Can Energy Transfer Occur with Zero Total Energy Change in Newtonian Mechanics?

[fortaq]

Hello,

The statement in question is:
"In Newtonian mechanics, changes in the energy state of a point particle are described by the equation dK + dU = dE, whose terms are the (infinitesimal) changes in kinetic, potential, and total energy, respectively. The total energy change dE describes the energy transfers to and from the particle (and is zero for conservative processes)."

My question is:
Is the last sentence true, i.e., may energy transfer to or from the particle only be associated with a total energy change dE of the particle? Doesn't it also represent an energy transfer to or from the particle if dE = 0 but dU (and so also dK) is different from 0?

tnx, fortaq

 

[PhanthomJay]

Hello,

The statement in question is:
"In Newtonian mechanics, changes in the energy state of a point particle are described by the equation dK + dU = dE, whose terms are the (infinitesimal) changes in kinetic, potential, and total energy, respectively. The total energy change dE describes the energy transfers to and from the particle (and is zero for conservative processes)."

My question is:
Is the last sentence true, i.e., may energy transfer to or from the particle only be associated with a total energy change dE of the particle? Doesn't it also represent an energy transfer to or from the particle if dE = 0 but dU (and so also dK) is different from 0?

tnx, fortaq

I see your point, i think, because I do not like the form of this equation. I prefer to see it written as dk + dU + dE_other = 0, where dE_other is the (infintesimal) changes in other forms of energy (like heat, sound, chemical energy, etc.). In this way, the equation states that there can never be any changes of the total energy of a system, that is, total energy (U + K + E_other) is always conserved, it just takes on different forms. When only conservative forces act, dk + dU =0; when non conservative forces act, du + dK = work done by non conservative forces, where the work done by the non conservative forces represents the transfers to or from the U and K of the system , in the form of heat, sound, etc. When E_other is 0, energy transfers are between K and U only, with no heat, sound ,etc. forms of energy involved.

 

[Bhumble]

I think the idea is that a conservative process dE = 0 => any change in dK results in an equal but opposite change in dU and vice versa.

PhanthomJay is showing the form of a non-conservative process. I don't believe they really exists... you just have to adjust what is included in your system...

 

[fortaq]

Okay, but it is still not so clear for me. Let me rephrase the question:

What is energy transfer to or from the particle?
a) dK + dU = dE, which is the particle's total energy change.
b) dK = -dU + dE, which is the total work performed on the particle.

@ PhanthomJay: you seem to prefer answer a), right?

 

[PhanthomJay]

Okay, but it is still not so clear for me. Let me rephrase the question:

What is energy transfer to or from the particle?
a) dK + dU = dE, which is the particle's total energy change.
b) dK = -dU + dE, which is the total work performed on the particle.

@ PhanthomJay: you seem to prefer answer a), right?

The equations are equivalent, in that dE , in equation a), represents the energy transferred to or from the system by non conservative forces that do work, and that in equation b), dK represents the total work done on the system by both conservative and non-conservative forces that do work. In trying to understand the concept of energy transfer, equation a) is more understandable, as long as you realize that dE represents the energy transferred to or from the system by non conservative forces that do work, and represents a loss or gain of energy to the system in the forms of heat, light, electric, chemical, sound energy, etc, in other words, a gain or loss of energy in the form of energy other than kinetic or potential energy. And that is why I prefer to use an equation (c), that is, dU + dK + dE_other = 0, where dE_other is forms of energy other than kinetic or potential energy, which, in my mind, helps in the understanding that total energy is always constant, it cannot be changed or created, only transformed to different forms. I guess it is a matter of preference as to how to view these equations.

 

[fortaq]

But you talk about a system, whereas I talk about a (single) particle that interacts with other particles. That is, dE_other does not exist in my equations. But also in the case of a system those other forms of energy (heat, electric, chemical, sound etc.) are in fact kinetic and potential energies of the individual particles in the system, therefore we can get along with only kinetic and potential energies also in the case of a system.

Anyway, i think we agree that non-conservative forces account for energy transfers between particles, in the sense that a particle's total energy increases while that of an other particle decreases (e.g. during a two-particle collision, where one particle is at rest). But do also conservative forces account for energy transfer between particles? E.g., if a ball falls down in the Earth's gravity field (without atmospheric friction), then is there energy transfer between Earth and the ball, i.e., does the total energy of the Earth decrease while the total energy of the ball increases?

