# Work done by non conservative force

1. Jul 28, 2011

### pagmax

If the total work done is calculated using the area of the closed curve (Force vs Displacement), then the formulation doesn’t care if the force is conservative or not. Is that right? For instance, if I understand it right, spring force is a conservative force and hence work done by the spring in bringing a mass back to its original position should be zero. However, if we are able to plot spring force against displacement, we surely have a non-zero area under it.
My essentially question is can the work done by a non-conservative can be calculated by area under Force vs Displacement curve? Is that right? So when we calculate work done by a gas in piston as area under PV curve of the thermodynamic cycle, this is because gas force is a non-conservative force?

2. Jul 28, 2011

### gsal

hhhmmm...I don't remember my physiscs that well and what the difference between conservative and non-conservative force is...

...but are you sure you have a resulting work at the end? I mean, it may seem like there is area under the curve, but don't forget that your distance travel on the way back is of opposite sign as when going forward, and so, if you integrate that "area under the curve" is going to cancel out ...

does this make sense?

3. Jul 28, 2011

### Dickfore

I think you have a misunderstanding. The work would be the area in the coordinate plane Force vs displacement. But, this is true only in a one-dimensional case. In more than one dimension, the force is a curvilinear integral:
$$W = \int_{C}{\mathbf{F} \cdot d\mathbf{x}} = \int_{C}{F_{x} \, dx + F_{y} \, dy + F_{z} \, dz}$$
The force is non-conservative if the work is different for different contours in three-dimensional space between the same two points. There is no area that you can construct in this case.