# Law of Conservation of energy and Wnc

• guyvsdcsniper
In summary, the law of conservation of energy states that the total energy in a closed system remains constant, with the change in all forms of energy equalling zero. However, in reality, dissipative forces can cause energy to be lost from a system. This can be accounted for by the equation ΔKE+ΔPE=Wnc, where the change in mechanical energy is equal to the work done by nonconservative forces. It is important to consider the definition of a closed system and to determine where the lost energy has gone, rather than questioning conservation of energy itself.

#### 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.

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.

• guyvsdcsniper
quittingthecult said:
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
kuruman said:
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?

anorlunda said:
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?

quittingthecult said:
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}}##.

Last edited:
kuruman said:
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

quittingthecult said:
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.

## 1. What is the Law of Conservation of Energy?

The Law of Conservation of Energy states that energy cannot be created or destroyed, it can only be transformed from one form to another. This means that the total amount of energy in a closed system remains constant.

## 2. How does the Law of Conservation of Energy relate to Work-Energy Theorem?

The Work-Energy Theorem is a specific application of the Law of Conservation of Energy. It states that the net work done on an object is equal to the change in its kinetic energy. This is in accordance with the Law of Conservation of Energy, as the work done on the object is transformed into kinetic energy.

## 3. What is the significance of the Law of Conservation of Energy in everyday life?

The Law of Conservation of Energy is a fundamental principle in physics and has many practical applications in our daily lives. It helps us understand how energy is transferred and transformed in various processes, such as chemical reactions, electrical circuits, and mechanical systems. It also allows us to make more efficient use of energy resources.

## 4. Can the Law of Conservation of Energy be violated?

No, the Law of Conservation of Energy is a universal law of nature and has been proven to hold true in all physical processes. However, it may appear to be violated in certain situations, such as when energy seems to disappear, but this is due to our limited understanding or measurement of all the forms of energy involved.

## 5. What is the relationship between the Law of Conservation of Energy and Wnc (work not done by conservative forces)?

Wnc is the work done by non-conservative forces, which are forces that do not follow a specific path and cannot be represented by a potential energy function. According to the Law of Conservation of Energy, the total work done on an object must equal the change in its total energy. This includes the work done by conservative forces (Wc) and the work done by non-conservative forces (Wnc), such that Wc + Wnc = ΔE. Therefore, Wnc can be seen as the work that is not accounted for by potential energy, and it is in accordance with the Law of Conservation of Energy.