# 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. Oct 15, 2016 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. Apr 16, 2017 at 22:32

The work-energy theorem can be proven directly from Newton's 2nd law, without any reference to conservative or nonconservative forces.

The relations between conservtive forces and their potential energy (and in fact, the existence of a scalar function satisfying these relations) is an entirely mathematical theorem. See here, here, and here.

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.

• So, the main reason behind defining the work done by conservative forces as the negative of the change in potential energy is to comply with the law of conservation of mechanical energy when the work done by non-conservative forces is zero. Right?
– MrAP
Oct 16, 2016 at 14:19
• Exactly. But one must remember that only the sign convention is really a definition. The existence of a potential energy function which satisfies that relation is a nontrivial mathematical theorem, not a definition. Oct 16, 2016 at 14:32
• I could not understand the last part of your comment.
– MrAP
Oct 16, 2016 at 14:40
• Hey what happened?
– MrAP
Oct 17, 2016 at 9:36
• Sorry, I've been busy. I was just saying that it's not trivial that when a force is conservative, the work it does is equal to the positive/negative difference of some function of position. So I linked to some articles explaining how it was proven (and now I've added another useful article). Now, the sign convention (taking the negative difference) has been chosen for physical reasons (conservation of energy), but it's completely equivalent to define it with either sign - because if you have a function that satisfies the relation with a positive sign, just take its minus, and vice versa. Oct 19, 2016 at 14:48