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Conservation of Momentum - true?

  1. Dec 3, 2005 #1
    I’m currently involved in a debate with a few folks regarding the validity of the Conservation of Momentum principle.

    One demonstration of its failure was dropping a book. The book begins with no momentum, gains momentum, then looses it and again has none (when it hits the floor). Of course, a rebuttal to this demonstration was to note that, even though we cannot readily detect it, the earth is in fact simultaneously being drawn up towards the book. This results in a net zero momentum for all points in time and momentum is conserved.

    However, this rebuttal assumes some kind of frame of reference independent of the Earth and the book. What if the reference is the Earth? For the book then, we see a positive and negative change in momentum. For the Earth, however, there is no change in momentum as there is obviously no change in velocity for the Earth relative to the Earth. This means there is a production and destruction of momentum within the book! Interesting…

    All of this raises questions regarding valid frames of reference. Certainly something needs to be said regarding taking the book and Earth as the system and the frame of reference as the Earth – is that kind of system definition “legal” ? If it is, there’s certainly a problem…

    I welcome input. :biggrin:
  2. jcsd
  3. Dec 3, 2005 #2


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    Well think of this, say your driving along in your car at 100mph (crazy kids!). Relative to yourself, you're going 0mph. If youc rash into a brick wall, relative to yourself, you're still going 0mph. The problem is that its non-sensicle to say "relative to yourself".
  4. Dec 3, 2005 #3
    Welcome to PF, TiBaal89. Glad to have you on board. You've asked an interesting question and of course you are quite right. In the Earth's non-inertial frame, the total momentum of the book and the Earth will not remain constant. Unfortunately, you have not happened upon a new phenomenon. It is already quite accepted that the laws of physics, such as the conservation of momentum, apply to inertial frames only. This is not so much a flaw in the laws of physics as a peculiarity of non-inertial frames, or at least our description of them.

    The truth is, you can quite easily choose a sequence of frames whose velocity, or even acceleration, change to the extent that physical law seems to be broken. Witness the Coriolis force for a good example. However, these yield little in the way of trustworthy results. Many phenomena in our Universe will suddenly appear very different when viewed from a different inertial frame. Even the speed of light changes when viewed from a non-inertial frame.

    One thing is to consider how the rest of the Universe appears when choosing such transient frames. For instance, imagine a body that exists at a fixed radius from the Earth. As the book falls towards the Earth and the Earth budges slightly towards the book, the Earth would also budge slightly towards this fixed body, yet this fixed body has not moved at all. Such is the disadvantage of working from non-inertial frames.

    It is a matter of debate, but a fair assumption, that momentum is conserved within the Universe as a whole, even in changes of inertial frame or non-inertial frames.

    For illustration, imagine a Universe consisting of only one particle. Imagine that you have a frame of reference such that this particle is initally at rest. However, this is an accelerating (i.e. non-inertial) frame of reference and so, in this frame, the particle appears to accelerate despite the fact that no force is acting on it since there are no other entities in this (hypothetical) Universe. Thus momentum of the constituents of this Universe (the single particle) disobey the law of conservation of momentum in this frame.
  5. Dec 3, 2005 #4

    D H

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    First of all, conservation of momentum does not apply in the case of a book released a meter above the surface of the Earth, and the reason is not because of the choice of reference frames. Conservation of momentum states that the momentum of a system is constant if there are no external forces acting on the system. The Earth' gravity is an external force acting on the book.

    A reference frame independent of the Earth and the book is not needed to find a system in which momentum is conserved. The Earth-book center of mass frame is dependent upon both and is approximately inertial (ignoring the rotation of the earth, the sun, the moon, and the rest of the universe). These other influences can be ignored given the short time it takes for a book to drop one meter and then hit surface of the earth.

    In reality, there is no such thing as an inertial reference frame. Gravity and inertia are everywhere and are rather difficult to measure. The best one can do is is to find a set of axes with a sufficiently small rotation rate and an origin with a sufficiently small acceleration. "Sufficiently small" is context-dependent -- it means that the non-inertial nature of the reference frame does not render invalid the results achieved by assuming the frame is inertial.
  6. Dec 3, 2005 #5
    Thank you for the welcome and response El Hombre. It is very interesting to consider. When this debate first came up, I offered that I could change the kinetic energy of my freind by moving my head around! :rofl: Which, while silly, does raise some interesting questions as well - just another segment of our problem of reference frames.

