# Interpreting F=ma

1. Sep 1, 2004

### suffian

I've asked a couple of professors, how can you say F=ma is empirically true unless you define what F and m are. Each basically said they didn't have a perfectly good answer to that question. The most obvious answers (in my opinion) lead to circular definitions. Well, I was thinking about this and I can up with the following (perhaps sloppy) way of making sense of things.

We define the mass of any uniformly composed rock to be one kilogram. Then we say that the mass of any similar rock that is twice as large has twice the mass (or one half the size has half the mass). We now say that the mass of two bodies are equal if they balance an equal-arm balance near the earth. By empirical observations we note that mass is time-invariant and transitive (i.e. if m1 = m2 and m1=m3, then m2 = m3). Now we know what it means to say the mass of a particular system is x kilograms.

Let us now define the center-of-mass of a body of uniform composition to be at its geometrical center. The center-of-mass of any other body is then the limit of breaking up this body into smaller pieces of approximately uniform composition and summing up their mass-weighted geometrical centers. Now when we talk about the acceleration of a body, we are really talking about the acceleration of its center-of-mass.

Finally, we define the net force on a body to be equal to the product of its mass and acceleration (perhaps it is more appropriately defined in terms of momentum). Now we can state Newton's Law (an empirical observation) in this form [edit: actually, i'm not so sure this is a 'form' of his law]: The net force acting on a body (the central body) when surrounded by other bodies (the surrounding bodies) is equal to the sum of the net forces that would act on the body if the each surrounding body were considered in turn as an isolated system with the central body.

I know some of the definitions (such as the one involving an equal-arm balance) aren't entirely precise, but at least its something, which is always better than nothing.

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2. Sep 1, 2004

the actual definition if force is:

$$F = \frac{dp}{dt}$$

3. Sep 1, 2004

### suffian

I'm afraid my professors are not that dumb (i.e. not just say so it if the answer were so simple). The question would now become "what is momentum"? And even if you answer this, then you would not be able to say what the force is acting between two particular bodies when the system is compromised of three or more bodies because only the net force acting on a body equals the rate of change of momentum.

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4. Sep 1, 2004

### pervect

Staff Emeritus
Why not just use the SI definition of mass? It's quite similar to yours, except that we don't use rock, we use the international prototype of the kilogram in Paris. Another differences is that masses are compared by using balances. This seems like a better experimental way to compare masses to me. Perhaps it's not as philosophically "pure", because balances use forces, but it seems more practical.

Secondary standards are made by using a balance to compare the secondary standards to the primary standards. The most precise standards are made out of the same alloy (platinum-iridium) as the primary standard (more uniform and easier to control than rock). Stainless steel standards are also used, but are less precise due to buoyancy problems with air displacement. Apparently cycling the standards to and from vacuum has its own problems, though I don't quite understand why.

http://www1.bipm.org/en/si/si_brochure/appendix2/mass.html [Broken]

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5. Sep 1, 2004

### suffian

hmm, my point wasn't to come up any sort of precise standard for mass. Just to make a definition without the use of forces so that none of the definitions are circular.

6. Sep 2, 2004

### Galileo

I'm afraid it is not possible not to be circular.
Newton did not just formulate his laws, he also had to define the notions like momentum and force.
Ofcourse people had a intuitive feel for what a force was at that time, but it wasn't defined quantitatively until Newton.
Momentum ofcourse is defined as mv, which Newton called "quantity of movement" or "quantity of motion." (can't remember which).

These are fundamental introductions of physical quantities as mathematical objects, so we can work with it. You cannot derive it from anything else, so you cannot help but be circular in your reasoning.

Even mathematics is circular in it's roots. There exists no formal definition of a 'set' (but everyone has a feel for what it is). If you try to define it, you end up using words like 'collection' or so, but it doesn't solve the problem: how would ou then define 'collection'? The point is that these concepts are intuitively clear enough.
Think of it like this: Mathematics/physics is a language and any language is circular in reasoning. Take a dictionary of say the English language. If you search for a definition of a word, it is explained in other words which must be part of the dictionary. You can look for the meaning of those words, but the dictionary has only finite size, so you'll end up with circular reasonings eventually.