 

[Bhumble]

dK is the kinetic energy, dU is the potential energy, and dE is the total energy. Let's limit it to just the particle for now. This means that it is impossible for dE to change since nothing else exists. This means that dE = 0 since E is constant. So in order for dK + dU = dE =0 any change in kinetic energy must have an equal but opposite change in potential energy.

Now let's look at the case of a ball and the earth. We have a kinetic energy and a potential energy the sum of which is the total energy of the system.
K + U = E

Let's say we drop the ball from rest so K = 0. And we are at some height "h", so U = mgh. Let's just say the m = 1 kg, and initial h = - 1 m (just for convention to keep U positive).
g = -9.8 m/s^2.
Initially K + U = 0 + 9.8 (kg*m^2/s^2) = E = 9.8 J

Now say that after some time the ball hits the earth. Now h = 0 but we know that the ball is now moving or in other words it has kinetic energy.
E = 9.8 J = K + U = K + 0

dE = 0, dU = -9.8 J
since dE = dK + dU => 0 = dK - 9.8 J => dK = 9.8 J

So the situation you are describing shows that the total energy remains the same it just changes it's form from potential energy to kinetic energy. This could happen in the opposite scenario too. Let's say you throw a ball into the air giving it some amount of kinetic energy but at some point it is going to fall back down so you know the kinetic energy is zero when the ball comes to a peak height and it will have a potential energy proportional to its height.

I think the key concept you want to get out of this is that in a closed system (ignore everything external) that the total energy is constant and that it is only possible to change whether it is kinetic or potential energy.

 

[fortaq]

Oh yes, sorry, in my previos post I've asked if the ball's total energy changes or not (and of course it does not), but in fact I wanted to ask if Earth's gravity transfers energy to the ball while gravity performs work (F.dx > 0) on the ball.

 

[PhanthomJay]

Oh yes, sorry, in my previos post I've asked if the ball's total energy changes or not (and of course it does not), but in fact I wanted to ask if Earth's gravity transfers energy to the ball while gravity performs work (F.dx > 0) on the ball.

dU + dK + dE_other = 0

Since dE_other = 0, then

dU + dK = 0

rearranging

dk = -dU

and since W_gravity = -dU , then

dK = W_gravity

That is, the work done by gravity changes the kinetic energy of the ball, and that increase in its kinetic energy was transferred to it by its loss of potential energy.

 

[fortaq]

So you mean, merely the ball's potential energy transforms into kinetic energy, therefore no energy is transferred to the ball?

 

[PhanthomJay]

So you mean, merely the ball's potential energy transforms into kinetic energy, therefore no energy is transferred to the ball?

No, the ball's change (loss) of its potential energy results in a change (increase) of the ball's kinetic energy. Total energy in itelf is constant ...the sum total of its changes in forms is 0.

 

[PhanthomJay]

Hello,

The statement in question is:
"In Newtonian mechanics, changes in the energy state of a point particle are described by the equation dK + dU = dE, whose terms are the (infinitesimal) changes in kinetic, potential, and total energy, respectively. The total energy change dE describes the energy transfers to and from the particle (and is zero for conservative processes)."

My question is:
Is the last sentence true, i.e., may energy transfer to or from the particle only be associated with a total energy change dE of the particle? Doesn't it also represent an energy transfer to or from the particle if dE = 0 but dU (and so also dK) is different from 0?

tnx, fortaq

I think I see the source (besides me ) of the confusion. We know that total energy of a system can never change, thus total energy of a system is always constant.
The equation dU + dK = dE seems to say that total energy does change when du + dK is non zero, that is, when non conservative forces that do work are acting. This seems like a paradox, but you were correct about particle energy vs. system energy.

Supposing a particle of mass m was moving at constant velocity v along a frictionless level surface, with a KE of 1/2mv². Then it continues on until it hits a rough surface with friction, and it comes to a stop. It now has lost all it's energy. But since energy can never be lost, where did it go? It went to other forms of energy (especially heat and sound) that was transferred to the particle/surface/surrounding air system from the work done by the non conservative friction force . Similarly, supposing the particle of mass m and velocity v was moving on a frictionless level surface when suddenly an external force is applied that causes it to accelerate and gain speed. In this case, the particle energy has increased, so it's increase in energy was transferred into the system from the work done by the applied non conservative force .

