Still don't understand why we need F=dp/dt

[vanesch]

You simply observe that if the horses are indistinguishable there is no reason to believe that they pull any differently. Do the experiment with 10 horses and a large ship. Differences between horses will be statistically much less significant. If the boat is massive enough the horses won't move much, initially, and you should be able to establish the linear relationship between dp/dt and number of horses.

Yes, but you don't seem to get the gist of what I'm trying to say. In order to even be able to say that "identical horses pull in the same way" you first even have to establish that horses do something such as "pulling", hence you already have intuitively introduced a notion of "force". This is what is a good idea in intro courses, but it doesn't allow you to set up a totally formal definition. Why should "horses pull" on a boat, and why should this "pulling" be an additive quantity ? Why doesn't the "pull" by the horses go, say, as the square of their number or something ? This already pre-supposes that there is some thing like a vectorial quantity, that only comes about from the single "horse-boat" relationship, and that this vectorial quantity is additive.

It is very easy to show that the force produced by weights is additive. Take a 1kg, 2 500 g, and 10 100g. weights. Have them pull against each other by hanging them over a pulley. 10 100g. weights balance the 1 kg. weight, 5 100 g weights balance the 500g and 2 500 g balance the 10 100g and 1kg weights. You quickly conclude exactly what your muscles tell you: weights are additive.

Well, that assumes already that "to balance" the rope over a pulley, they have to "pull in same amounts". It is then simpler to say that the dp/dt must be zero, and that this dp/dt is the sum of the dp/dt that the rope would undergo by each weight individually!

Again, I'm not arguing against all these approaches (which are often used in intro courses and don't pose any intuitive problems). I'm only saying that they are at a certain level dependent on some intuitive notions based upon everyday experience, and that you can hence not build a formal system around that (but of course you can do practical stuff that way). And that the simplest way to define force purely formally, is to say that it is nothing else but another name for dp/dt, and that dp/dt has a very special property (namely its additiveness in different situations: from "sub-situations" up to a more involved situation). If you do so, you can get rid of the intuitive notion behind a force, that it is some "kind of pushing or pulling". In fact, you can start to understand our intuitive notion of "pushing and pulling" as a consequence of this special additive property of dp/dt.

 

[Andrew Mason]

Yes, but you don't seem to get the gist of what I'm trying to say. In order to even be able to say that "identical horses pull in the same way" you first even have to establish that horses do something such as "pulling", hence you already have intuitively introduced a notion of "force". This is what is a good idea in intro courses, but it doesn't allow you to set up a totally formal definition. Why should "horses pull" on a boat, and why should this "pulling" be an additive quantity ? Why doesn't the "pull" by the horses go, say, as the square of their number or something ? This already pre-supposes that there is some thing like a vectorial quantity, that only comes about from the single "horse-boat" relationship, and that this vectorial quantity is additive.

The simplest assumption is that they are additive, which seems to accord with our observations. There is no reason to assume it is proportional to the square of their number or some other non-obvious relationship. You then test that assumption against what is observed and see if there is a reason to change your assumption.

You would agree that empirically dp/dt \propto \sum{F}. Now you could assume that the pulls are not additive. You would then conclude that the force of their pulls is proportional to the square of their number but you would then observe that dp/dt \propto \sqrt{F}. That is not the simplest relationship and in the absence of some evidence requiring a more complicated relationship, you use the simplest explanation.

AM

 

[vanesch]

You would agree that empirically dp/dt \propto \sum{F}. Now you could assume that the pulls are not additive. You would then conclude that the force of their pulls is proportional to the square of their number but you would then observe that dp/dt \propto \sqrt{F}. That is not the simplest relationship and in the absence of some evidence requiring a more complicated relationship, you use the simplest explanation.

Yes, but in doing so, you needed to refer to dp/dt in order to define what it means, the "total force of seven horses equals the sum of the forces of each horse". This is all what I want to point out: that the deep, fundamental meaning of force, is not some "pulling or pushing" or "stressing materials" or the like, but is simply "dp/dt". Although we intuitively see it the other way around, and although in intro courses, one first does it that way (by first going to statics, and then seeing dynamics as "statics out of equilibrium"). It was also the historical way (the "d'Alembert force" = -dp/dt in order to have "equilibrium" again).

 

[Andrew Mason]

This is all what I want to point out: that the deep, fundamental meaning of force, is not some "pulling or pushing" or "stressing materials" or the like, but is simply "dp/dt".

