Empirical and Definitional Content of Newton's Laws

[madness]

As the distance increases the measured angle decreases but the hypotenuse increases. If you work it out you will find that the measured linear acceleration of distant objects is independent of the distance. If you are using a non-inertial frame then the distant stars indeed fall, they do not stay fixed.

Which angle? I was referring to their angle(s) in polar coordinates with Earth at the centre. Certainly this doesn't decrease as the distance from Earth increases. We can essentially only measure these two angles (azimuth and elevation). The acceleration of azimuth and elevation goes to zero as the distance of the object goes to infinity, even if we apply linear acceleration here on Earth. As DrStupid says, today we may be able to make more accurate measurements, but only insofar as the stars aren't sufficiently distance to be considered "fixed".

 

[Dale]

Which angle? I was referring to their angle(s) in polar coordinates with Earth at the centre. Certainly this doesn't decrease as the distance from Earth increases.

The angle subtended by their change in position over time. Yes, measured in polar coordinates with Earth in the center.

The acceleration of azimuth and elevation goes to zero as the distance of the object goes to infinity

Yes, but the hypotenuse goes to infinity. Their product is constant and independent of the distance.

 

[madness]

The angle subtended by their change in position over time. Yes, measured in polar coordinates with Earth in the center.

Yes, but the hypotenuse goes to infinity. Their product is constant and independent of the distance.

Sure, but you can't use observations of the night sky to construct an inertial frame using the coordinates of the distant "fixed" stars. Unless you use something like redshift to estimate their radial acceleration.

 

[Dale]

Sure, but you can't use observations of the night sky to construct an inertial frame using the coordinates of the distant "fixed" stars. Unless you use something like redshift to estimate their radial acceleration.

Yes, of course. You definitely need radial information also! If you don't have radial information then how could you apply Newton's laws? You wouldn't have velocities or accelerations but something quite different and not meaningful in the context of the laws.

 

[madness]

Yes, of course. You definitely need radial information also! If you don't have radial information then how could you apply Newton's laws? You wouldn't have velocities or accelerations but something quite different and not meaningful in the context of the laws.

If we have access to detailed measurements of their three-dimensional motion then it ceases to make sense to refer them as being "distant" or "fixed". My impression was that the term "distant fixed stars" was intended to convey something about their usefulness for constructing an inertial frame in contrast to nearby moving objects.

 

[vanhees71]

Today a much simpler frame of reference is the restframe of the cosmic microwave background, which you can establish using local observations. It's very accurately done already with, e.g., the CMBR satellites WMAP and Planck.

 

[Dale]

If we have access to detailed measurements of their three-dimensional motion then it ceases to make sense to refer them as being "distant" or "fixed". My impression was that the term "distant fixed stars" was intended to convey something about their usefulness for constructing an inertial frame in contrast to nearby moving objects.

The term “distant fixed stars” is just a historical term. I agree that the term doesn’t make sense given modern astronomical knowledge. But for historical reasons it is still used when discussing constructing an inertial reference frame from astronomical objects. I believe that it was originally intended to refer to any actual stars (besides the sun), as opposed to planets and comets which were seen as wandering stars rather than fixed stars.

 

[f todd baker]

My goodness, this is a long chain of discussion! I haven't read it all but what I did read did not mention something which I always taught my intro students in the first class. Newtonian physics requires that we somehow understand intuitively without derivation four concepts: length, time, force, and mass. Force and mass are the tricky ones for students, force is a push or pull and mass somehow measures how much stuff you have. But suppose that we come up with an operational definition of mass--a kilogram is the mass of some standard chunk of stuff in some vault in Paris. And suppose we imagine that, although we do not have an operational definition of force, we can imagine having a machine (maybe a spring) which will reliably exert a constant force; then we can imagine doubling the force (two machines), tripling it, etc. Now we interact with the real world and do an experiment and easily discover that a∝F/m. We make this into an equation by adding a proportionality constant C which we can choose to be anything we want because F has not been defined. I choose C=1 and voila, F=ma and F is now operationally defined as the force which causes a 1 kg mass to have an acceleration of 1 m/s^2.
But, although that is what we like to do as physicists, it is not unique because F and m are not unrelated. The other way to approach the problem is to choose F rather than m to build our system of units. This is exactly what the Imperial units do, based on the pound, foot, second rather than kilogram, meter, second. Experiment still finds a∝F/m and I still choose C=1 and I still have the same Newton's second, m=a/F, and now a unit of mass is the mass which will experience an acceleration 1 ft/s^2 if pushed with 1 lb of force. (I realize that the more conventional Imperial unit for mass is the mass of a 1 lb weight, but then the second law is no longer F=ma. My goal here is to illustrate that one only needs three intuitive concepts to start physics, F and m not being independent.)

