Is Newton's second law somewhat arbitrary

[himanshu2004@]

Is Newton's second law somewhat "arbitrary"

I am trying to undestand something basic here.

Newton formulated his second law as: The rate of change of momentum of an object equals the force acting on it in (considering througout this discussion only intertial frames to keep things simple).
If its mass is assumed constant, we get F = ma.

However, it seems to me, that this Law does NOT establish a relationship between F, m, and a, i.e. between 3 quantities previously well understood by themselves, but rather it is the very defintion of F. So far, so good.

It seems Netwon could have chosen to say F = 2 m a, and the rest of the laws of physics would get changed (in their mathematical representation) approriately with this modified definition of F. ------------- (a)

However, going by a similar reasoning, could Force have been defined, for example, by F = (ma)³ ? i.e. Force acting on a body is the cube of the rate of change of momentum? (with the rest of the laws being modified appropriately based on this new "definition" of F, albeit now in a much more complicated manner) --------------- (b)
[I had earlier posted this as squared in stead of cubed, but then realized that would mean Force would be positive even if acceleration was negative. And so I have now changed squared to cubed]

So finally, here are my specific questions:
1) If (a) correct, then is (b) also correct?
2) If not, why. Alternatively, what (experiments, reasoning, etc) lead to Netwon formulating his second law the exact way he did? Obviously I understand it makes sense to keep laws/formulations as simple as possible. But is that all there is to the way the second law has been stated, and, in theory, is this formulation just arbitrary?

(I have chosen the F = ma version for ease of writing the equation here, but my question applies to the original rate-of-change-of-momentum formulation just as well)

 

[member 11137]

Perhaps would it be useful to come back to something more basic: the fundamental law of (attraction) gravitation between two masses.
F = G. masse n°1. masse n°2 / distance(between m1 and m2) x distance (between m1 and m2). You immediately recognize the formalism: F = masse n°1 (or n°2 - it does not matter) x something (which is an acceleration). So: the formalism itself is not somewhat "arbitrary".

 

[AlephZero]

However, it seems to me, that this Law does NOT establish a relationship between F, m, and a, i.e. between 3 quantities previously well understood by themselves, but rather it is the very defintion of F. So far, so good.

Indeed, that is a good way of looking at it, with hindsight - though Newton probably thought that "force" was something "real", not just a mathematical way of keeping score.

It seems Netwon could have chosen to say F = 2 m a, and the rest of the laws of physics would get changed (in their mathematical representation) approriately with this modified definition of F.

Sure, that is just a matter of working in consistent units. In the customary American units, replace your 2 by factors of 32 or 386.4.

However, going by a similar reasoning, could Force have been defined, for example, by F = (ma)³ ? i.e. Force acting on a body is the cube of the rate of change of momentum? (with the rest of the laws being modified appropriately based on this new "definition" of F, albeit now in a much more complicated manner)

I can't see why not, but as you say it just makes everything more complicated without adding any physical understanding of what is going on, so why would anybody want to do that? It would seriously mess up any math that involved dF/dt or dF/dx - and that includes pretty much everything in phyiscs.

As Einstein is supposed to have said, "Everything should be as simple as possible..."

 

[himanshu2004@]

Perhaps would it be useful to come back to something more basic: the fundamental law of (attraction) gravitation between two masses.
F = G. masse n°1. masse n°2 / distance(between m1 and m2) x distance (between m1 and m2). You immediately recognize the formalism: F = masse n°1 (or n°2 - it does not matter) x something (which is an acceleration). So: the formalism itself is not somewhat "arbitrary".

I really don't think you understood what I asked. I don't see how the law of gravitation is "more basic"; in fact it uses the concept of force which is defined itself by the second law. Correct me if I am wrong.

I can't see why not, but as you say it just makes everything more complicated without adding any physical understanding of what is going on, so why would anybody want to do that? It would seriously mess up any math that involved dF/dt or dF/dx - and that includes pretty much everything in phyiscs.".

Hmm, its interesting that you agree that the second law is somewhat arbitrary, even with respect to changing the powers of the 'm' and 'a' terms. I was, and still am, half expecting, almost hoping, that someone would point out to me something basic that I have missed.