7. Sep 2, 2004

### arildno

Newton used "quantity of motion", or at least, that was the word used in the first English translation of "Principa"
(Incidentally, the Norwegian equivalent of "quantity of motion" (bevegelsesmengde) is the most common word we use (occasionally using "momentum" instead))

8. Sep 2, 2004

### Galileo

In the Netherlands we use 'Impuls' for momentum.
For rotational equivalent with respect to some point we add the word 'moment'.
So angular momentum is 'impulsmoment', which translates to 'moment of momentum'. It's actually quite logical.
Torque translates to 'moment of force'.
'Moment of inertia' follows quite logically.

'Impulse', in English is the integral of Force with respect to time. Actually, I don't know what that is called in Dutch (couldn't be Impuls).

9. Sep 2, 2004

### suffian

I don't think the tricky part is in defining either force or acceleration.

The problem lies in defining mass. Now most of you guys would seem to claim that mass is something we can simply "feel". Perhaps you percieve of mass as the amount of "stuff" in the system. In that case, we might wonder about single particles (like an electron, quark, etc.), do they have different mass because their is more stuff in one than the other? Does the amount of "stuff" in a body increase as it approachs the speed of light? If a photon has no mass, then how can it exist (should be no "stuff" there)? I think clear definitions can help us from frustrating over these kinds of questions that are, in a sense, meaningless.

In my definition of mass (and I'm not trying to claim it's perfectly clean), I am trying to define mass as a property of a body [free from forces]. Nothing more, nothing less. We could then notice some empirical observations about mass (time-invariance, transitvity, additivity, etc.). This would, in a way, rebuild our conceptual understanding of mass, but on a more fundamental level. Before we simply intuited the fact that the mass of a system consisting of several bodies was their sum, but now we would see that this is really an observation (well, i guess that might depend on how you actually define mass).

In any case, even if you disagree, I just hope you see my point.

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10. Sep 2, 2004

### ZapperZ

Staff Emeritus
1. I didn't realize the definition of "existence" MUST include this quantity call "mass". Is this a universally accepted definition? Who made this up? It certainly isn't in any of the NIST standard, nor in CODATA.

2. Your definition of mass ("... as a property of a body [free from forces]. Nothing more, nothing less....") is so vague, you could fit in a dozen trailer trucks in there. This does nothing in advancing of our understanding of what it is.

3. Were any of your professor condensed matter physicists? If they were, I'm surprised they did not point out to you that another definition of mass (effective mass) DOES exist as defined by the dispersion relation between energy and crystal momentum in a solid. Lest you think this has no relevance in defining what the mass of a "bare" particle is, let's not forget that Peter Higgs took this concept right out of condensed matter to fomulate the Higgs field as the origin of mass in the fundamental particles.

4. If you have a problem with "mass", how come you don't have a problem with "charge"? After all, there is a completely analogous expression for charge: F = qE. Can you completely decouple all those quantities from each other to arrive at an independent determination of "q"?

5. If you think you have a problem with "mass", just wait till you get to "spin".

Zz.

11. Sep 2, 2004

### da_willem

F=qE is not at all analogous to F=ma, as the first relates a field to the force on an object in that field, while the latter is a fundamental relation describing the motion of an object under the infuence of a force.

12. Sep 2, 2004

### DarkEternal

i think ZapperZ's point about F=qE was that it is similarly difficult to define the concepts of charge and the electric field without circular definitions.

