Can you prove Newton's Second Law from this experiment?

In summary, the conversation discusses an experiment conducted in school to explore Newton's Second Law of Motion. The set up includes a weight suspended from a string connected to a trolley on a flat table, with a pulley to reduce friction. The experiment aims to measure the acceleration of the trolley using the force exerted by the falling weight. However, it is argued that this experiment does not prove the Second Law as the force used is determined using the same law. Alternative methods, such as using a Newton-meter or elastic bands, are suggested to demonstrate the law. The conversation also touches upon the circular argument often found
  • #1
titaniumpen
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As far as I know, you can't prove laws, but anyway...

We got to do an experiment at school. A weight is suspended vertically from a string which is connected to a trolley which is placed horizontally on a flat table. There's a pulley at the edge of the table to reduce friction. Then we let the weight fall due to gravity and pull the trolley across the table. The trolley pulls along a tape as it moves, and the tape has to go through a ticker timer, which automatically dots the tape every 0.2 second.

The set up looks like this:
http://www.mathsrevision.net/alevel/pure/using%20Newton.JPG [Broken]

Anyway, we studied the tape and tried to measure the acceleration. We know the force exerted by the falling weight, and we also know the acceleration of the trolley from the tape. We also know the mass of the trolley. So if we put the values into F=ma, which is Newton's Second Law of Motion, we should find that both sides of the equation is the same. Which proves that Newton's Second Law is true!

Now, I don't think you can prove the Second Law like that. How do we know the force exerted by the falling weight on the trolley? The force is mg, right? (m=mass of weight, g=9.81m/s^2) But that is determined using Newton's Second Law. We cannot prove a law by using the law itself. Isn't that a circular argument?

Thanks for reading, this got me thinking for some time...
 
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  • #2
You cannot prove the force law ... it the definition of what the Newton means when he uses the word "force". "Force" is defined to be a shorthand word for "the rate of change of momentum".

You are correct that you have to use the force law to determine the force due to the falling weight. Thus the experiment has set up a circular argument.

The best this shows is that falling motion will couple to horizontal motion in a manner consistent with Newton's law. So the rules are internally consistent.

What you are seeing is a very common mistake in junior physics classes.

You could also use a Newton-meter to drag an object and record the force you use and the acceleration you produced ... no falling weight and you measure force directly... but wait: a Newton-meter actually measures the extension of a spring, converting to force by Hook's Law. The calibration is done by using the 2nd Law, so once again.

I'm sure you can think of others.
 
  • #3
Thanks, what you just said is exactly what I was thinking, especially the spring part!
 
  • #4
The force on the trolley is not equal to the weight at the end of the pulley. The force on the trolley equals the tension in the string (assuming there is no friction)
The weight on the end of the string is accelerating and the resultant force = weight - tension
 
  • #5
^ Whoops, just realized what a silly mistake I made! I keep repeating my old mistakes... :/

I guess that doesn't affect the main question, right?
 
  • #6
Yeah I was just going to say - titaniumpen can completely scupper the experiment described by varying it to include a range of applied forces (by using a number of small masses). Then a plot of the applied force (assumed to me nmg for n small masses m) against the observed acceleration should yield a line with slope M (the test mass). This is what I meant about the coupling and internal consistency.

In fact you get a curve - because it is the sum of the masses that counts.
The data already produced should show it up anyway if it was sufficiently carefully gathered. If the applied force was small though, it may have been small enough for the applied mass to slip under the uncertainties.

Usually, when this experiment is performed in class, a mass is moved from one to the other, keeping the overall mass the same while changing the applied force.
So - titaniumpen may be able to finesse the experiment by suggesting this as an improvement :)
 
  • #7
The best way to use this set up to find out about Newton's laws is to pull the trolley with elastic bands, trying to keep the stretch equal, 2 bands will give 2x force, 3 bands
3x and so on. With care and repeated results it is p[ossible to confirm that acceleration is proportional to force.
By stacking trolleys it is possible to cinfirm that acceleration is inversely proportional to mass.
These are great experiments in junior physics.
 