 

[mishrashubham]

Well I am not a physicist, but what our hight school textbook talks about force is that:
According to Second law of motion,

Force is directly proportional to the rate of change of momentum
implies that F=k*(momentum_final-momentum_initial)/time
=k*(mv-mu)/t
=k*m(v-u)/t
=k*ma

where k is constant of proportionality and the units of force are defined so that k=1.
So yes may be you could play around with constants but exponents would violate the statement of proportionality given by the law.

 

[himanshu2004@]

Well I am not a physicist, but what our hight school textbook talks about force is that:
According to Second law of motion,

Force is directly proportional to the rate of change of momentum
implies that F=k*(momentum_final-momentum_initial)/time
=k*(mv-mu)/t
=k*m(v-u)/t
=k*ma

where k is constant of proportionality and the units of force are defined so that k=1.
So yes may be you could play around with constants but exponents would violate the statement of proportionality given by the law.

Ok, when you say
F = k * ma (proportinality),
how do you define the F on the left hand side, without referring to right hand side, which is something you clearly ought to be able to do before you can say that they are "equal" or "proportional". If you define F as what is on the right hand side, then all you are saying in this "law" is:
ma = ma
So, seemingly, this "law" is just a giving a term to the important concept that is ["the rate of change of momentum" OR "mass times acceleration"].
And hence its seems more like a "definition" than a "law". My question was could there have been other "definitions" of force as well, for example based on the cube of (mass times acceleration), which would of course have changed the way we express the other laws as well. And if not, why?

 

[AlephZero]

Hmm, its interesting that you agree that the second law is somewhat arbitrary, even with respect to changing the powers of the 'm' and 'a' terms. I was, and still am, half expecting, almost hoping, that someone would point out to me something basic that I have missed.

You can do a lot of useful mechanics without using "forces" at all. You can argue (following Lagrange, Hamilton, Noether, etc) that the ideas of conservation of momentum, conservation of energy, and relativity (the principle that different observers "see" the same laws of physics even, if they are moving relative to each other) are the really basic ideas, but it took 350 years after Newton's Principia to get all that sorted out.

In Newton's time the concept of "energy" was not understood at all, so he didn't have that option to work with and used forces instead.

If you chose to define
force = (rate of change of momentum)^3
or whatever, that would mess up simple results like Hooke's law for the force in a stretched spring. (Hooke and Newton were alive at the same time, and knew each other). It could be done, but why bother?

It would be similar to the first attempts to define scales for measuing temperature, which were based on a few "nice round numbers" (like the boiling and freezing point of water), but other "temperatures" were just numbers read from straight line graph produced by some measuring instrument. But once it was understood that heat was a form of energy, and therefore temperature was related to the amount of internal heat energy in a body, it was possible to define a rational temperature scale, with an absolute zero.

 

[himanshu2004@]

Thanks, that acutally makes a lot of sense. In fact, thinking about this for a while now, I'd realized something on similar lines myself - that the concept of force isn't really needed, and that the law of conservation of momentum is much more basic instead. But I also reasoned that "force", the way it is defined, is a useful concept since it relates well to the "feeling" of "pushing against" that one can subjectively feel (which would cause a proportional amount of acceleration) or for that matter one could relate it to the tension in a string or compression in a spring, etc.

In fact, not only does F = ma, define F itself, it is what also defines m! So very circular indeed, but definitely a very pathbreaking kind of thing to have done at that time, given the understanding of Physics then.

Also, thanks for pointing out a more complete set of fundamental principles more important than the concept of force, and the people who also argued about their primacy relative to ideas like force.

 

[0xDEADBEEF]

Well force is not arbitrary. The reason are the other two laws. Forces add, and that is the main point. From this and the third law you can prove the conservation of momentum and angular momentum. I don't think that you can prove the conservation of angular momentum from the conservation of momentum alone. The same goes for the conservation of energy in conservative fields. The third law gives a lot of meaning to the whole force idea.

 

[himanshu2004@]

Well the way I look at it now, is that Newton's second law isn't so much a a "law", as it is a statement of the importance of momentum and its rate of change.