13. Sep 2, 2004

### ZapperZ

Staff Emeritus
I knew this would happen.... :)

I should have qualified what I meant by "analogous", even though I think I explained it in the next sentence of that posting. I was pointing out the similarities in terms of the inability to "decouple" the three quantities, the same way the original argument was made on the apparent inability to independently determine F, m, and a in F = ma. The concept of "charge" is irrelevant unless it is in an EM field, and so are intimately tied to it. So this is the "analogous" argument that can be made similar to a measurement of "mass" and "force".

Zz.

Ah, thanks DarkEternal. I'm glad I just didn't imagine the point I was trying to make there. I could have been clearer though, so I'm glad da_willem gave me the opportunity to clarify.

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14. Sep 2, 2004

### suffian

1. your right. when did i say it had too? in fact, i was trying to argue that we should avoid just letting mass be an intuivitive concept (if we can) because it allows many people to fall into such a trap.

2. sorry, I suppose this is my fault, but when i said "my definition of mass", I was referring to my first post. you might want to read that a little more carefully.

3. okay.. that's fine. indeed it could serve as a good basis. as a first year student in physics, i wasn't about to think about that one, so i'll just stick with my equal-arm balance approach for now. it's like the trig functions, initially they are defined in terms of a unit circle, but as your knowledge increases they are defined in terms of a series.

4. okay, i didn't think about charge simply because i haven't gotten there. so, yes, maybe i am jumping the gun by considering these concepts so early. you would probably have to consider it all together, but i wouldn't rule out the possibility that you could decouple them.

5. this is just a guess, but "spin", "mass", "charge".. these are all similar properties, namely they are a quantification of a property that a system appears to possess. if you come up with a good way to define one of them, then you could possibly use analogous techniques to define the others.

I'm not trying to cause any mayhem here. So please don't get so agitated. I'm just saying "hey, we could do better than this." I think many people do sometimes ask meaningless questions because these definitions are not clear. So i think it's to everyone's benefit.

15. Sep 2, 2004

### ZapperZ

Staff Emeritus

You implied that anything with no mass should not exist.

I did! That's why your 2nd "definition" of what a mass is was astounding.

Not sure what you're trying to say here. Trig functions are not defined as a series. The series (if we're talking about Taylor series) are just expansions about a value. The fact that they can converge to a finite value or expression does not mean the series IS the definition!

Then you also shouldn't rule out the possibility you can decouple F=ma, which would then make this whole thread moot.

You seem to have a rather strange concept of what a "property" is. We say one of the property of an electron is that it has a "charge" to mean that when it is placed in an electric field, it will behave in certain ways that are well defined. This IS the definition of what it means to have a "charge"! You seem to think the "quantification of a property that a system appears to possess" isn't a definition. If you think about it carefully, what defines you is how you look, how you behave, how you smell, etc.. etc. These are all your "properties". An electron will have a set of properties that are RELEVENT to its interactions. We do not define an electron with properties that we do not detect upon measurement (i.e. electron is extremely shy and has sensitive feelings).

So to criticize the fact that a "mass" is not a valid "property" simply because it cannot exist by itself but has to be in conjuction with other quantites is missing the WHOLE POINT of assigning those properties in the first place. We give things a set of properties BECAUSE we want to know how they would interact and behave in conjuction with other quantites! If not, who cares what properties that thing has, because it does and affects nothing.

In any case, shouldn't you be busy batting down the hatch before Francis shows up?

Zz.

16. Sep 2, 2004

### da_willem

You're welcome . No serious, sorry I interpreted your claim wrong. I only scanned the topic when my eye fell on that sentence. I should have known better and read all of it...

17. Sep 2, 2004

### suffian

Sorry. I'm just not a good writer then. Let me clarify. I certainly do not mean that anything without mass shouldn't exist. What I meant was is that if we are not clear about what we mean by mass (as would seem in most texts, basic ones at least), then many people will start asking somewhat meaningless questions (like how can a massless particle exist?) because they want their intuitive conception of mass to jive with what they are being told.

