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In the derivation of the work-energy theorem

[tex]W_C = \int_C \mathbf{F} \cdot d\mathbf{r} = T_2 - T_1[/tex]

Newton's second law of motion is used. Hence, the force appearing in the line integral is actually the

*sum*of all forces. Correct?

Now, on the other hand, if we have a conservative force [tex]\mathbf{F}_G[/tex] then we know that it can be represented by the negative of the gradient of its potential function, or

[tex]\mathbf{F}_G = - \nabla U[/tex].

But, then

[tex] W_C = \int_C \mathbf{F}_G \cdot d\mathbf{r} = U_1 - U_2, [/tex]

by the fundamental theorem of line integrals. If we equate the two expressions for work, we have

[tex] T_1 + U_1 = T_2 + U_2[/tex],

which leads to conservation of mechanical energy. However, doesn't this only apply if the

**ONLY**force working on the particle in question is conservative? It seems like we're comparing apples and oranges here. In the work-energy theorem, we're talking about the work from the sum of all forces, whereas in the argument dealing with a conservative force, we're dealing with the work done only by that force. So if we have two different works (one from the sum of all forces and the other from just the conservative force), how can they be equated? It seems to me that this can only happen when the sole force working on the particle is the conservative force!

I have a feeling I've made a huge error in logic somewhere, but I can't seem to find it. Maybe I'm completely misunderstanding the arguments. I've consulted Kleppner's book and Marion and Thornton and am still not comfortable with the topic.