8.2 Conservative and non-conservative forces

In Potential Energy and Conservation of Energy , any transition between kinetic and potential energy conserved the total energy of the system. This was path independent, meaning that we can start and stop at any two points in the problem, and the total energy of the system—kinetic plus potential—at these points are equal to each other. This is characteristic of a conservative force    . We dealt with conservative forces in the preceding section, such as the gravitational force and spring force. When comparing the motion of the football in [link] , the total energy of the system never changes, even though the gravitational potential energy of the football increases, as the ball rises relative to ground and falls back to the initial gravitational potential energy when the football player catches the ball. Non-conservative forces are dissipative forces such as friction or air resistance. These forces take energy away from the system as the system progresses, energy that you can’t get back. These forces are path dependent; therefore it matters where the object starts and stops.

[In [link] , we use the notation of a circle in the middle of the integral sign for a line integral over a closed path, a notation found in most physics and engineering texts.] [link] and [link] are equivalent because any closed path is the sum of two paths: the first going from A to B , and the second going from B to A . The work done going along a path from B to A is the negative of the work done going along the same path from A to B , where A and B are any two points on the closed path:

You might ask how we go about proving whether or not a force is conservative, since the definitions involve any and all paths from A to B , or any and all closed paths, but to do the integral for the work, you have to choose a particular path. One answer is that the work done is independent of path if the infinitesimal work $\overrightarrow{F}·d\overrightarrow{r}$ is an exact differential    , the way the infinitesimal net work was equal to the exact differential of the kinetic energy, $d{W}_{\text{net}}=m\overrightarrow{v}·d\overrightarrow{v}=d\frac{1}{2}m{\text{v}}^2,$

when we derived the work-energy theorem in Work-Energy Theorem . There are mathematical conditions that you can use to test whether the infinitesimal work done by a force is an exact differential, and the force is conservative. These conditions only involve differentiation and are thus relatively easy to apply. In two dimensions, the condition for $\overrightarrow{F}·d\overrightarrow{r}=F_{x}dx+F_{y}dy$ to be an exact differential is