Expert: Daniel Mazur - 6/30/2009

Question
QUESTION: Hi Daniel,

I want to understand what a "force" actually is and what a mass actually is.

This is how I tried to understand it:

Suppose we have two things, say two rocks (let's assume them to be electrically neutral). Let's name them R1(rock 1) and R2(rock 2).
Suppose that it's only these two rocks that are present in space and no other material thing.

Now I try to do an experiment. First I keep the rock R1 in space. I just keep it. I don't push it or anything so it stays where it is. Nothing happens. Now I take this rock out, and keep only the rock 2 i.e. R2 in the universe. Again nothing happens.

But now suppose I keep R1 in the empty universe and let's clamp it for the moment so it can't move. Now I bring in R2 in the universe.
I will observe that R2 starts moving towards R1.
So what we observe is that when we keep only R2 in the universe, nothing happens. But when R1 (which is clamped) is present, R2 starts moving towards R1.
So we draw the conclusion that "it must be R1 that is doing SOMETHING on or to R2 so that it starts moving towards it."

We don't really know what this something is, and probably it is nothing, it is just the human way of thinking about it that since R2 moves only in prescence of R1, R1's gotta be doin something on R2.

We call this SOMETHING, a FORCE.
Also now we try to figure out the reason for this SOMETHING that R1 does on R2. Again, just a human way of thinking. We name the REASON for this SOMETHING or the force as "MASS".

So we say that there's something called mass that is responsible for the force. Also the same thing will happen to R1 too if we hadn't clamped it.

Am I right in the way I thought about it? If not, please tell me what a force and what a mass basically are.

Also I'd like to know how did newton reach the conclusion that force is mass times acceleration. Why is F=ma?? How do we know??

Also, in the above example we took electrically neutral rocks. So the case I discussed was essentially that of Gravitational interaction.
If the rocks weren't electrically neutral, they would have flied away or towards each other depending on the situation. Also in that case the "REASON" for the motion would have been charge and not mass. But still the force would have been "mass" times acceleration. Why??

Are forces and masses just the human way of interpretting what goes on. What happens is just that whwnever two material particles or a system of particles are present in space seperated by some distance, they move towards or away from each other. Are forces and masses just our way of interpretting this fact??

Thanks,
Shikhin



ANSWER: Dear Shikhin,
you are asking deep questions here, but perhaps some of them can be satisfactorily answered. Allow me to give you a more fundamental "lecture" first and turn to your questions later.

I have these comments to your thought experiment:
1) An experiment always starts with the observer. I don't mean to say that nothing happens without an observer, I am saying that in EVERY experiment there is an observer and he/she is part of their universe. I will come to this later.
2) When you say "there is only rock R1 in the universe", presence of the Observer in the Universe is (to our knowledge) negligible, because by a "rock" we understand something so substantial that it won't annihilate/transform by simple process of being observed. The moment he "clamps the rock down", the Observer significantly influences the picture of the Universe: Motion of objects is always relative to another object, so even zero motion is relative to something - in this case relative to the Observer, or else relative to an inertial system of coordinates of Observers choosing. This is the Galilean principle of relativity.
3) When you clamp R1 and see R2 move (or vice versa) it just means that you have pinned a system of coordinates to R1 - without any knowledge, whether the system will be inertial or non-inertial. I hope you can still follow me, it sounds a lot like play on words but it is something at the roots of our Universe. I am only transforming your thought experiment to terms of standard mechanics in 1:1 correspondence.

What is Force? There is time for that question. First you need to define system of coordinates R, time T and velocity V = dR/dT. The crucial concept then to understand is momentum P (or inertia, or impetus), which is "something proportional to velocity" and where the proportionality parameter is called mass M, i.e. P = M*V. Force on a free (!) object was defined by Newton as F = dP/dT, not so much as F=m*a. Perhaps Newton didn't believe that mass is independent of velocity (indeed it isn't, Einstein's special relativity tells us), perhaps he just preferred the differential notation expressed without unnecessary parameters of proportionality. Either way, the definition F = dP/dT is a direct follow-up of defining R, T, V, P and M as described above. You ask, how do we know F=M*a=M*dV/dT. The answer is: This directly follows from establishing measure for space (distance R) and time T. Space and time are axioms, they are taken on face value - or value of everyday experience. Force is then a direct product of those axioms, there is nothing mysterious about it.