 

[vanhees71]

Well, yes, that's of course a good way to start, but we are discussing the problem that for doing all this you need to start with some spacetime model. It all starts with "kinematics" before it comes to "dynamics", and the issue with the Newtonian system of postulates was known from the very beginning, i.e., how to operationally determine either Newton's absolute space and absolute time (Newtonian point of view) or a set of bodies defining an arbitrary inertial reference frame (Leibnizian point of view), but we've discussed this above in great detail already.

Of course FAPP your approach is the right one and the only way to get started in physics.

 

[avicenna]

First, we must know that our textbook explanation of Newton's laws is our accepted approach which may be different in some ways to Newton's original; e.g. space and frame.
1) The Latin of the Principia is "Three Axioms" or accepted truths to start with, not laws as with testable laws.
2) mass - The Principia also assumed mass as a defined concept/axiom using the weighing scale and the pendulum to support its concept of mass.

The first law defines the inertial reference frame. A "perfect" orbiting space lab with no rotation with respect to the Earth is an inertial reference frame. An object can be made to stay still or move with uniform motion. The inertial reference frame is only an ideal and never testable to an absolute precision.

The second law is an axiom/definition of force. I think only within an inertial reference frame.

Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

The third law need no testing as it is a consequence of the 1st and 2nd law.

 

[Trying2Learn]

I am not sure if I can help, as most of the time, I ask questions.

Still, this confused me too, until I learned that aside from the three laws, Newton also put forth some definitions, BEFORE the laws.

Here are four
The quantity of matter is the measure of the same, arising from its density and bulk, conjointly.

The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter, conjointly.

The force is a power of resisting, by which every body, as much as in it lies, continues in its present state, whether it be be of rest

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.

Then, I do remember reading somewhere (I do not know where) that some physicists feel that the first law should not be a law, but a definition (or something). I do not recall. This is probably not much help.

 

[A.T.]

The third law need no testing as it is a consequence of the 1st and 2nd law.

How does the 3rd law follow from the 1st and 2nd?

 

[avicenna]

How does the 3rd law follow from the 1st and 2nd?

If m₁ acts on m₂ with a force f, then f=m₂ a₂; a₂ is relative to m₁. Then m₁ too has relative acceleration a₂ relative to m₂. Thus there is a force acting on m₁; the magnitude of the force is also the same f = m₂ a₂.

I think Newton's law applies only in inertial reference frames; it cannot be otherwise. We assume that m₁ and m₂ are initially at rest in an inertial reference frame. Then a force f appears.

 

[Dale]

The third law need no testing as it is a consequence of the 1st and 2nd law.

This is certainly not correct. In fact, it is the third law that contains the empirical content of Newton’s laws.

If m₁ acts on m₂ with a force f, then f=m₂ a₂; a₂ is relative to m₁.

No, a₂ is relative to any inertial frame. The acceleration with respect to m₁ may be very different from a₂

 

[DrStupid]

In fact, it is the third law that contains the empirical content of Newton’s laws.

And that's why Newton supported it with experimental evidence.

 

[A.T.]

...m₁ too has relative acceleration a₂ relative to m₂ ...

No, the acceleration in the 2nd law is not relative to a second mass. The 1st and 2nd laws do not even mention a second mass, only the 3rd does. So it doesn't make any sense that the 3rd follows from the 1st and 2nd.

 

[avicenna]

No, the acceleration in the 2nd law is not relative to a second mass. The 1st and 2nd laws do not even mention a second mass, only the 3rd does. So it doesn't make any sense that the 3rd follows from the 1st and 2nd.

The OP used this form of the 3rd law, action and reaction between two bodies.

Third law: When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

The Principia original may be: "For every action, there is an equal and opposite reaction"

Let's consider m₁ acts with the force f on m₂. Assume the inertial reference frame with m₁ as origin. From the 2nd law, m₂ has an acceleration a₂ = f/m₂ in this frame.

Now we take all motion relative to frame (inertial?) with origin at m₂. In this frame, m₁ has an acceleration which magnitude is the same a₂= f/m₂. If this frame of m₂ is inertial, then there is a force of magnitude f acting on m₁.

 

[A.T.]

Assume the inertial reference frame with m₁ as origin...

The rest frame of m1 is not inertial, if there is a force acting on m1.

...If this frame of m₂ is inertial,...

See above.

 

[avicenna]

The rest frame of m1 is not inertial, if there is a force acting on m1.See above.

