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 non-constant force acts on an object along a curved path. [A non-constant 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 straight-line 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:

1. 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.

2. 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.

3. 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.