Conceptual problem with conservation of energy

This really isn't a homework problem with specific computations, rather it's a conceptual problem I'm struggling with; hence, the template doesn't really apply. (Please don't delete my thread!)

In the derivation of the work-energy theorem

$$W_C = \int_C \mathbf{F} \cdot d\mathbf{r} = T_2 - T_1$$

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 $$\mathbf{F}_G$$ then we know that it can be represented by the negative of the gradient of its potential function, or

$$\mathbf{F}_G = - \nabla U$$.

But, then

$$W_C = \int_C \mathbf{F}_G \cdot d\mathbf{r} = U_1 - U_2,$$

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

$$T_1 + U_1 = T_2 + U_2$$,

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.

Answers and Replies

vela
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You're right. To put it simply, energy is only conserved when you're dealing with conservative forces. That's why they're called conservative forces. :) If you have a non-conservative force, like friction, you don't have $T_1+U_1=T_2+U_2$. Instead, you have $T_1+U_1+W_{NC}=T_2+U_2$, to account for the work of the non-conservative force.

How do we justify applying 'conservation of energy' to problems involving blocks sliding down (frictionless) planes? What about the normal force? Doesn't that throw a wrench into the whole works?

vela
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The normal force doesn't do any work because it's perpendicular to the direction of motion. So perhaps it would be more precise to say that energy is conserved when only conservative forces perform work.

Holy buckets. There it is. How did I overlook that?

Thanks for the help!

Just wondering that energy is always conserved if you know where it went.
But besides frictional forces,can you justify energy conservation if external forces perform work?
Change in mechanical energy=W(non-conservative)+W(external).