Vector Fields and
Line integrals
Part 4: Approximating the Work Done Along a Curved Path
In
Part 2 you calculated work done when a constant force acts on an object
that moves along a straight path. Using the applet in Part 3, you found
the work done when a nonconstant force acts on an object along a curved
path. [A nonconstant force
is one for which the direction or the length (or both) of the arrows changes from
point to point.] In this part
you will learn one way to calculate the work done when the force is not
constant and the path is not a straight line.
We can approximate a curved path
by a collection of straight lines. The figures below show a quarter of a circular
arc and some straightline approximations to the arc connecting points at equally
spaced x values.

Approximation with
1 line segment 

Approximation with
4 line segments 

Approximation with
10 line segments 
You
will calculate the work done by the force on
an object moving from the point to the point along this circular arc:
 Calculate the work done if the
circular arc is approximated by one line segment and the force field is approximated
by the value of F at the start of the segment.
 Refine your estimate of the work
done by using four line segments to approximate the circular arc and four
starting points to approximate the force field.
 Refine your estimates further
by approximating the circular arc with greater numbers of line segments until
you feel confident that you have found the amount of work done.