I think I am wrong. The 3rd law is another necessary axiom. It definitely is not as a testable law. It is the 3rd axiom that allows Newton's gravitational law to act mutually between two bodies.

 

[Dale]

It definitely is not as a testable law.

It is testable:

Start with two bodies interacting with each other but otherwise non-interacting. Measure their momentum over time wrt some inertial frame. If their total momentum changes then the third law is experimentally falsified.

 

[avicenna]

It is testable:

Start with two bodies interacting with each other but otherwise non-interacting. Measure their momentum over time wrt some inertial frame. If their total momentum changes then the third law is experimentally falsified.

Your experiment cannot be said to be a test of the third axiom alone, but rather a test of Newton's three axioms together; i.e. Newton's three laws of motion holds when there is no change of momentum over time.

The reasoning is this; the law of conservation of momentum is a consequence of applying the three laws together. If anyone of the three axioms cannot be satisfied or fails, then there would be a change in momentum over time; .e.g. if the 2nd law fails, then force will not be proportional to rate of change of momentum; this would also imply f ≠ ma and your experiment would fail to verify the three laws of motion.

 

[Dale]

Your experiment cannot be said to be a test of the third axiom alone, but rather a test of Newton's three axioms together

That is a world of difference from your blanket statement that it is definitely not testable.

I frankly don’t care if you consider the laws separately or together. Either way, you definitely do end up with something that is testable.

And in any case it certainly “can be said” to be a test of Newton’s 3rd law alone by considering the first two laws to be definitions and therefore not testable. Perhaps that is not your preferred formulation but “cannot be said” is vastly overstating the case.

 

[avicenna]

That is a world of difference from your blanket statement that it is definitely not testable.

I frankly don’t care if you consider the laws separately or together. Either way, you definitely do end up with something that is testable.

And in any case it certainly “can be said” to be a test of Newton’s 3rd law alone by considering the first two laws to be definitions and therefore not testable. Perhaps that is not your preferred formulation but “cannot be said” is vastly overstating the case.

I think there is good reason why some would translate the Latin "Lex" as axiom and not law. The conventional use of the word law in physics is a standalone testable law as in Coulomb's law, Ohm's law.

Newton's three laws of motion are closer to axioms in mathematics; they are the basic framework wherein calculations may be made in mechanics. There are no priority in the three axioms; they are the set of axioms. If the momentum experiment fail, there is no way to determine which of the axioms failed; it could be the 3rd axiom is ok, but the 2nd axiom failed.

So I would prefer to view Newton's three laws together and as the starting foundation of mechanics.

 

[Dale]

The conventional use of the word law in physics is a standalone testable law as in Coulomb's law, Ohm's law.

In that sense Newton’s 3rd law is indeed a law. In Ohm’s law current and voltage are defined elsewhere. Similarly in Newton’s 3rd law inertial frames and forces are defined elsewhere. The law (Ohm’s or Newton’s 3rd) is then testable. If you allow Ohm’s law as a law then you have little basis to exclude Newton’s 3rd law as a law.

I think there is good reason why some would translate the Latin "Lex" as axiom and not law. ...

Newton's three laws of motion are closer to axioms in mathematics; they are the basic framework wherein calculations may be made in mechanics. There are no priority in the three axioms; they are the set of axioms. If the momentum experiment fail, there is no way to determine which of the axioms failed; it could be the 3rd axiom is ok, but the 2nd axiom failed.

All of this, at best, is a matter of personal preference, and it is a preference that I do not share. Particularly the claim that “Newton's three laws of motion are closer to axioms in mathematics“. Since at a minimum together they are experimentally testable, such a claim seems problematic at best.

I have not even seen a professional scientific reference which makes this claim. Do you have such a reference or is this pure personal speculation?

So I would prefer to view Newton's three laws together and as the starting foundation of mechanics.

So say that. Your blanket claims like “it cannot be said ...” are simply wrong.

 

[fightingphysics]

There is no empirical content. Newton's laws are axioms, that not goes from reality, like axioms of mathematics or church dogmatic or something like that

 

[vanhees71]

You have axioms in mathematics. There are no axioms in physics, just some fundamental laws which are based on observations.

 

[fightingphysics]

You have axioms in mathematics. There are no axioms in physics, just some fundamental laws which are based on observations.

It's not true for modern physics at all and "classical" physics in particular. Where do you "observe" Newton's laws? In other words, how you deduce it from reality concretely?

 

[weirdoguy]

Where do you "observe" Newton's laws?

In the lab during second semester of first year of studying physics at Warsaw University