8.2 Conservative and non-conservative forces  (Page 2/6)

You may recall that the work done by the force in [link] depended on the path. For that force,

Therefore,

which indicates it is a non-conservative force. Can you see what you could change to make it a conservative force?

Before leaving this section, we note that non-conservative forces do not have potential energy associated with them because the energy is lost to the system and can’t be turned into useful work later. So there is always a conservative force associated with every potential energy. We have seen that potential energy is defined in relation to the work done by conservative forces. That relation, [link] , involved an integral for the work; starting with the force and displacement, you integrated to get the work and the change in potential energy. However, integration is the inverse operation of differentiation; you could equally well have started with the potential energy and taken its derivative, with respect to displacement, to get the force. The infinitesimal increment of potential energy is the dot product of the force and the infinitesimal displacement,

Here, we chose to represent the displacement in an arbitrary direction by $d\overrightarrow{l},$ so as not to be restricted to any particular coordinate direction. We also expressed the dot product in terms of the magnitude of the infinitesimal displacement and the component of the force in its direction. Both these quantities are scalars, so you can divide by dl to get

This equation gives the relation between force and the potential energy associated with it. In words, the component of a conservative force, in a particular direction, equals the negative of the derivative of the corresponding potential energy, with respect to a displacement in that direction. For one-dimensional motion, say along the x -axis, [link] give the entire vector force, $\bar{F}=F_x\widehat{i}=-\frac{\partial U}{\partial x}\widehat{i}.$