And now comes the bombshell: The Mass. From our level of physics knowledge there is NO REASON, why the mass M obtained in course of our simple mechanics should itself be also a SOURCE of some force! It simply just so happens in this Universe that the inertial mass M is identical with the gravitational mass. The theory of general relativity does not have an explanation - it simply takes the fact M(inertial)=M(gravitational) as a postulate. According to the sources I trust people to this day don't know, why our Universe works like this.

Then you ask:"...Also in that case the "REASON" for the motion would have been charge and not mass. But still the force would have been "mass" times acceleration. Why??" This is the thing about the quantity Force that it can arise from one thing and have effect on another. There are 3 known sources of force in our Universe: the gravitational, the electro-weak and the strong interaction. The effects (!) of these forces can be varied: mechanical F=M*a (linear motion), Mom=J*alpha (rotational motion), F=k*dL (deformation), F'=F (action-reaction), then electrical, caloric, piezoelectric,... And our saying that electric force F=k*q1*q2/R^2 is also equal to F=M*a is just saying that the electrical interaction (former equation) has in some cases mechanical consequences (latter equation). Again, this is no mystery. Saying that F from the first and F from the second equation are the same allows us to link an observed effect (acceleration) to its cause (charge): a=(k*q1*q2)/(M*R^2)
Again, no tricks involved.

To your last question I say: Note that Force is not about objects moving towards or away from each other. Force is about objects ACCELERATING towards or away from one another. There is all the difference in the world between the two... It is important to be pedantically precise in the use of words here, otherwise we easily can run into mysteries or apparent paradoxes.

I hope I helped a bit, don't hesitate with a follow-up.
Cheers,
Daniel

---------- FOLLOW-UP ----------

QUESTION: Hi Daniel,

Thanks for the expalnation but still it remains sort of unclear to me.
First
I understand that F=dP/dT, and P itself depends upon the frame we chose.
You say that momentum is "something propotional to velocity" and mass is the constant of propotionality.

What is that "something" that we call momentum. how was it defined. If it was just defined as Mass times velocity (measured from some inertial frame), then we can ask what is "mass" and then according to the definition, the answer will be P/v. This leads to a circular definition where we have just chosen something called mass at our will and multiplied it with velocity and called the whole product the momentum. Both momentum and mass are then defined in terms of each other but none of them is defined seperately, fundamentally. I'd like this point cleared.

Also, I really don't know the difference between "INERTIAL MASS" and "GRAVITATIONAL MASS".
Would be great if you could tell me. But what I get from what you say is that Mass is not necessarily the "reason" for a force, it's just that in the case of gravitational interaction, it  is responsible for the force.

Also, like in the electrical interaction, charge is responsible for the force and this interaction has a mechanical consequence, i.e. acceleration and you showed that this acceleration can be linked to the charge as
a=k*q1*q2/M*R^2

But in this equation we also see "MASS" in the denominator. So the electrical interaction has a mechanical consequence acceleration. But what has mass got to do with it?? why isn't there a link only between charge and the mechanical consequence acceleration?? why does mass have to enter the equaton???

Thanks,
Shikhin



ANSWER: Hi Shikhin,

momentum is defined as the measure of object's "tendency to keep its state of motion". In other words, the ability to resist influences that are trying to change its velocity... The "influence" is a more general term than "force", it is more philosophical and hence doesn't have a mathematical recipe: even if you smile or scowl at your rock R1, philosophically you are trying to influence it :-). In the next step we define "force" as a subset of "influences" that are capable to change object's state of motion and there comes the first mathematical definition.

I suspect you are stuck at the point that in F=m*a we define mass at the same time we define force, or in p=m*v we define mass at the same time we define momentum. Why does that disturb you? Isn't it normal that definitions proceed by model
<new quantity>=<parameter of proportionality>*<previously defined quantity> ? This is not a circular definition, it is a SIMULTANEOUS definition. Why are you not satisfied that inertial mass is defined as a constant of proportionality in force (or momentum) definition?

Look at magnetic field B produced by a coil carrying current I: here B=L*I. If we wanted to, we could take this equation as the fundamental definition of magnetic field B.  There is a constant of proportionality L (inductance), which is then defined at the same time with B, by the same equation. In reality the fundamental definition of B uses an integral, because our world is not primarily made of magnetic coils. But as an integral is nothing more than a sum, then B=Sum_of_all_bits[L_bit*I_bit] and piece-by-piece the induction L is indeed defined by the same equation as B.

