# Why is work done by non-conservative force equal to change in mechanical energy?

In some places, it is stated that the work done by a conservative forces is equal to the negative of the change in the potential energy (without any proof). Then it is derived that the work done by non-conservative forces is equal to the change in mechanical energy using work energy theorem. In some other places it is just the reverse; it stated that the work done by non-conservative forces is equal to the change in mechanical energy and using this the work done by conservative force is derived.

This seems circular to me and I don't understand where to start from. If I start with conservative forces, why is the work done by conservative force equal to the negative of the change in potential energy or if I start with non-conservative forces why is the work done by non-conservative forces equal to the change in mechanical energy?

• Most conceptual structures in physics can be viewed as explanatory in either direction. There is nothing unusual or troubling about that. The whole work and energy conceptual structure is self-consistent, but has to be boostrapped from somewhere. The best place to perform the bootstrap depends on the audience. Nor should you worry that the bootstrap seems arbitrary: the real test of these idea is how well they survive experimental testing. – dmckee --- ex-moderator kitten Oct 15 '16 at 21:07
• Note that the definition of potential energy is: the negative of internal work done by conservative internal forces. Thus no proof can be given. But note also the importance of the work "internal" here. – garyp Apr 16 '17 at 22:32

After both of these results have been proven, one can separate the total work appearing in the work-energy theorem into the conservative and nonconservative parts. Since the conservative work is minus the difference in potential energy one can move it to the other side of the equation and get a positive difference in the total mechanical energy. In fact, the potential energy is defined so that the work equals the negative difference in the potential energy exactly because we want to get the difference in the mechanical energy (and not the difference in kinetic energy minus the difference in potential energy). That way, when the nonconservative forces do $0$ work we get conservation of energy.