Problem: A force which acts on a particle moving in the xy plane is given by:

a) Find the work done in moving along the indicated path.
b) Is the force conservative?

Solution:

a) The work done between two points is a sum of path integrals:

where each path differential ds has to be expressed in terms of x and y and the unit vectors. Along path AB,

Substituting these into the first integral,

The path for the second integral has a different equation:

The second integral is similar to the first:

Now for the third integral - this is a long problem! Skipping some of the steps, the equation of the path is

The integral is:

Adding the three values of the work, the result is -13.3 J.

b) The force is conservative if the work is independent of the path. Another route to point D from A is a straight line drawn directly from A to D. That line has a slope of -2 and a y-intercept of 2:

This is different from the work done on path ABCD; thus, the force is nonconservative.