    Ah, but i specifically designed this quesiton so that this is not the case! The system is the book and the Earth, while the frame of reference is the Earth. There is no transport over my system boudary and thus it should work, we'd say.
    Last edited: Dec 3, 2005
  7. Dec 7, 2005 #6
    While D H is correct in summarising Newton's first and second laws, in essence that a change in momentum is brought about only by unbalanced force, he forgets Newton's third law: "All forces occur in pairs, and these two forces are equal in magnitude and opposite in direction." As a result, his answer is incorrect.

    In the case of a book raised above the Earth's surface and released, the Earth with apply a gravitational force to the book, changing the book's momentum. The book will also change the momentum of the Earth such that the change in the Earth's momentum is equal and opposite to the change of the book's momentum.

    As such, your initial post is correct in stating that the apparent breach of the law of momentum conservation is entirely explained by the change of the Earth's momentum. The total change in momentum in the system is zero, and momentum is conserved.

    The choice of a reference frame in which the Earth is always at rest, i.e. a non-inertial frame that is very approximate to an inertial frame, will give the illusion that momentum is not conserved, just as it does in the so-called pseudo-forces such as the Coriolis force. That's accelerated frames for you.
  8. Dec 7, 2005 #7

    D H

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    El Hombre Invisible, I did not forget Newton's Third Law. Conservation of momentum does not apply to the book as observed from an Earth-fixed position, not because the principal is violated but because external forces act on the book. The observer has to move beyond the Earth-fixed frame to see that the momentum of the Earth and the book are conserved.
  9. Dec 7, 2005 #8
    If, by external forces, you mean external to the book, then that is of no consequence - any force causing the book to move is an external force (or for any apparent force it may be a pseudo-force as in this case).

    If you mean external to the system, TiBaal89 stated the system to be the Earth + the book, so no force from outside the system is being applied to book (the force exerted on the book is exerted by the Earth and vice versa).

    What you said was:

    You seem to be stating to me that, in general, where an unbalanced force acts on the book the law of conservation of momentum does not apply. This is wrong. It disregards Newton's third law. Conservation of momentum does apply (total momentum of Earth + book remains constant), however the non-inertial frame of the observer gives rise to pseudo-forces acting on the book (and on the Earth, since the observer may calculate it should be accelerated and wonder why it seems not to be). The choice of reference frames is very pertinent.

    Now you seem to be agreeing that choice of reference frame is important.
  10. Dec 7, 2005 #9


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    It isn't a choice of "frame", it is the choice of the "system".

    The original post confuses the application of the conservation of momentum. This always happens when people read off a statement of a principle without understanding the physics surrounding the principle (example: Thermodynamics 2nd Law being bastardized as proving that evolution cannot happen). The classical conservation of momentum clearly indicates that in an ISOLATED system (i.e. no net external force acting on what you consider to be your system), then momentum is conserved.

    When you let go of the book, as in the OP, if you JUST look at the book as your whole system, then one can CLEARLY see that this is NOT an isolated system. There is an external force acting on the book, due to the gravitational forces from the earth. So OF COURSE, momentum of that book is not conserved, and it shouldn't!

    However, if one looks at the system as being the book + earth, and assuming that all other forces external to those two are negligible, then this system can be considered as isolated, and the momentum of that system is conserved at all times.

    (This is a perfectly valid assumption. Gravity is extremely weak in most instances. In particle physics and solid state physics, gravity isn't even considered in the potential term of the Hamiltonian that describes all the relevant dynamics and state of the system. So while gravity is responsible for the force between book and earth, other gravitational forces from other bodies exert negligible influence on the book-earth system.)

    Take note that the whole Newtonian mechanics is a consequence of the conservation of momentum. You can't say that conservation of momentum can be violated, and then turn around and use F=ma. If the momentum of an object can change by itself and violates the conservation of momentum, then this change is not due to an external force, and thus, F=ma is also not valid! This is just one of many examples in physics where things are connected and inter-related. Modifying something can create a wide-ranging ramification. It is why physics cannot be studied in bits and pieces only.

  11. Dec 7, 2005 #10
    ZapperZ, the thread-starter had already clarified the choice of both the system and the reference frame in their second post:

    So I disagree - this is not about the choice of the system. The apparent non-conserved momentum under question is due to the non-inertial frame of the observer.
  12. Dec 7, 2005 #11

    D H

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    The Earth-centered and Earth-book center of mass frame differ by about [itex]10^{-20}[/itex] meters. To all practical purposes, they are identical. It is much more practical to look at the book accelerating due to an external force (and hence conservation of momentum does not apply) than it is to look at the momentum-conserving Earth-book system.

