ... ... .. ..

.... .... ... ..... .

C

z

y

x

b

r

r+∆ r

∆ r

..

.

..

...

..

...

..

...

..

...

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...

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Figure 16.22