The concept of momentum is perhaps more basic than that of mass itself. In those days, people didn't understand what mass really was (as we do today), but the "motion of a body" i.e. its momentum was a concept better understood. So a heavier body had more "motion" than a lighter body moving with the same speed (where speed, and eventually velocity was another concept well understood.)

It was gradually observed, perhaps beginning with Galileo or earlier, that ignoring effects of friction, the "motion" (momentum) of a single body or a system (e.g. two interacting balls) seemed "conserved".
Mass was perhaps then more clearly understood as the part of the momentum of the body that was not its velocity. Empirically, this mass-entity also seemed to be a measure of how heavy the body was. And it was seen that a body twice as heavy as another has twice as much momentum, given equal speeds. Hence heaviness (mass) x velocity seemed to be a measure of the "motion" of a body.
[That the mass-entity which was a factor in the "motion" (momentum), i.e. intertial mass, was also found to be a measure of heaviness, i.e. gravitational mass, has to do with the equivalance of intertial and gravitational mass. This being so just made things simpler, but even if this were not so, there would have been observed the importance of the idea of momentum]

So what Netwon did was to establish the importance of the idea of the rate of change of momentum, by giving it a name (Force).

Well force is not arbitrary. The reason are the other two laws. Forces add, and that is the main point. From this and the third law you can prove the conservation of momentum and angular momentum.

That force can be added and the fact that when the net force is zero there is no acceleration, are not "consequences" of the second law; but rather just stating that if the rate of change of momentum (i.e. force) is zero, there is is no change in the motion, something which follows from the very defintion of momentum.

Also, then, the third law is just a consequence of the observed conservation of momentum applied to the specific case of two interacting bodies.

There has been no mention of concept of interial frames yet, and this is simply the idea of a non-accelerating frame (relative to another inertial frame), or the frames in which Netwon's laws hold. The first law, being a very simple case of the second law, serves as the test/defintion of an inertial frame.

To to summarize the main point, Netwon's second law isn't a "law" but rather the giving of a name (i.e. Force) to the very important idea of the rate of change of momentum, and reinforcing the already unerstood importance of momentum. Why these ideas are importance is primarily due to the fact that this momentum-entity, which seems to be a measure of the "motion", is seemingly conserved in closed systems.

Since posting the question, this has been my understanding of how these ideas evolved. I found it much more instructive instead of just "reasoning" that since force seems proportional to both mass and acceleration, Netwon's second law makes sense (because crucially, not only is Force a concept that this law newly defines, at least mathematically, but so is Mass itself). Obviously there would have been many more details in the devlopment of these ideas, spanning several years, including perhaps Netwon's attempts to find the basic principles underlying Keplers' laws, and other people's ideas before Netwon that he built upon.

 

[HallsofIvy]

I am trying to undestand something basic here.

Newton formulated his second law as: The rate of change of momentum of an object equals the force acting on it in (considering througout this discussion only intertial frames to keep things simple).
If its mass is assumed constant, we get F = ma.

However, it seems to me, that this Law does NOT establish a relationship between F, m, and a, i.e. between 3 quantities previously well understood by themselves, but rather it is the very defintion of F. So far, so good.

I would say it is not so much a definition of F (we already have a good idea of how hard we push or pull something) but rather a definition of "m" (mass as opposed to weight).

It seems Netwon could have chosen to say F = 2 m a, and the rest of the laws of physics would get changed (in their mathematical representation) approriately with this modified definition of F. ------------- (a)

But then changing the value of the mass an object has would get us back to the simpler fromula F= ma.

However, going by a similar reasoning, could Force have been defined, for example, by F = (ma)³ ? i.e. Force acting on a body is the cube of the rate of change of momentum? (with the rest of the laws being modified appropriately based on this new "definition" of F, albeit now in a much more complicated manner) --------------- (b)

No, that's a different matter since now you are changing "proportions". That would say if we applied 8 times the force to an object, it would accelerate at only twice the acceleration- and "experimental evidence" shows that does not happen,.