Again, forgive me for not being clear. In no was I trying to redefine mass the second time. What I was trying to say is that in my first definition I did not try to visualize mass in any way. I simply labeled as a property that a system posesses. We then come up with a way to say that when this property is equivalent between systems, etc. From there we make empirical observations to build up a set of principles to describe how mass behaves.

Haha. I think I got you on this one. it IS the definiton! Simply because it makes for a more concrete definition.

I never said you couldn't! That's what my whole first post was a try at doing.

I agree, but you can't say "well, this is how it reacts in the presence of an electric field" when you don't know what an electric field is. You can't say this is how it responds to a force when you don't know what a force is. Imagine your trying to explain all of this to an alien. We might expect he will understand the concept of space, time, and motion, but forces, energy, fields, etc., he likely wouldn't know. Maybe he devised a way to predict the motion of objects without the concept of forces. We would have to explain what we mean by these things, and in doing so, we would be defining them. So at the beginning, I think you really only have a couple basic ideas to deal with. One is the motion of a system and the other is the concept of a system itself. Whether you can reconstruct all the other cocnepts in a way that coordinates with how we think of them now from these basic concepts combined with empirical observations, i'm not so sure. Maybe it can't be done. In that case, your right, you might have to leave a few other concepts undefined as well. But wouldn't we like to leave as few undefined concepts as possible?

yeah.. we lost a tree and had a leaky roof to charley. but any disadvantages are outdone by school being off. :-)

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18. Sep 2, 2004

### ZapperZ

Staff Emeritus
Let's try this ONE more time...

Try again. How does the Taylor expansion of sin(theta) DEFINES what sin(theta) is?

Alien: what is mass?

ZapperZ: OK, see this thing? See how it is oscillating up and down at the end of that spring? Now, let me replace that and put another thing at the end. Now see how it is still oscillating, but at a slower rate? That 2nd thing has a larger quantity that we call "mass" than the first.

Alien: Are you sure? I mean, your earth could simply be pulling more on the object because .... er.... it loves that object more than the first one.

ZapperZ: OK, we can do this in space. Let's use your spaceship and go to the middle of nowhere where gravitational force isn't a factor.

Alien: You got a deal!

<Alien and ZapperZ get into a spaceship and fly to Detroit>

<OK, OK, that was a low blow, but you get the idea>

Alien: OK, we are in the middle of nowhere, now where's that bouncing thingy?

ZapperZ: here it is... now here's the first mass... see how it oscillates the same way?

Alien: Yes.

ZapperZ: OK, now substitute that with the 2nd mass. See that it is also oscillating slower? The only difference we have here and on earth is that the equilibrium position is different. But other than that, nothing has changed.

Alien: Yes.

ZapperZ: You are seeing the quantity that we call inertial mass, and that the 2nd thing has more of it than the first.

Alien: OK, I understand. But I'm not the one who you should convince.

ZapperZ: I know. But maybe he's busy with Francis.

Zz.

19. Sep 2, 2004

### suffian

Lol, sounded like a great conversation to me. Really, I think that works well. But I don't see it as a refutation of anything I've said. In fact, if you look at it, it's pretty similar to my own definition of mass. It's free of the force concept. Also, notice that you and the alien did take two ideas as implicitly understood (like a "set" in math). Both of you knew what constituted a system (the 'thing' in your scenario) and both of you understood what slow and fast was. But, you did not explicitly mention forces (and certainly not energy, etc.) because it is a more advanced construct. Instead, you made sure to explain (in fact, define) mass without the use of forces. Now your in a good position to explain (and again, define) what a force is.

http://mathews.ecs.fullerton.edu/c2002/ca0504.html [Broken]
Here is a link to a page in complex analysis. I'm not proficient in it, but it shows you right at the top (Definition 5.5) that the definition of sin(x) is in terms of a series. And you can really be sure this is a definition and not a statement of its series because they go on to derive the basic properties of sin(x) from the series. I'm very certain they do this in all classes in modern analysis. :-)

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20. Sep 2, 2004

### ZapperZ

Staff Emeritus
But...but.. I was using YOUR rules that the alien already understands the concept of time and space! You said that, didn't you?

