# Conservation of Energy and Momentum Regarding Forces - clarification needed

The other day, my teacher stated something along the lines of, "Conservation of momentum is not violated by the actions of internal forces, but the conservation of energy is violated. Energy is conserved only when/if internal forces are conservative."

Can anyone explain this is simpler terms or reiterate it in some fashion?

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The real difficulty here is that your instructor hasn't really told you what is meant by "violating conservation of energy" in this context. There isn't a real violation going on, just a removal of some of the energy for considered forms (probably macroscopic kinetic energy and gravitational potential) into unconsidered forms (heat, sound, light). Some of this will presumably be clarified latter in the course. – dmckee Apr 4 '12 at 14:49

Consider the simple case of a system with two objects $A$ and $B$.

The internal force $\mathbf{F}_{BA}$ acted by $B$ on $A$ changes the momentum of $A$ by $\Delta \mathbf{P}_{A}=\mathbf{F}_{BA}t$, where $t$ is the duration of the force. Similarly, the internal force $\mathbf{F}_{AB}$ acted by $A$ on $B$ changes the momentum of $B$ by $\Delta \mathbf{P}_{B}=\mathbf{F}_{AB}t$.

Newton's third law tells us that $\mathbf{F}_{AB}=-\mathbf{F}_{BA}$, which implies that the change in total momentum $\Delta \mathbf{P}_{A}+\Delta \mathbf{P}_{B}=0$. This is why the actions of internal forces conserve momentum.

However, internal forces do not always conserve the energy of a system. A conservative force is defined to be one that conserves energy. Examples are gravitational force, spring force, and electric force. Such forces convert the kinetic energy of the system to potential energy and vice versa, without any loss of energy. Non-conservative forces, such as friction, cause the system to lose energy to other forms, such as heat.

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+1. Succinct and easy to understand. Go Newton! And you! :) – Joe Apr 6 '12 at 2:37

A conservative force is one in which the work done by the force on a body is independent of the path taken. For example, we can move a ball one meter up in multiple ways. We can just move it up, or we can move it to two meters and then let it fall. The net energy supplied to the system by you is the same, it is $mgh$. Now, lets look at processes where the ball comes back to where it is. You can move it to a height of one meter, and let it fall, but you won't be supplying any net energy. Whatever energy you supply will be released during the fall of the ball.

On the other hand, friction/drag/etc are nonconservative. Take a block on a rough surface. Lets say that the kinetic friction force has constant magnitude $f$. Now, move the block $x$ forward, and take it back. You will do work $2fx$ against friction (So friction does work $-2fx$). Even though there was no net change of position, there was work done. Now, work done=change in PE. But, potential at a point must be constant, so change in PE=0! So, potential is not definable.

So, to answer your question, if you have internal friction forces, energy will not be conserved since it will dissipate as heat. This also holds for other nonconservative forces (most dissipative forces are nonconservative, as are certain EM fields like $B$ field and induced $\bf E$ field )

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Induced EM fields are nonconservative? I don't believe I've heard that before... – David Z Apr 4 '12 at 4:47
@DavidZaslavsky: Induced E field is circular.. Nonconservative. You can't define an absolute $\phi$(potential) for it, since the path integral varies. Faraday's law: $V=-\frac{\rm d \phi_B}{\rm d t}$ Since it's flux, it works for a closed loop. You can take an electron around the loop, and have a change in energy, even if it comes back to the initial position. Also $\nabla\times\bf E_{ind}\neq 0$ – Manishearth Apr 4 '12 at 4:55
@DavidZaslavsky: Though since you're saying this, I feel unsure about it now :P – Manishearth Apr 4 '12 at 4:56
Unless you got confused by the "induced"-- I was talking about induced E and B. Fixed now. Yes, B fields aren't exactly nonconservative, since they do no work on charges. Just listing it as I'm quite sure that B's "conservativeness" is slightly controversial – Manishearth Apr 4 '12 at 4:58
Ah, yeah, you're right. I was mixing up "nonconservative" and "dissipative" (since usually when I'm talking about these concepts, it's in the context of teaching intro mechanics, where the only nonconservative forces are dissipative). – David Z Apr 4 '12 at 5:09

Energy is always conserved if the system considered is closed. So, depending upon the reference considered you can state whether energy is conserved or not relative to that reference.

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