  • #8
The rubber band is being used as a Newtonmeter and has the same problem: you know 2 bands (in parallel) give twice the force because of Newton's laws. However, I'll agree it's a better demonstration ... you can pretend to define force in terms of how much a spring stretches and then show this is proportional to the rate of change of momentum: it'll hold em. (But involve a bit of a rework in how Newtonian physics is usually taught.)

It's not like confirming that the period of a pendulum is proportional to the length of the string ... those are independently defined thingies. For that matter, it is possible to demonstrate that an object continues it's motion indefinitely (absent external influences).

Here's another example:
http://galileo.phys.virginia.edu/Education/outreach/8thgradesol/Newton2.htm
... notice the wording though - there is no claim that the experiment is proving anything, just exploring ideas about force and motion described in terms of the 2nd law. They do the "number of rubber bands" thing there too.

Newton1 and Newton3 at the same place cover the other laws.
 
  • #9
titaniumpen said:
[..] Now, I don't think you can prove the Second Law like that. How do we know the force exerted by the falling weight on the trolley? The force is mg, right? (m=mass of weight, g=9.81m/s^2) But that is determined using Newton's Second Law. We cannot prove a law by using the law itself. Isn't that a circular argument?

Thanks for reading, this got me thinking for some time...
That's a good question! :smile:
I agree, that experiment makes use of Newton's second law and does not demonstrate it.

Newton's second law, according to one translation:
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.
"Force" was apparently an existing concept in his time, and his use of the term "impressed motive force" suggests to me a definition of force based on impression (for example if you have a weight of 1 kg on a scale with an an appropriate spring and you add another weight then you have double the force on the scale as measured by double the deflection, according to Hooke).
Even if you use a nonlinear spring, you can calibrate it so that you know which impression corresponds to double the force, based on the assumption (or definition) that pushing twice as much corresponds to double the force. With such a tool you could set up an experiment to verify Newton's second law: double the force as measured with your calibrated spring should according to him result in double the alteration of motion, and in the direction that you push.

[Edit: I now see that technician already mentioned that approach; and indeed, these are nice experiments!].
 
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  • #10
Simon Bridge said:
You cannot prove the force law ... it the definition of what the Newton means when he uses the word "force". "Force" is defined to be a shorthand word for "the rate of change of momentum".
I disagree somewhat. Force is implicitly defined in the first law as "that which causes the state of motion or rest of a body to change". I think one can prove that Newton's second law follows from the premises of Galilean relativity ie.

1. Newton's first law: a body will continue in its state of motion or rest unless a (net) force acts on it.
2. All inertial frames of reference (ie. frames of reference defined by the motion of a body on which no net forces are acting) are equivalent in the sense that the laws of motion are the same in all inertial frames of reference.

With those two premises you can prove (ie that it mathematically or logically follows from the premises) that Force = mass x acceleration.

AM
 
  • #11
I agree - it demonstrates the concept of force as Newton re-visualized it. The word was in use in his time, but the concept had a somewhat different common application. The ideas codified in the three laws had, after all, been around since Galileo and before.

He goes on, after the passage you quote, to specify what he means by "alteration of motion".

So long as the systematic application of force is being demonstrated, as in the virginia.edu link, there is nothing wrong with this. As soon as you try to prove F=ma the thing gets unstuck.

So starting out with defining force in terms of the extension of a spring is fine, it matches the intuitive ideas students will have around the word after all ... we don't want to be too nit-picky. We want to demonstrate this since we want students to get a tactile feel for how Newton's laws work.
 
  • #12
Andrew Mason said:
I disagree somewhat. Force is implicitly defined in the first law as "that which causes the state of motion or rest of a body to change". I think one can prove that Newton's second law follows from the premises of Galilean relativity ie.

1. Newton's first law: a body will continue in its state of motion or rest unless a (net) force acts on it.
2. All inertial frames of reference (ie. frames of reference defined by the motion of a body on which no net forces are acting) are equivalent in the sense that the laws of motion are the same in all inertial frames of reference.