Now back to your question, about when dE =0 (only conservative forces acting) and the transfer of energy of the particle is from kinetic energy to potential energy or vice versa. In this case, I would not call it a transfer of energy, it is rather a transformation of energy of the particle into different forms. The term transfer, per the original equation, apllies to energy when non conservative forces act that is transferred from the particle to the system. When conservative forces only are acting, dE is zero, thus , the transferred energyto the system is 0, and the changes of the particle energy are changes in its KE and PE only transformations, with no effect on the system in terms of gains or losses from other forms of energy.

Confusing?

 

[fortaq]

We know that total energy of a system can never change, thus total energy of a system is always constant.

This may only be true if we impose constraints on the system, e.g., isolation. But nature itself does not require a system to be isolated so that its total energy is constant, but rather allows energy transfer to or from a system. (Of course, we may always include the energy transferring particles into the system, so that we end up with a bigger system that is isolated. But this method after all just leads to the rather trivial statement that the universe's total energy is constant.)

I think the other part of the discussion is clear now.

 

[PhanthomJay]

This may only be true if we impose constraints on the system, e.g., isolation. But nature itself does not require a system to be isolated so that its total energy is constant, but rather allows energy transfer to or from a system.

We're probably saying the same thing, but I am not sure. If, in the example of a particle sliding on a surface with friction, we consider the system as the particle and surface only, isolated from everything else in the Universe, then the energy within the system cannot change. As you noted, we have imposed constraints on the system. But the constraints are arbitrary. If instead we choose the system to be the particle, surface, and surrounding air, isolating it (imposing constraints) from the rest of the Universe, then energy within this new system cannot change. Now if, instead, we consider the particle and surface as System A, and the surrrounding air as system B, and we do not isolate A from B, then yes, energy is transferred from A to B, and system A plus B becomes our isolated system. And this can continue on to an infinite number of systems, showing that the total energy of the Universe cannot change. Are we saying the same thing?

 

[fortaq]

Yes, we are saying the same thing, but I consider this statement (that the universe's total energy is constant) as the statement with the lowest possible information content about energy conservation. Since the universe is the only system that is actually isolated and any other subsystem may just be a more or less good approximation of an isolated system, we have a rather "unsubstantial" definition of energy conservation if we define it by means of an isolated system.

 

[PhanthomJay]

. Since the universe is the only system that is actually isolated and any other subsystem may just be a more or less good approximation of an isolated system, we have a rather "unsubstantial" definition of energy conservation if we define it by means of an isolated system.

Yes, I agree, except when only non conservative forces are acting within the system.

I consider this statement (that the universe's total energy is constant) as the statement with the lowest possible information content about energy conservation

And it has not been ruled out that that constant might be zero (as in 0).

 

[fortaq]

Yes, I agree, except when only non conservative forces are acting within the system.

Why? What happens in that case? I mean, if a system is isolated, then its energy should remain constant, no matter if the forces within the system are conservative or non-conservative.

 

[PhanthomJay]

Why? What happens in that case? I mean, if a system is isolated, then its energy should remain constant, no matter if the forces within the system are conservative or non-conservative.

If a system is isolated, then yes, the energy remains constant regardless of whether the forces are conservative or non conservative. But I am talking transfer of energy away from the system to another system, or into the system from another system. When only conservative forces act, there is no transfer across systems...that's why dU + dK = 0 for such systems, the dE = 0 implying that no energy is transferred away from or into the system. When non conservative forces act in your system A , I can sense the heat and sound etc, in my system B, it is transferred across systems. But if only conservayive forces act, your energy does not transfer to me , i can't sense your change in speed or position, such cahnges of which are internal to your system only.

I noted in the beginning that i did not like the form of the equation dU + dK = dE; om 2nd thought, it is not all that bad, and quite correct.

 

[fortaq]

So, we have to distinguish, on the one hand, between conservative and non-conservative forces, and, on the other hand, between system-internal forces and external forces on the system.
I think the problem results from our different approaches to the energy equation: you considered a system, where there are internal forces and therefore internal energy changes, whereas I considered a (point) particle, where there are only external forces acting on it. Putting these two approaches together, obtaining an open system, where there are both internal and external forces acting, we get a more comprehensive version of the energy equation:
dK + dU_internal + dU_external = dE_internal + dE_external,
where dE_internal, the system's total energy change caused by the system-internal non-conservative forces, is always zero. After all, this equation includes both dK + dU = dE (for the external forces) and dK + dU +dE_other=0 (for the internal forces), and dE_internal is similar to your dE_other.