[I had earlier posted this as squared in stead of cubed, but then realized that would mean Force would be positive even if acceleration was negative. And so I have now changed squared to cubed]

So finally, here are my specific questions:
1) If (a) correct, then is (b) also correct?
2) If not, why. Alternatively, what (experiments, reasoning, etc) lead to Netwon formulating his second law the exact way he did? Obviously I understand it makes sense to keep laws/formulations as simple as possible. But is that all there is to the way the second law has been stated, and, in theory, is this formulation just arbitrary?

(I have chosen the F = ma version for ease of writing the equation here, but my question applies to the original rate-of-change-of-momentum formulation just as well)

No, the formula is not arbitrary- however, "constants of proportionality" are, in fact, "arbitrary", and change with the units we usee. If we measured F in "Joules", mass in "kilograms", but a in "centimeters per second per second", F= ma would become F= 100 ma.

 

[A.T.]

So very circular indeed,

One should differentiate between circular stand-alone definitions and sets of definitions which only make sense together and reference each other. If you are establishing a new theory starting out with nothing, how would you be able to define any new terms? You have to postulate some concepts and provide the definitions that relate them.

 

[SpectraCat]

For your original questions,

a) is fine .. just change units to see this. Choice of units is always arbitrary, provided you don't try to change the dimensionality of the quantities (i.e. you can't substitute length for mass, but substituting inches for meters is fine).

b) is completely wrong. Just look at the dimensional analysis to appreciate this. Remember Newton's laws are observationally based, meaning that he used experimental evidence to derive them.

 

[0xDEADBEEF]

Well the way I look at it now, is that Newton's second law isn't so much a a "law", as it is a statement of the importance of momentum and its rate of change.

The concept of momentum is perhaps more basic than that of mass itself. In those days, people didn't understand what mass really was (as we do today),[...]

So what Netwon did was to establish the importance of the idea of the rate of change of momentum, by giving it a name (Force).


That force can be added and the fact that when the net force is zero there is no acceleration, are not "consequences" of the second law; but rather just stating that if the rate of change of momentum (i.e. force) is zero, there is is no change in the motion, something which follows from the very defintion of momentum.

Also, then, the third law is just a consequence of the observed conservation of momentum applied to the specific case of two interacting bodies.
[...]

Well, I can tell you that your idea is very popular and there is a modern school of physics teachers that try to teach physics to students without a force concept. It just feels so enlighted and modern. I consider it completely misguided, because forces are a concept too power full to give up.

1) There can be tremendous forces without a change of momentum, for example when things are squeezed. This cannot be accounted for by momentum conservation.
2) Force fields look clumsy when derived formulated with impulse flows.
3) Potentials look even more ugly
4) You have not shown that you can derive angular momentum conservation from momentum conservation alone, but from Newton's laws it is possible.

Basically, these concepts had almost three hundred years to be thought about, and they are some of the most perfect coherent theories physics has ever produced. The way it is taught in textbooks today is the result of a long development and in most cases things are exactly where they need to be.

There is a way to describe classical mechanics completely with toruses of action variables. Some people think this is closer to the truth of the real world (Hamilton–Jacobi equation (HJE)), but usually this description is completely useless for calculations.

Conclusion: Just because you understood a new concept that is useful in some cases, this doesn't mean it is the best way to look at the world in all cases. Forces are more useful than momentum flows.

 

[himanshu2004@]

Well, I can tell you that your idea is very popular and there is a modern school of physics teachers that try to teach physics to students without a force concept.

Well I am surprised that's the case. Anyhow, I certainly don't think the concept of force should be done away with at all because it is very useful. My original question wasn't about whether force is useful, but about how Netwon came up with the second law, which requires more than just reasoning that force is proportional to mass and acceleration, etc because force and mass were both vaguely understood concepts then. I have since then been able to make sense of things. The process of understanding I went through, which I have documented here over a couple of posts, I found very instructive. And I think more people should understand the how early understanding of Physics evolved the way it did, than directly being introduced to Netwon's laws as many textbooks and teachers do.

Unfortunately many of the posts trying to "explain" things to me seemed to lack depth of reasoning, and while I felt a little impulse to reason with them, I have resisted the temptation to do so. Having said that, some of the posts were certainly helpful, and to those people my thanks.

 

[pete20r2]

Maths is the definition of tautology, you know that don't you.