# Conservation of momentum and energy

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I was wondering if there are cases in which momentum or energy are not both conserved together ie energy conserved, but momentum is not. The source of my confusion stems from the explanation that momentum is conserved in a "closed system" (no matter exchange, no external forces) while energy is conserved in an "isolated system" (no mass or energy exchange with environment).

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Momentum and energy are always concerved, noting mass-energy conversion possibility. In those systems where it is not called concerved, it means that some of it is lost to or gained from something outside the system of interest.

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Yes, there certainly are.

Consider an object of mass m moving at a speed v in a circle of radius r. Energy will be conserved; here we simply have kinetic energy. 1/2 mv2. But momentum will not be conserved, because circular motion requires a centripetal force, and forces change momentum. The force does no work, however, because it is is always perpendicular to the motion.

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Yes, there certainly are.

Consider an object of mass m moving at a speed v in a circle of radius r. Energy will be conserved; here we simply have kinetic energy. 1/2 mv2. But momentum will not be conserved, because circular motion requires a centripetal force, and forces change momentum. The force does no work, however, because it is is always perpendicular to the motion.

What is keeping the object moving in the circle? If it is attached to something, the momentum calculation must include the momentum of this something.

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What is keeping the object moving in the circle? If it is attached to something, the momentum calculation must include the momentum of this something.

No, you don't have to include the source of the force in your problem. That's the whole concept behind solving physics problems — you can define your system in such a way that a problem can be solved. You may not know enough detail about the source of the force, or including it may obscure the detail you're trying to find.

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What is keeping the object moving in the circle? If it is attached to something, the momentum calculation must include the momentum of this something.

The initial confusion was over the fact that the description of when momentum is conserved is not identical to the description of when energy is conserved. Swansont provided an example of a system that is defined in such a way that it conserves one but not the other.

Obviously any lost energy/momentum must go somewhere, but that isn't what the question was about.

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What is keeping the object moving in the circle? If it is attached to something, the momentum calculation must include the momentum of this something.

If you consider no external forces applied, ideal system, everything is conserved. If we are doing calculations you have to remember we don't care about the loses.

Eg example above is what is the centrifugal force of a rotating object. Then from there we can dtermine the thing you are talking about. the required force to keep the path of the object in circle which is something needs to hold it. We assume the system is ideal. The string must be able to handle a force equal to that of the centrifugal force to be able to maintain same radius of the circle.

in the case of energy. if there are no loses eg: heat friction air resistance gravity. It will continue to move on forever if no energy is transferred or transformed to other forms of energy. But we have to remember if it is a rotating body with centrifugal force acting, there is a force trying to deviate the path of the object then there are energy are loses.

Edited by gabrelov
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No, you don't have to include the source of the force in your problem. That's the whole concept behind solving physics problems — you can define your system in such a way that a problem can be solved. You may not know enough detail about the source of the force, or including it may obscure the detail you're trying to find.

Fair enough, but in an absolute sense, for any and every inertial frame extended without limits, momentum is conserved...correct?

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Fair enough, but in an absolute sense, for any and every inertial frame extended without limits, momentum is conserved...correct?

Yes, for a large enough system, there are no external forces.

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I'm sure the OP was referring to energy and momentum conservation of a closed system.

Otherwise we should be speaking of an energy continuity equation where we would include both the energy of the system as well as the energy flux in or out of the system. The same thing goes for momentum.

At the Newtonian level these can be shown to be a result of Noether's theorem and the properties of Euclidian space and absolute time.

The same can be said of charge conservation vs. charge continuity.

Stepping outside the Newtonian scope, energy to date, is not well defined on the curved manifold of general relativity. Noether's theorem does not apply. Without a proper definition of the energy of a system, nobody really knows.

It is interesting to note that charge continuity, however, is well defined on a curved manifold. It is most elegantly stated as simply dJ=0 in the language of differential forms, and is a direct result of applying an electromagnetic potential field to spacetime. The same elegance is not to be found for either energy or momentum. The apparent reason for this is that neither is a fundamental quantity.

So why do the conservation and continuity equations forr energy and momentum fail while those for charge do not?

Total charge is to total energy as electric current is to momentum.

Why do we know of a continuity equation for one pair and not the other? The short, rough answer is that charge-current obtains from a 3 dimensional object and energy-momentum from a 4 dimensional object.

The details involves some modern math beyond the scope of this forum.

Edited by decraig
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• 2 years later...

The initial confusion was over the fact that the description of when momentum is conserved is not identical to the description of when energy is conserved. Swansont provided an example of a system that is defined in such a way that it conserves one but not the other.

Obviously any lost energy/momentum must go somewhere, but that isn't what the question was about.

That question is now placed.

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That question is now placed.

If momentum is not conserved it's because the system has been defined in such a way that there is an external net force on the system.

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• 1 month later...

Yes, there certainly are.

Consider an object of mass m moving at a speed v in a circle of radius r. Energy will be conserved; here we simply have kinetic energy. 1/2 mv2. But momentum will not be conserved*, because circular motion requires a centripetal force, and forces change momentum. The force does no work, however, because it is is always perpendicular to the motion.