    An inertial reference frame (in classical physics) is a theoretical construct in which Newton's Laws are valid. There is no such thing as an inertial reference frame in practice (if it exists, the origin is so far away as to make computations intractible). In practice, we make do with the best definition available (J2000 or ICRF) and make ammends (e.g., third-body perturbations) when needed.
  13. Dec 7, 2005 #12
    As El Hombre points out, the system and frame were specifically designated with the intention of creating a "problem." The earth+book system is inertial. The problem lies in the fact that the frame of reference (attached to the earth) is not.

    Whats interesting, and I think this is something near to what D H is alluding to, is to compare the earth as just sitting there to the earth after it has undergone the acceleration due to the book's 'pulling up on it.' I hope we can all agree that these are extremely similar - which raises some interesting questions.
  14. Dec 7, 2005 #13


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    But this is getting to be rather silly. At what point do you stop considering the 1200th decimal places?

    When you "let go" of the book and earth, they will both "fall" and meet at the center of mass of the system. As has been pointed out (and one can easily do this oneself), this location is extremely, extremely close to the center of the earth's mass, meaning the earth, for all practical purpose, does not move. If you care that much about this that this is causing you to be in a "non-inertial" frame, then I'd ask why stop there? Include that round thingy you see at night in the sky, and the big bright thingy you see during the day too, why don't you?

    The original question made a faulty use of the conservation of momentum. That's what I was addressing. I don't know why there is now an issue with simply considering the earth-book system.

  15. Dec 8, 2005 #14
    What is practical and what is being asked are not the same thing. Both of you would do well to stick to the scope of the question, irrespective of whether you are more comfortable discussing a different one.

    Momentum conservation is the issue whether it is practical or not. Earth + book is the system whether it is practical or not. The thread-starter has made this clear before the argument over it, and has clarified it since. Both of you: please stop trying to derail the conversation.

    That's fine. We're discussing a theoretical question about Newton's laws.
  16. Dec 8, 2005 #15


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    But it IS relevant. If what you're saying is correct, then we, on earth, cannot apply the conservation of momentum, just because our "reference frame" is inappropriate for it! Would you like to stand by this claim?

  17. Dec 8, 2005 #16
    If you read the thread, Zz, you'll find that's quite the opposite of what I was saying.
  18. Dec 8, 2005 #17


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    Really? Let's review, shall we?


    [My bold]

    So explain to me where I got what you're saying wrong.

  19. Dec 8, 2005 #18
    ZapperZ, I said to read the thread, not quote the whole thing! :smile:

    That second quote was not describing the situation in the Earth's rest frame (read on in the same post to see why), so you're misquoting me. All I'm saying there is that the change in momentum of the Earth is equal and opposite to the change in momentum of the book. That's all. Not a huge controversy there, just two of Newton's laws: that force is the derivative of momentum and forces occur in equal and opposite force pairs. What's the problem?

    The first quote says you may choose a non-interial frame in which an object not subject to a force appears to be accelerating and so subject to a force - i.e. the so-called pseudo-forces. Again, what's the problem? How do either of these state that momentum is not conserved on Earth?

    It was D H that said we cannot apply conservation of momentum on Earth, not me. I was arguing against that point. Again, had you read the thread you would have noticed that.

    I thought I had stated in an earlier post that the Earth is a good approximation of an inertial frame, but alas that did not make the final draft. Anyway, no that is not my claim so no I do not stand by it.
  20. Dec 8, 2005 #19


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    OK, let me understand this correctly. If I am IN the earth's frame, then your statement that

    "As such, your initial post is correct in stating that the apparent breach of the law of momentum conservation is entirely explained by the change of the Earth's momentum. The total change in momentum in the system is zero, and momentum is conserved."

    .. is now VALID? If I am performing this experiment on earth, I have ZERO ability to explain the "violation" of the conservation of momentum of that book?

    And I asked to how many decimal places do you always keep in your numerical answers? To what extent is the earth not an inertial frame ENOUGH that a book moving from rest in such an OBVIOUS manner requires the often negligible non-inertial forces of the earth?

    I don't see it. I did see DH saying that there really isn't true inertial frame, but this isn't a surprise. However, to argue that something this obvious CAN be attributed to the non-inertial Earth, that is a major puzzler! This is something you did, and not something he did. D H repeatedly said something to the effect that you only need to look at the book ALONE, without having to look at the source of the force, to get a complete explanation on why the book changes momentum.

