Law of Conservation of energy and Wnc

[guyvsdcsniper]

This is my understanding of the law of conservation of energy and the role non conservative forces factor into it. Could someone confirm if I have this right or explain where I am going wrong if I am? I would appreciate it.

With the law of conservation of mechanical energy, ΔKE+ΔPE=0. This essentially mean energy is converted from KE to PE or vice versa and the amount of energy converted should be equal. There is no energy lost. We can generalize this and turn this into the law of conservation of energy by saying ΔKE+ΔPE+[change in all other forms of energy]=0. So the change in all energy done in a system should equal zero. Because energy is conserved.

That is ideal. But in reality we have dissipative forces that cause us to loose energy from a system. We can then derive an equation to find the quantity of these dissipative force are by saying ΔKE+ΔPE=Wnc. This essentially says that the change in the mechanical energy will be equal to the work done by nonconservative forces. So this means the work done by dissipative forces will be equal to the change in mechanical energy.

 

[kuruman]

Personally I prefer your first statement "ΔKE+ΔPE+[change in all other forms of energy]=0" although it, in itself, is not entirely correct. The other side of the equation should not be zero but W, the work done by forces external to the system. But let's say you have a closed system in which there are no external forces.

Consider a weightlifter who lifts a weight over his head by stretching up from a crouching position at constant speed. The system is the weightlifter + weight + Earth, so it's closed and there are no external forces doing work. There is no change in kinetic energy, only an increase in potential energy: $\Delta U=Mg\Delta h_{\text{cm}}+W\Delta H$ where $M$ is the weightlifter's mass and $W$ the weight he is lifting. You can write that down as $\Delta U$, but you cannot write it down as $W_{\text{nc}}=\int \vec F_{\text{nc}}\cdot d\vec s$ because you have no idea what $\vec F_{\text{nc}}$ looks like unless you have a biomechanical model that is detailed enough to sort out how the non-conservative forces are split between displacing his CM and the weight.

So you might as well leave $\int \vec F_{\text{nc}}\cdot d\vec s$ on the other side of the equation and consider it a change in the internal energy of the system. Hence my preference as stated above because it is applicable to all cases, including deformable systems. Your formulation is applicable only to cases where $W_{\text{nc}}=\int \vec F_{\text{nc}}\cdot d\vec s_{\text{cm}}$ as in, for example, a block sliding across a table and stopping because of friction. The center of mass subscript makes a difference.

 

[anorlunda]

But in reality we have dissipative forces that cause us to loose energy from a system.

I think your confusion relates to your definition of "a system". If a sliding block dissipates energy via friction with the surface, then the surface must be included in "the system" to account for conservation of energy.

That's why physics likes to talk about a "closed system" emphasizing that nothing comes in or goes out. Dissipative processes inside the closed system conserve energy. Dissipative processes that allow energy to cross the border of the system are not closed.

So rather than question conservation of energy (which is a consequence of Noether's Theorem if you want to study more), you would be better off figuring out "where did the energy go?" That can be very difficult in some cases.

 

[guyvsdcsniper]

Personally I prefer your first statement "ΔKE+ΔPE+[change in all other forms of energy]=0" although it, in itself, is not entirely correct. The other side of the equation should not be zero but W, the work done by forces external to the system. But let's say you have a closed system in which there are no external forces.

Consider a weightlifter who lifts a weight over his head by stretching up from a crouching position at constant speed. The system is the weightlifter + weight + Earth, so it's closed and there are no external forces doing work. There is no change in kinetic energy, only an increase in potential energy: $\Delta U=Mg\Delta h_{\text{cm}}+W\Delta H$ where $M$ is the weightlifter's mass and $W$ the weight he is lifting. You can write that down as $\Delta U$, but you cannot write it down as $W_{\text{nc}}=\int \vec F_{\text{nc}}\cdot d\vec s$ because you have no idea what $\vec F_{\text{nc}}$ looks like unless you have a biomechanical model that is detailed enough to sort out how the non-conservative forces are split between displacing his CM and the weight.

So you might as well leave $\int \vec F_{\text{nc}}\cdot d\vec s$ on the other side of the equation and consider it a change in the internal energy of the system. Hence my preference as stated above because it is applicable to all cases, including deformable systems. Your formulation is applicable only to cases where $W_{\text{nc}}=\int \vec F_{\text{nc}}\cdot d\vec s_{\text{cm}}$ as in, for example, a block sliding across a table and stopping because of friction. The center of mass subscript makes a difference.

So ΔKE+ΔPE+[change in all other forms of energy]=0 should technically be ΔKE+ΔPE+[change in all other forms of energy]=Wnc to account for external forces. But that makes things difficult cause there could be so many different factors of dissipative forces that it would be hard to account for them. So therefore for closed systems ΔKE+ΔPE+[change in all other forms of energy]=0 would be ok?

But if we have a closed system we can consider Wnc the change in internal energy/thermal energy. We could really only use ΔKE+ΔPE+[change in all other forms of energy]=Wnc for simple problems like the box sliding and stopping due to friction?

Am I interpreting your response correctly?

 

[guyvsdcsniper]

I think your confusion relates to your definition of "a system". If a sliding block dissipates energy via friction with the surface, then the surface must be included in "the system" to account for conservation of energy.

That's why physics likes to talk about a "closed system" emphasizing that nothing comes in or goes out. Dissipative processes inside the closed system conserve energy. Dissipative processes that allow energy to cross the border of the system are not closed.

So rather than question conservation of energy (which is a consequence of Noether's Theorem if you want to study more), you would be better off figuring out "where did the energy go?" That can be very difficult in some cases.

Ok so in a closed system energy is just conserved into another form like heat. That doesn't mean the energy was lost but it is now in a different form. is that correct?

 

[kuruman]

Am I interpreting your response correctly?

Perhaps you might wish to read this insight to see where I am coming from regarding $W_{\text{nc}}$.

 

[guyvsdcsniper]

Perhaps you might wish to read https://www.physicsforums.com/insights/is-mechanical-energy-conservation-free-of-ambiguity/?preview_id=26278&preview_nonce=2d38f097b9&post_format=standard&_thumbnail_id=28730&preview=true to see where I am coming from regarding $W_{\text{nc}}$.

It says I am not allowed to preview drafts

 

[kuruman]

Sorry I put in the wrong link by mistake. Here it is in full.
https://www.physicsforums.com/insights/is-mechanical-energy-conservation-free-of-ambiguity/
Please try again.

 

[anorlunda]

Ok so in a closed system energy is just conserved into another form like heat. That doesn't mean the energy was lost but it is now in a different form. is that correct?

Yes. Energy is conserved, not a particular kind of energy. Think of an electric motor. It's purpose is to convert electric energy to mechanical energy.