What happens to the momentum there*? Is it safe to say momentum is continuously transferring from x to y axii & visa versa (when z is the rotational axis)? In other words we never have a (permanent) stable value of momentum for any coordinate. Instead we have a varying flow of momentum. The momentum value is always changing, but in a complementry way, so that what 1 axis looses, the other axis gains, & visa versa. ? Edited by Capiert
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What happens to the momentum there*? Is it safe to say momentum is continuously transferring from x to y axii & visa versa (when z is the rotational axis)? In other words we never have a (permanent) stable value of momentum for any coordinate. Instead we have a varying flow of momentum. The momentum value is always changing, but in a complementry way, so that what 1 axis looses, the other axis gains, & visa versa. ?

There has to be a force on an object in order to have it move in a circle.In the presence of a net force, momentum is not conserved. F = dP/dt

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There has to be a force on an object in order to have it move in a circle.In the presence of a net force, momentum is not conserved. F = dP/dt

If we equate flow of momentum with force, could you elaborate there?
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If we equate flow of momentum with force, could you elaborate there?

Really this is a definition of force: 'the rate of change of momentum'. If the momentum is not constant then there is a force. And vice versa, if the net force is not zero, there there is change in momentum.

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If we equate flow of momentum with force, could you elaborate there?

Momentum is not force, change in momentum per unit time is. You can't just equate things because you want to.

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Flow of momentum is pressure, one of the components of the stress-energy tensor. (At the risk of sending Capiert off on another tangent!)

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• 4 weeks later...

Momentum is not force, change in momentum per unit time is. You can't just equate things because you want to.

"No one" said momentum is force.

It seems ajb #16 understood the question & answered it better, & more appropriately than you.

Physicists have been equating things (=the things they can) for centuries,

simply because they want to, e.g. Newton.

Without that creativity, we would be nowhere, a big zero!

Change of charge per time is current "flow".

I don't find my question was extraordinary,

but the answer seemed a bit rude.

(If you can't, it looked like ajb could.)

Perhaps "accumulation" was not obvious,

& that irritated you.

e.g. if we charge a capacitor,

(the charge accumulates at a final destination

inside the capacitor).

If momentum flows into a mass (in 1 direction),

then I would expect the mass to accelerate.

If the mass stays in equilibrium,

with no accumulation (of momentum),

(e.g. current flow analogy similar to a wire conductor)

then maybe it (the situatation)

is as Strange described #18,

as pressure (force per area).

Recognition is the 1st step to improvement.

Edited by Capiert
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"No one" said momentum is force.

You did, just a few posts back. "If we equate flow of momentum with force". Momentum doesn't flow, so if you meant something different, you failed in expressing it.

It seems ajb #16 understood the question & answered it better, & more appropriately than you.

No, our answers are basically the same.

Physicists have been equating things (=the things they can) for centuries,

simply because they want to, e.g. Newton.

Without that creativity, we would be nowhere, a big zero!

Um, no, not because they "Want" to.

Change of charge per time is current "flow".

Current is the amount of charge passing a point per unit time. If the current is constant, there is a constant amount of charge in any given region. How would you have an amount of momentum passing by you?

I don't find my question was extraordinary,

but the answer seemed a bit rude.

(If you can't, it looked like ajb could.)

Perhaps "accumulation" was not obvious,

& that irritated you.

e.g. if we charge a capacitor,

(the charge accumulates at a final destination

inside the capacitor).

If momentum flows into a mass (in 1 direction),

then I would expect the mass to accelerate.

If the mass stays in equilibrium,

with no accumulation (of momentum),

(e.g. current flow analogy similar to a wire conductor)

then maybe it (the situatation)

is as Strange described #18,

as pressure (force per area).

Recognition is the 1st step to improvement.

Momentum doesn't really "accumulate" as if it were a substance.

The bottom line here is that you are using non-standard terminology. Don't place the blame elsewhere if you are misunderstood, because by using your own version of the language you are making it easy to misunderstand you.

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"No one said momentum is a force."

You did, just a few posts back.

"If we equate flow of momentum with force".

I don't recognize momentum's "flow" (a statistic, e.g. change)

& momentum as the same thing.

I reject your accusation, as not true. I did not say momentum is a force.

I said the "flow" of momentum could be seen as a force, in that sentence, not momentum (as no flow, at all).

You obviously missed:

flow of momentum=(momentum's) "flow".

That is actually unmistakeably clear.

Please don't push the blame on me,

that wouldn't be fair.

Momentum doesn't flow,

Does it not?

What is a change of e.g. momentum

wrt to a location (or position)?

I would say flow.

so if you meant something different, you failed in expressing it.

ajb & Strange (seem to have) got the most out of it,

you apparently the least.

No, our answers are basically the same.

The same?

You started with the anti_thesis,

boalstering as thought I said something wrong,

(as a correction)

when you were not even on topic (thesis);

& then say it's basically the same.

You came in the back door,

& are covering it up.

Bravo: Anti_thesis, then thesis.

Not a word false there;

only the sequence is mixed up (=reversed),

giving away the (correcting=correction) intent.

(You obviously misinterpreted. No tragedy. Mistakes happen.)