With those two premises you can prove (ie that it mathematically or logically follows from the premises) that Force = mass x acceleration.

AM
I thought that in Newtonian mechanics, Galilean relativity is a direct consequence of the laws; did he postulate it somewhere?
Anyway, I'm riddled as I don't see how the second law logically follows from those premises: assuming that with "=" you meant "~", please explain how those premises exclude for example that Force ~ m (and/or proportional to some other term that I did not think of).
 
  • #13
Simon Bridge said:
I agree - it demonstrates the concept of force as Newton re-visualized it. The word was in use in his time, but the concept had a somewhat different common application. The ideas codified in the three laws had, after all, been around since Galileo and before.

He goes on, after the passage you quote, to specify what he means by "alteration of motion". [..]

So starting out with defining force in terms of the extension of a spring is fine, it matches the intuitive ideas students will have around the word after all ... we don't want to be too nit-picky. We want to demonstrate this since we want students to get a tactile feel for how Newton's laws work.

Thanks I had not noticed that - indeed, "added to or subtracted from the former motion" is a specification of "alteration of motion".

But sorry I was in fact a little nit-picky when pointing out that Newton referred to "impressed force": not exactly in terms of the extension of a spring but as measured with a spring or something flexible that has been calibrated with multiples of weights (as I imagine that may have been the standard, but I was only guessing the historical context of measurement standards for "force"). For example there still is (or I knew) the old unit "kgf", which provides a direct (obsolete) standard for force based on weights.
 
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  • #14
harrylin said:
I thought that in Newtonian mechanics, Galilean relativity is a direct consequence of the laws; did he postulate it somewhere?
Anyway, I'm riddled as I don't see how the second law logically follows from those premises: assuming that with "=" you meant "~", please explain how those premises exclude for example that Force ~ m (and/or proportional to some other term that I did not think of).
Galilean relativity can be observed in nature. So we can start with that as a postulate or premise: we observe that the motions of bodies behave the same locally whether we are traveling on a uniformly moving train, standing at rest on the ground or in a uniformly moving spaceship. Since we find that we cannot do a local experiment with bodies to distinguish between inertial frames of reference, we start with the premise that inertial frames of reference are equivalent. Call it the Galilean principle of equivalence.

It is implicit in Galilean relativity that measurements of distance and time are the same in all inertial frames of reference. This is really the only difference between the postulates of Galilean and Special Relativity.Newton's first law implies that "force" is something that changes the motion of a body: in the absence of force, there is no change in a body's motion. So a body at rest in one inertial frame of reference moves at constant speed relative to another inertial frame of reference. We could call the mathematics of translating between frames of reference the Galilean Transformation: t' = t and x' = x + vt where v is the relative speed of the respective origins of the two frames of reference.

It follows from the first law and the principle of Galilean equivalence that the same interaction between bodies will have the same effects in any inertial frame of reference. Otherwise the frames would not be equivalent. This means that the same interaction between bodies will cause the same change in motion locally in all inertial reference frames.

Suppose I have a body, M, at rest on a frictionless surface and I pull it the same way (let's say I pull it with a spring stretched the same length - I want to avoid the use of the term "force") for the same amount of time, a time Δt. I observe that the body now has speed v. The change in speed that I observe is v-0 = Δv. So now, by the Galilean transformation, it is at rest relative to a frame of reference moving at velocity v relative to the initial frame of reference.

I then consider the new frame of reference as the "rest" frame and I repeat the same experiment: i.e. I pull M the same way for a time Δt. Again the change in motion must be the same as in the first experiment in the original reference frame, since the two frames are inertial and are equivalent. So the velocity again changes by Δv, relative to the second inertial reference frame. I now define a third reference frame as the rest frame of body M. I apply the Galilean transformation and I see that Body M's rest frame is now moving at a speed 2Δv relative to the original reference frame. I repeat the experiment n times in succession and observe that Body M moves at a speed that is nΔv relative to the original reference frame (ie. after applying the pull for time nΔt). So I observe: (1) nΔv/nΔt = Δv/Δt = constant for all Δv and Δt (ie. for all n)Then I take another body identical to the first so I have Body M times 2. I apply the same pull to each of the two Body Ms simultaneously ie. the magnitude of the total pull is 2F. This is identical to the first experiment except that I have two identical bodies and two identical pulls. So the change in motion in time t has to be the same as the first: v. Pretty soon we realize that number of pulls, i, varies as the number of Body Ms I have. So I conclude that (2) iF/iM = constant. So I combine (1) and (2) and get F/M = constant x Δv/Δt. And that is pretty much the second law.