Secondly, the idea of "mass" was NOT decoupled from force! What did you think the spring was? And that was my whole point! The "property" that is associated with anything is what is relevant upon interaction with other quantities. By itself, a "property" is meaningless!

Thirdly, you are confusing a math identity with a "definition". I define the polar angle in spherical coordinate to be measured from the z-axis. I did NOT derive, nor logically prove that! The properties of an electron are NOT derived logically from some First Principle. They are defined via experimental observation. On the other hand, the trigonometric expansion series are DERIVABLE! They are the logical outcome of each other! These are NOT definitions!

Zz.

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21. Sep 2, 2004

### suffian

Yes, it was decoupled. I think I'll extend your spring-idea to give you a show of how this is possible. Again, I begin by assuming everyone knows what constitutes a system and what motion is. From here I go hunting for an ideal spring (a system). After getting that done, I tell someone to bring me any object (a system) and I will define what its mass is. I simply attach the object to the spring and let it oscillate. I then define the mass of the object to be the inverse of the square of the angular velocity. At this point we could verify experimentally some observations about mass. For instance, if I combine object A and object B into object C, then the mass of object C is the sum of the mass of object A and object B. I refer to mass as a property of an object, because it does not vary from day to day with the object nor does it change if I reshape the object. It appears to be some intransient "characteristic" of the object.

Jolly good. So then I go on to tougher things, like forces. I define the force acting on a system A by a system B to be the product of system A's mass and the acceleration that would result on system A if system A and system B were isolated from any other systems. By isolated I mean in the limit that other systems are infinitely far apart.

Okay, now you seem to say, "hey, your definition of mass involves forces because that's what causes the body to oscillate." This is only true because I defined force in such a way that it would make it true. Had I defined it some other way, we would not neccesarily be able to say this is true.

Example:

Let's suppose I attach an object to a spring and subsequently extend the spring by pulling on the object with my hand. If I want to determine the force acting on the object by the spring, I simply resort to the definition. What would happen if only the spring and object were there? Well, all I I have to do is release my hand. I enthusiastically observe that there is indeed a force F. I run the same experiment again, this time by removing the spring and observe the force is -F. Finally, I observe the force acting on the object is zero when both my hand and spring are considered as a single system. This leads me to hypothesis the principle of superposition. With later testing I determine this is a smashing success and I'm on my way to building rockets.

Example:

When no other systems are present, there can be no forces acting on an object. I then make the observation that when no forces are acting on an object, it's acceleration is always zero. Wonderful, Newton's first law!

These examples are simply meant to show that I can begin reconstrucing classical physics (slowly but surely) with definitions of force and mass that are not circular. The mass of any system in my ficticious world does not depend on the forces acting on it. The two are correlated, but don't confuse the concept of dependence and correlation. It is the forces that, by definition, require knowledge of the mass. It might be possible to make another ficticious world where I define what force and mass are completely independently. In that case F = ma becomes an empirical observation.

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22. Sep 3, 2004

### metacristi

The problem you [raise] here it's very interesting going well beyond mere science,[belonging rather to the] epistemology of science (not too interesting for a great number of scientists,who takes for granted strong realism,namely that science approach,at least,truth and that all frutful theoretical scientific constructs are also real).Well a 'perfect',undisputed,definition of mass would require an explanation,based finally on some basic 'truths',deduced directly from empirical data.Unfortunately,epistemically speaking,we do not have such undisputed truths,from where all other 'truths' about the world be deduced further,step by step,the dream of Aristotle.Unfortunately science is fallible,we can never be sure of the truths of our theories,'raw' empirical observations,sensorial or measurements,can only be interpreted;we cannot deduce absolute truths from raw data (moreover currently does not even exist an argument,proving beyond all reasonable doubt,counting as sufficient reason,that is in a very clear,'objective',way,that our inferences from observations are approximatively correct,true,or at least more probable to be correct).Thus there might be no such 'perfect' definition (known to be correct in absolute,at least approximatively correct) if underdetermination of theories irrespective of data gathered is correct and if some final [measurement] limits exist (here already we have enough reasons to think that HUP puts such a limit,in spite of the fact that there is still enough room for a rational,solid skepticism).Anyway the point is that science does make assumptions that goes well beyond mere empirical data.