And all I wanted to point out that this realization was the great "discovery" of Newton. It was not about insight into creating a "definition" of force.

Newton did this by showing that a force - which we intuitively understand and feel - is measured by the change in motion that it produces in a given time interval: \Delta v \propto F and that these changes in velocity are additive:

"Law II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both."

AM

 

[vanesch]

And all I wanted to point out that this realization was the great "discovery" of Newton. It was not about insight into creating a "definition" of force.

Newton did this by showing that a force - which we intuitively understand and feel - is measured by the change in motion that it produces in a given time interval: \Delta v \propto F and that these changes in velocity are additive

Sure, and it is also taught that way, initially. But it is based on intuitive, and hence ill-defined concepts which make it difficult to be entirely rigorous when giving a more formal definition.

 

[Aero]

This all depends on what you consider obvious.

Newton considered it obvious that F_total = F1 + F2 + ...

And therefore for him it definitely was a law, to say that F = dp/dt.

But the modern view is that it is silly to think something like the fact that forces are additive obvious before you have actually defined force. For Newton, defining force was easy: force is the amount by which something is pushed or pulled.

But this is not rigorous enough for modern physicists: they want to define it in terms of what they already know i.e. measuring things. And so the only way to define force is to define F = dp/dt.

And so while Newton considered it a law relating two quantities that he already knew about, nowadays it is just a definition in preparation for the modern law F_total = vector sum of individual forces.

One can either be intuitive and concrete, and explain informally what a force is, and then uncover the law F = dp/dt.
Or one can be formal and rigorous, and define a quantity called F = dp/dt, and show that it has this remarkable additive-when-split-up-into-simpler-situations property.

 

[vanesch]

Yes, aero, that sums it up nicely

 

[Andrew Mason]

Sure, and it is also taught that way, initially. But it is based on intuitive, and hence ill-defined concepts which make it difficult to be entirely rigorous when giving a more formal definition.

I am not so sure they were ill-defined concepts. Force, including gravity, was well understood as a concept long before Newton related force to dp/dt. But until one realizes that force is proportional to the change in momentum it produces in an object, the only way to measure it is by how many bricks/springs etc. are used to produce the force.

Newton considered it obvious that F_total = F1 + F2 + ...

And therefore for him it definitely was a law, to say that F = dp/dt.

I don't think he just considered it obvious that forces were additive, he observed it.

Newton observed that if you apply a given (constant) force to a object with mass m, the quantity p = mv of the object will increase linearly with time as: \Delta p = constant\Delta t. He observed also that if you double that force (by applying two of the things that gave rise to the intial force), the proportionality constant between \Delta p and \Delta t doubled. So he concluded that the force is proportional to dp/dt.

AM

 

[vanesch]

Imagine you live in a Newtonian universe, and there are particles around you.
However, you don't know their interaction laws at all. How do you define force ?
Remember, everything bites the tail of everything else: first of all, you have to establish what is an inertial frame, because you know somehow some must exist somewhere. But in order to do so, you need to have particles on which there are no forces acting, to take them as reference points on which to build an inertial frame. You don't know which ones (how do you determine whether a force is acting on a particle or not ?). You don't know what force laws (if any) are valid in your universe. How do you go about to define such a thing as force ?

On earth, we are somehow lucky: a stupid frame attached to the surface of the Earth is already not such a bad inertial frame, without thinking. Also, there's a simple law of gravity which gives you already a nice measure of mass. But you can't count on that in every Newtonian universe, where other force laws may exist. So how do you go about finding a definition of force which holds in any Newtonian universe ?

I claim that the "trick" is, to find a coordinate system (which is a transformation of a "feasible" coordinate system based upon existing particles, which may, however, undergo certain forces), such that the dp/dt of all particles can be written as relatively simple sums of contributions from expressions containing the coordinates and velocities of other particles, in that transformed frame. This is a kind of complicated fit, in which the masses of particles, the transformation to the "trial" coordinate frame, and the expressions in the dp/dt are all "free fit variables/functions".

In most frames this will be a complicated affair, and in a certain set, this will vastly simplify (or so we hope). We take it that *that* is then an inertial frame. We might even observe that for some particles, dp/dt in that frame is 0 or nearly so. We take it that that are then particles on which no forces act (free particles) and we can use them in the future as references to build directly an operational reference frame which is inertial. The expressions occurring in the dp/dt are then the "forces" due to the interactions with other particles, and we have to hope to find some general rule for them.