As far as there being a distinction between [itex]F=Ma[/itex] and [itex]F\propto Ma[/itex], the difference is simply the choice of units or the constant of proportionality.

Alternatively, you could postulate that F≠ma and you could do the reverse thought experiment to prove that inertial reference frames were not equivalent. Since we observe that inertial frames are equivalent, we conclude that the premise is false, which necessarily means that F=ma.

AM
 
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  • #15
Interestingly, I was concerned that I was being a tad pedantic and nit-picky :) particularly about what counts as a "proof".

The distinctions about how you avoid a circular argument in an empirical "proof" can be a bit of hair-splitting (pilleoquadrasection?) in philosophy. In practice we don't normally try to prove things empirically anyway... not how empiricism works. When we think about it like this, we start talking about an experiment "demonstrating" a concept or "providing support for" a theory, rather than "proving" it.

I think the original question has now been answered :)
 
  • #16
Thanks for the numerous replies. If you're wondering why I didn't reply, I'm still trying to figure out what you guys are saying! I don't know that a question concerning grade 10 physics would bring on so many more questions! I'll spend more time reading your replies, hope I can learn something from them.
 
  • #17
Andrew, thanks for the elaboration - I had overlooked how the first law already refers to force. Probably the Galilean transformation follows in a similar way from the second law plus the first law. Note that since one century ago we know that the Galilean transformation is inexact, but that's beyond the scope of this forum.

And titaniumpen, as Simon remarked we certainly think that your original question has been answered*, we just continued the discussion as it triggered some more thoughts. :tongue2:

* we basically agree with you: your test allows to establish (for low speeds) that the trolley accelerates with a constant acceleration when you apply a constant force.

However, by using an old definition of force such as Newton likely used (for example kgf or kilogram-force), and with a modification of your experiment as discussed earlier, you can verify how well the second law of Newton works for accelerated trolleys: you should find that the acceleration is proportional to the impressed force and inversely proportional to the weight of the trolly.
In equation: a ~ F/m
 
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  • #18
Simon Bridge said:
Interestingly, I was concerned that I was being a tad pedantic and nit-picky :) particularly about what counts as a "proof".

The distinctions about how you avoid a circular argument in an empirical "proof" can be a bit of hair-splitting (pilleoquadrasection?) in philosophy. In practice we don't normally try to prove things empirically anyway... not how empiricism works. When we think about it like this, we start talking about an experiment "demonstrating" a concept or "providing support for" a theory, rather than "proving" it.

I think the original question has now been answered :)
The experiments I was referring to are thought experiments, the results of which are determined entirely by the premise. If a result is purely a logical consequence of the premise I should think that qualifies as a proof. Whether the premise is true or not is not a matter of proof, of course.

In this case if the same effect on a body created different changes in velocity for the same object in different inertial frames of reference then the premise would not be true. Therefore, Δv must be proportional to Δt for constant mass and constant force. And for different numbers of unit bodies, numbers of unit forces must be proportional to the number of unit bodies in order to achieve a given Δv in a given Δt. That, it seems to me, is the essence of the second law.

AM
 
  • #19
Simon Bridge said:
You cannot prove the force law ... it the definition of what the Newton means when he uses the word "force". "Force" is defined to be a shorthand word for "the rate of change of momentum".

You are correct that you have to use the force law to determine the force due to the falling weight. Thus the experiment has set up a circular argument.

The best this shows is that falling motion will couple to horizontal motion in a manner consistent with Newton's law. So the rules are internally consistent.

What you are seeing is a very common mistake in junior physics classes.