Ad "gravitational mass" versus "inertial mass":

Inertial mass is defined as a constant of proportionality in momentum or force definition. It is consequently the object's "tendency to keep its state of motion PER UNIT VELOCITY". That's that, this is inertial mass, which we can even call "Bob".

Gravitational mass is the "ability of a measurable object attract another measurable object" and it is defined at the same time (again) with the gravitational interaction.  The question of what the gravitational mass really is does not have any meaning in a universe without gravitation. We can call grav. mass "Jack".

Here I have defined two quantities, Jack and Bob, by completely different definitions. Bob can exist even in a universe without gravitation, i.e. without Jack, so they are different! Even in a universe without gravitation you may push a cart with force F and thereby give it acceleration a. Bob exists as a constant of proportionality between the F and a. So because Bob can exist without simultaneous presence of Jack, these two must be different quantities. As far as we know, it may be an ACCIDENT that, where Jack exists, Jack=Bob. Ever since Einstein people have been trying to find the reason, why Jack=Bob. No success so far. It has several deep implications about our Universe... and we have no explanation for it.

Now to the charge-motion effect. As you yourself repeated: "charge is responsible for the force and this interaction has a mechanical consequence". In my new terminology a=k*q1*q2/Bob*R^2. There is no place for Jack in this equation. And of course Bob is there, it is the constant of proportionality of the "mechanical effect" part of the equation.

Please try to first accustom to the fact that Bob and Jack are different, and what Bob really is (nothing more than a parameter of proportionality). Then perhaps it will all be clear - apart from the great mystery of Universe "Why does Jack=Bob?"

Cheers,
Daniel


---------- FOLLOW-UP ----------

QUESTION: Hi Daniel,

Thanks for the great explanation and I think I understood it.
This is what I make of it.
If something accelerates w.r.t. some inertial frame, there must have been some sort of interaction of this thing with some other particle which and we call this interaction the force.
Now if the thing has to be accelerated at 1m/s^2, the amount of the "interaction" needed relative to an amount of this "interaction" defined (i.e. 1 Newton)will depend on the mass of the paricle in question or the no. of atoms present in the particle.
Or in other words, if we consider a single atom, the acceleration that can be produced on this atom by a certain amount of that interaction is fixed in nature.
Or in other words the inertial mass of an atom is fixed in nature.

And in simple terms mass just links the amount of the "interaction" to the consequence, i.e. the acceleration.

now coming back to  a previous statement i wrote in my first question, that it's just that when two material particles are present seperated by some distance in space, they acclerate with respect to each other.(may be electric forces,gravitational, electroweak or a combination). I don't know about the nuclearforces, whether they also produce an acceleration. I know they fall off rapidly with the distance.
But forgetting them for a moment, we can say that material particles accelerate w.r.t. each other whenever they are seperated by a distance. Now it's how our brain thinks about it that it's only in the prescence of one another that these two particles accelerate, so the two particles must be "doing something" on or to the other particle, so that it accelerates.
isn't it that we call this "something" the force???

Though i got your point and really understood your last answer, but would just like to know whether i am wrong in the above definition of force.

thanks,
shikhin

Answer
Hi Shikhin,

yes, you've nearly got it - just some word polishing needed. When two particles (objects) do something to each other, we call the whole phenomenon an "interaction". This interaction is frequently expressed by a force... Only quantum mechanics (as far as I am aware) has interactions without related forces - roughly speaking due to the indistinguishable nature of elem particles of the same kind. In classical mechanics and large-scale electrodynamics an interaction always has a force connected to it. Some prefer to express the forces by fields, but the nature of the beast does not change with the way we look at it.

I have a feeling that you are circling around the question "How does an object 'know' that there is another object nearby and that it should start accelerating towards or away from it?" Well, theories try to explain that with the help of curved spacetime and the exchange bosons of fundamental interactions (gluon, W and Z boson, photon and graviton). One can imagine that interactions curve spacetime and as an object accelerates under the influence of the curvature, it radiates corresponding bosons. This is used in synchrotrons, where high-energy electrons are circulating on a polygonal path: in each apex they are suddenly "accelerated sideways", which causes them to release a burst of x-ray photons. These photons are then extracted from the ring and used for (mostly) materials science experiments. I wouldn't feel qualified to explain beyond this level - my specialization doesn't deal with any other exchange bosons than photons.

Like I said, for all physics outside quantum effects you may say that "this 'something' between two objects is the 'force'".

Cheers,
Daniel