    Look, put an object of mass m in space that has gravitational field. Are you telling me that you have ZERO ability to explain why it is changing its momentum without invoking the SOURCE of that field and the frame you are in? Honestly? I look at the original post and this is what is being asked. I don't need to explain that the earth also moves by a miniscule amount to explain the overall conservation. I only just require that someone pay attention to the fact that the book has AN EXTERNAL FORCE ACTING ON IT. Case closed! There should be no more talk about the book (not the book+earth) violating conservation of momentum. This is what DH was saying way in the beginning of this thread.

    Why is this very simple and straightforward explanation insufficient? And why is it so irrelevant to the question that you had to keep reminding the two of us to not derail the thread? If anything, I find that you were the one who is making it more complicated than it should.

  21. Dec 8, 2005 #20
    That is not what I said and you damn well know it. I already replied to you in my previous post that this quote has nothing to do with reference frames. I'm just describing action at a distance - IN ANY FRAME! All I say here is that the book's momentum will change despite no collisions, etc. I then say the Earth's momentum will change in an equal and opposite way and therefore momentum is conserved.

    Makes no difference. Not the scope of the question. Question could easily be reworded such that both bodies are of equal mass and the question would still hold!

    Well, I suggested you read the thread, you obviously can't bothered so I'll do it for you.

    I can only assume you have interpretted what I said in a way completely unintended because I cannot begin to figure what your beef is. You seem to be grasping at straws randomly, arguing one thing (say, the book as the system), then another (precision of measuring the Earth's change in momentum). A more consolidated argument from you might clear things up.

    Hey, or even better: don't look at the book or the Earth - a much easier way of not answering the question. The thread author has stated the Earth + book as the system. It was easily inferred from the opening post and explicitly stated in his second. You and D H continue to answer the question in terms of a one-body diagram. This is not answering the question.

    Nope. I know that's what you'd like me to be saying, but I have to disappoint you. The question (nor my answer) is not about why momentum changes. I never said we don't know why the book falls or that we can't calculate how it will fall. This is something you've pulled out of the air for want of a better argument. I explained why the conservation of momentum of the Earth + book system isn't apparent - that's all.

    No, it is not. Hence the thread author's reply to D H, and moreover his reply to myself. You and D H have intepretted have interpretted the OP one way, I did another. By the author's replies it is safe to say my interpretation was the correct one. Allow me to illustrate:

    Here the author has set the scene. He has explained a process whereby the gravitational attraction between the Earth and the book causes the book to gain (very noticeable) momentum towards the Earth and also causes the Earth to gain (undetectable - hence your arguement about precision is irrelevant) momentum in the direction of the book. The system has clearly been described as Earth + book, and has been described in such a way that momentum is conserved within this process. Groovy!

    The author then states that in the Earth's rest frame there is no change in momentum of the Earth, but there is in the book. The system is still Earth + book as stated above. All that has changed is the reference frame. Thusly our author asks if the frame is 'valid' or 'legal'.

    He does NOT ask if momentum must be conserved be there external forces or not (which, since Earth + book is the system, in this case = not). He is asking a question about frames of reference. I have answered the question about frames of reference. You and D H are answering different questions and not particularly well I might add.

    I have no doubt you're arguing now more for the sake of saving face than anything else, so I guess this will become a case of who stops restating and restating and restating their argument first.
  22. Dec 8, 2005 #21


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    No I don't, and that's why I ASKED! You keep accusing me of misinterpreting your responses, and before I send you another rebuttal, I wanted to make sure that what I understood is what you meant. You somehow took that as an affront! I am not responsible for making you this defensive.

    I can do even better than that. Look at my first two responses in this thread, and in particular, the SECOND post:

    My "beef" is with the EXPLANATION that one can explain the apparent non-conservation of either JUST the book, or the APPARENT NON-MOTION of the earth, via this "non-inertial" frame. This is like trying to squeeze blood out of a rock! I have clearly started above the the earth-book SYSTEM is the conserved system, not just the book. WE ALL AGREE ON THAT! However, I also stated that you simply cannot use the "non-inertial" frame argument as attributing to the observation of non-conservation. I have contined to ask you about this and you keep telling me I'm reading the wrong thing. Yet, when I said it isn't the "frame" but rather the "system", you complained!

    The OP's argument that the earth "doesn't move" and thus the earth-book system violates conservation of energy should in fact be explained by looking at the PROPORTION of the mass of the book when compared to the mass of the earth and then show where the center of mass of the system is! Each time you sit down or you get up, the earth doesn't become a non-inertial reference frame, or else someone's experiment elsewhere will go very goofy.