With this fact clarified we must ask if,whilst accepting some basic assumptions (the actual basic assumptions of science,for example),we can define mass.It must be made clear first that there is a difference between a purely phenomenological definition and an explanation deduced from some more basic assumptions.Mach,or Auguste Co[m]te,for example,old positivists in general,would have argued that the definition of mass based on the Paris etalon,even yours,is enough for all our practical purposes and that it is futile to try to derive it from more basic notions (which,upon them,would be a waste of time,for it might be not possible).Indeed once defined phenomenologically the notions and the etalons of mass,space and time,[kg],[m],,toghether with Newton's postulates,all classical mechanics can be derived.In our days these positivistic dogmas have been dropped of course,it is worth trying to find such deeper explanations (even the myth of 'unicity,the only rational possibility' of the copenhagen interpretation,positivist in attitude,has been seriously shaken).Moreover we have more choices,there are no restraints,there is no univocity,for example we can define force,space and time phenomenologically and then mass in function of force and acceleration (space,time).[Also the mathematical formalism of classical mechanics,assigned to the same interpretation of observed facts,is not univocal,D'Alembert principle is basically equivalent with Newton's second law,from here we can deduce both Euler and Hamilton's equations,which proves much more convenient in many practical situations].

Returning to the main point there are such 'deeper',potential,explanations of mass within the frame set by the actually accepted set of scientific enunciations and the actual version of the scientfic method going down till the hypothetical Higgs boson.Now if it is 'real',even 'confirmed' indirectly,[rather] than only another theory ladden theoretical construct,is totally another problem;again there is a huge room for skepticism.Some people (feyerbendists for example) would argue (there are enough logical resons to consider such a skepticism as being rational) that no,even the 'evident' notions such as space and time are theory ladden (I think Russell's 'The problems of philosophy' is still a very good account of the difficulties existing in the theory of knowledge,though,of course,as almost everything in philosophy,some of his points are controversial,rejected by some other (very serious) philosophers: http://www.popular-science.net/books/russell/chapter0.html).

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23. Sep 3, 2004

### da_willem

I think this is the only sensible thing we can say about Newtons second law. And it requires an independent description of a force like $F_g=-GmM/r^2$ to be of any use.

24. Sep 3, 2004

### ZapperZ

Staff Emeritus
This example completely escapes me.

You pulled on the object with your hand, and you are able to determine the force acting on the object by the spring? OK, this assumes you know either (i) the amount of force you apply with your hand, or you are able to measure the extension of the spring and Hooke's law is obeyed.

But the part where you let go, and the part where you remove the spring and observe a force of -F, and oh, "superpostion"? Whoa!

Er.. I haven't seen it yet how you have accomplished this. F=ma IS a phoenomenological equation. I've said that WAY in the very beginning of this thread, did I not? Each of your examples requires that you have a force acting on an object to determine its mass. If the object is just sitting there, as in your last example, you have ZERO ability to determine it's mass. I give you a bowling ball and an a tennis ball. They both just sit there on a table (zero net force) Now tell me their masses.

Zz.

25. Sep 3, 2004

### cartuz

Of cause, you are all know Mach's definition of the mass in Mach's Principle. According Mach, the mass is a map of the interaction objects with the ALL objects in Universe. If I'm a correct to remember. I think it is the start point to the mass definition to me.
E. Mach, Conservation of Energy, No 1, 1872

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