A Classical Approach of Newtonian Mechanics

5 CONSERVATION OF ENERGY 5.3 Work

X

' ·

A

̧

X ∫^

f
|f|
|s| cos 


Figure 36: Definition of work

that the vector displacement of the ith segment is ∆ri. Suppose, further, that N is

sufficiently large that the force f does not vary much along each segment. In fact,

let the average force along the ith segment be fi. We shall assume that formula

(5.14)—which is valid for constant forces and straight-line displacements—holds

good for each segment. It follows that the net work done on the body, as it moves

from^ point A^ to^ point^ B,^ is approximately^
N
W fi ∆ri. (5.16)
i=1
We can always improve the level of our approximation by increasing the number
N of the straight-line segments which we use to approximate the body’s trajectory
between points A and B. In fact, if we take the limit N
expression becomes exact:

then the above

W = (^) Nlim fi·∆ri =
B
f(r)·dr. (5.17)
→∞ (^) i=1 A
Here, r measures vector displacement from the origin of our coordinate system,
and the mathematical construct B f(r)·dr is termed a line-integral.
The meaning of Eq. (5.17) becomes a lot clearer if we restrict our attention to
1 - dimensional motion. Suppose, therefore, that an object moves in 1-dimension,
with displacement x, and is subject to a varying force f(x) (directed along the
x-axis). What is the work done by this force when the object moves from xA
s
|s|

N
→ ∞