    Now note that I said nothing about a system with two similar masses - you did! In none of the explanation I have so far did I say that it works for ALL cases. In fact, my reply that I quoted above would clearly indicate that this is strictly for when the center of mass of the system and the center of mass of the heavier object practically coincide. I never claim this is valid for similar masses. It would be silly for me to do this. Take two equal masses stretched by a spring, let them go, and they meet at the center of mass in between. WE ALL KNOW THIS. If you're sitting on one of this masses, then you will be an a non-inertial frame. But you will have apparent contradiction using the "non-inertial frame" argument to explain ordinary things dropping on earth. The fall of M towards M is different than the fall of M/2 towards M, and different from M/4 towards M. How does one keep using "non-inertial frame" as the "dropping mass" becomes smaller? We see no detectable difference in the free-fall acceleration of a tennis ball and a bowling ball. If one wants to argue that each one of this observation is due to the "non-inertial property" of the earth due to being attracted to the bowling ball or tennis ball, then there's something weird going on considering that each one exert different forces onto the earth, yet they fall the same rate!

    So then what do you do? You invoke the argument that the earth really doesn't move that much anymore since the dropping mass is so small. BINGO!!!!!

    That is what I've been arguing from the very beginning when I entered this thread that it isn't the "frame", but the "system". The SOURCE of all of this is the ratio of the mass of the two! It is why I keep asking that if you keep insisting of a "non-inertial frame" explanation, to what decimal places do you keep everything?

  23. Dec 8, 2005 #22
    OK, well lets make some crowbar separation then between the dispute between D H and I and the dispute between you and I so we can narrow things down somewhat.

    D H argued that momentum was not conserved since an external force is acting on the book. Since the system within which momentum is conserved is both the Earth and the book, I argued that momentum is conserved since the Earth will have an equal and opposite change in momentum. Feasibility of detecting this change aside, and ignoring any effects from real-life bodies effecting the Earth's momentum that were not included in the author's system, do you agree? If so, we can lay that one to rest. That was the only disagreement between D H and I and it came entirely down to being consistent about what is included within the system (and so what constitutes an 'external' force).

    So assuming we agree there (that momentum is conserved, however undetectable), the argument you seem to have is that the ratio of masses is so large that the centre of mass of the system is to all practical purposes the centre of mass of the Earth itself. You agree this is the dispute?

    So the difference here is that I gave a general answer that applies to a book of any mass, including the mass of the Earth itself (heavy reading indeed), whereas you gave a special answer for realistic cases of [tex]m_{book} << m_{Earth}[/tex].

    So then it comes down to the question of not which answer is correct, but which is relevant. Since the author stated in his opening post that in an inertial frame the change in momentum of the Earth is equal and opposite to the change in momentum of the book, it is quite clear that he does not have the practical coincidence of the centre of mass of the system and the centre of mass of the Earth alone in mind - he included the Earth's change in momentum from the get go. The argument you are giving is actually not against me at all - it's against the author for giving an example where the change in momentum of the Earth he refers to is not practically detectable.

    Look at the question the poster is asking. Ask yourself whether the practical implications of detecting or describing a change in the Earth's momentum due to a falling book is pertinent to what he wants to know. YES - I alone brought the example of a system with two similar masses... by way of illustration for you! The reason I brought this up is because you can ask the same question the author asked using a system of two similar masses. You don't see this because you're hung up on the negligible acceleration of the Earth - but this is besides the point. If you replace 'book' with 'planet of same mass' and read through the OP again, when you get to the question it will still hold. Not the question you're answering, but the question the author is asking.
  24. Dec 8, 2005 #23
    In fact, this nails the dispute down nicely. Reread the OP:

    Clearly, then, you are misunderstanding the OP's argument. The OP is not arguing that the earth "doesn't move" due to the PROPORTION of the mass of the book when compared to the mass of the earth since this is true in the inertial frame too - the frame in which the OP argues the Earth does move!

    You MUST be able to see that!
  25. Dec 8, 2005 #24
    The principle of conservation of momentum applies to total momentum of a closed system. You cannot dismiss the Earth in this case. The Earth will change its velocity. The change is so small as to not be detectable by your senses or perhaps even very precise instruments. The change in momentum of the Earth is that small. Give it a try. Try crunching the numbers and see what you get for the delta v of the Earth.

  26. Dec 8, 2005 #25
    Thank you, Pete. Nice not to be alone on this one. I'm sure we all agree the feasibility of detecting such a change is around the zero region, but there's a world of difference between that and it not moving at all.
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