5 CONSERVATION OF ENERGY 5.4 Conservative and non-conservative force-fields
∫BAFigure 40: Two alternative paths between points A and BSuppose that the same object moves along a different trajectory, labeled path
2, between the same two points. In this case, the work W 2 performed by the
force-field is
W 2 = (^) A B:path 2 f·dr. (5.27)
Basically, there are two possibilities. F
→
irstly, the line-integrals (5.2 6 ) and (5.2 7 )
might depend on the end points, A and B, but not on the path taken between
them, in which case W 1 = W 2. Secondly, the line-integrals (5.26) and (5.27)
might depend both on the end points, A and B, and the path taken between
them, in which case W 1 = W 2 (in general). The first possibility corresponds
to what physicists term a conservative force-field, whereas the second possibility
corresponds to a non-conservative force-field.
What is the physical distinction between a conservative and a non-conservative
force-field? Well, the easiest way of answering this question is to slightly modify
the problem discussed above. Suppose, now, that the object moves from point
A to point B along path 1, and then from point B back to point A along path 2.
What is the total work done on the object by the force-field as it executes this
closed circuit? Incidentally, one fact which should be clear from the definition of
a line-integral is that if we simply reverse the path of a given integral then the
2
1