Say it right the 1st time, please.

Um, no, not because they "Want" to.

To be fair, what then?

Newton had a talent for making formulas.

When Hailey visited Newton

with a problem,

in a manuscript.

I get the idea Newton worked on (some) things he wanted to,

because a contract for that problem had not existed.

Current is the amount of charge passing a point per unit time. If the current is constant, there is a constant amount of charge in any given region.

Yes, but it's moving.

How would you have an amount of momentum passing by you?

If someone shot at me, but missed.

Momentum doesn't really "accumulate" as if it were a substance.

But it is a "property" that can accumulate.

The bottom line here is that you are using non-standard terminology. Don't place the blame elsewhere if you are misunderstood, because by using your own version of the language you are making it easy to misunderstand you.

I'll try to improve, but I'm amazed how many understand, & also how many are not able to understand.

Standard terminology comes from the specialists. I look to them that they can bring things into context,

to make things understandable,

considering they know what their specialty is all about.

But can the specialists communicate with the public?

Nobody is perfect.

It is rediculous to expect that I know your specialty,

especially as good as you.

Otherwise I'd be sitting in your place.

Edited by Capiert
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Capiert, to save further argument over terminology I suggest you research

Momentum Transport

This is the technical term I think you are trying to portray.

https://en.wikipedia.org/wiki/Transport_phenomena

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Empirical evidence aside, unless you can find at least a few physics textbooks that teach momentum flow, I don't see how you can claim that you are being unmistakably clear.

Standard terminology of basic concepts is found in introductory textbooks. It may come from specialists but it's available to the novice.

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Capiert, to save further argument over terminology I suggest you research

Momentum Transport

This is the technical term I think you are trying to portray.

https://en.wikipedia.org/wiki/Transport_phenomena

Good suggestion. Thank you, for putting a name on it. e.g. translating, identifying it.

Empirical evidence aside,

& to hear that (line) from a scientist.

What a disgrace for science.

unless you can find at least a few physics textbooks that teach momentum flow, I don't see how you can claim that you are being unmistakably clear.

If someone says "flow" to me, I sure do have to ask myself, "of what?".

You didn't bother with the word (at all) as though it did not exist.

You ignored the description, even if a standard naming exists.

That's ignorance.

You don't have to budge,

if you don't want.

You don't have to do anything

if it makes no sense to you

(even if it does for others.

A question (just down the road) of puzzlement

did not come from you, to clarify misunderstandings.

I'd evaluate that as not really interested.)

MIT seems to have a meaning for momentum flow,

if you don't.

http://web.mit.edu/16.unified/www/FALL/fluids/Lectures/f07.pdf

"When material flows through the surface, it carries not only mass, but momentum as well. The momentum flow can be described as

momentum flow = (mass flow) × (momentum /mass)"

to say it awkwardly.

It doesn't look like anyone is bold enough to call momentum flow a force,

although the variable is swapped

from the acceleration (speed (velocity) per time)

to the mass per time.

With viscous friction,

speed won't remain constant.

e.g. a bullet thru air.

I still see a momentum transfer,

e.g. flow,

no matter what way you look at it,

mass or speed as constant.

Thus can be equated (somehow, someway).

https://www.av8n.com/physics/euler-flow.htm

"We (temporarily) assume there are no applied forces (i.e. no gravity etc.) and no pressure (e.g. a fluid of non-interacting dust particles). We also assume viscous forces are negligible. Then, the only way a momentum-change can occur is by momentum flowing across the boundary:

∂ Πi/∂ t = ∫(ρ vi) v · dS = ∫(ρ vi) vj djS (8)

We are expressing dot products using the Einstein summation convention, i.e. implied summation over repeated dummy indices, such as j in the previous expression."

It seems like some people can make sense

of the idea of momentum flow.

Why you couldn't or can't is beyond me.

(..other than doing a number on me. =Giving me the "treatment".)

Standard terminology of basic concepts is found in introductory textbooks.

Yes, a "sea" of liturature exists. You guys (& gals) have the roadmaps. (The shortcuts, to efficiency (I hope).)

It may come from specialists but it's available to the novice.

Yes, but then accessible, when known (e.g. where).

That's the difference, whether it can be used.

As technology advances, new nomenclature will be developed.

However, what names become popular, (especially when old names exist (multi_naming, multiple names),

or which become outdated: what's IN, & what's OUT,)

it is all written in the wind, what the future will bring.

E.g. Speed of light symbol

was once V,

now it's c.

Who changed it? When? Where? & why?

(I don't know that, but am curious.)

Identification, is where it's at.

Recognizing.

Edited by Capiert
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MIT seems to have a meaning for momentum flow,

if you don't.

"When material flows through the surface, it carries not only mass, but momentum as well. The momentum flow can be described as

momentum flow = (mass flow) × (momentum /mass)"

to say it awkwardly.

It doesn't look like anyone is bold enough to call momentum flow a force,

Because as described it's not a force. Your term is from fluids course, where there is actual flow of a fluid. There need not be a force exerted on a fluid. Motion at constant velocity involves no net force.

You can't equate momentum flow with force.

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