You could also use a Newton-meter to drag an object and record the force you use and the acceleration you produced ... no falling weight and you measure force directly... but wait: a Newton-meter actually measures the extension of a spring, converting to force by Hook's Law. The calibration is done by using the 2nd Law, so once again.

I'm sure you can think of others.

thats right. its not really a law, it is the definition of 'force'
 
  • #20
titaniumpen said:
As far as I know, you can't prove laws, but anyway...

We got to do an experiment at school. A weight is suspended vertically from a string which is connected to a trolley which is placed horizontally on a flat table. There's a pulley at the edge of the table to reduce friction. Then we let the weight fall due to gravity and pull the trolley across the table. The trolley pulls along a tape as it moves, and the tape has to go through a ticker timer, which automatically dots the tape every 0.2 second.

The set up looks like this:
http://www.mathsrevision.net/alevel/pure/using%20Newton.JPG [Broken]

Anyway, we studied the tape and tried to measure the acceleration. We know the force exerted by the falling weight, and we also know the acceleration of the trolley from the tape. We also know the mass of the trolley. So if we put the values into F=ma, which is Newton's Second Law of Motion, we should find that both sides of the equation is the same. Which proves that Newton's Second Law is true!

Now, I don't think you can prove the Second Law like that. How do we know the force exerted by the falling weight on the trolley? The force is mg, right? (m=mass of weight, g=9.81m/s^2) But that is determined using Newton's Second Law. We cannot prove a law by using the law itself. Isn't that a circular argument?

Thanks for reading, this got me thinking for some time...
Getting back to the original post here, what you are trying to do is not to prove Newton's second law in the mathematical sense but to merely verify it - to show that it gives a correct prediction of what happens in nature.

This experiment is sometimes used in high school physics to demonstrate the second law, but it has a particular flaw: the tension force on the string pulling the trolley varies not only with the mass of the falling weight but also with mass of the trolley. What you need is the same constant pull on the trolley for different trolley masses. You would need to pull the car with elastics or springs stretched a set amount.

If you do that, you can show that the acceleration is proportional to the number of springs or elastics and inversely proportional to the mass of the trolley, just as the second law predicts. That demonstrates that Newton's second law provides the correct result. And you will not have used Newton's second law in order to produce your data.

AM
 
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  • #21
Andrew Mason said:
If you do that, you can show that the acceleration is proportional to the number of springs or elastics and inversely proportional to the mass of the trolley, just as the second law predicts. That demonstrates that Newton's second law provides the correct result. And you will not have used Newton's second law in order to produce your data.

AM

1. How do you define mass?

2. "You will not have used Newton's second law in order to produce your data" But you have to use 2nd law to show that force is additive. That is to say, the force provided by two springs stretched dx each is twice the force provided by one spring stretched dx.

If F=ma^2 (or some other weird stuff), force will not be additive.
 
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  • #22
Getting back to the original post here, what you are trying to do is not to prove Newton's second law in the mathematical sense but to merely verify it
+1

The sort of discussion we read in Principia (Newton's, not Mal2s) is showing that the form chosen for the second law - the formal definition of force - is sensible in terms of common understandings of the time. I'd bet that most people would assume that two springs, as described, would provide twice the force without a formal definition.

The formalism, though, is why we have to be careful when we use words like force and work when talking to people unfamiliar with physics.

I think it has already been well-hashed out that the 2nd law can only be demonstrated, not proved.

(That A follows from assumptions B and C would be a logical, analytic, proof ... true, and not empirical - which is important since it is a synthetic statement which is to be proved. I think there's a thread in the philosophy forums for this?)
 
  • #23
Simon Bridge said:
I think it has already been well-hashed out that the 2nd law can only be demonstrated, not proved.

(That A follows from assumptions B and C would be a logical, analytic, proof ... true, and not empirical - which is important since it is a synthetic statement which is to be proved. I think there's a thread in the philosophy forums for this?)
It is called mathematics. All I am saying is that the statements:

1. the first law of motion defining inertial frames of reference is true and
2. all intertial frames of reference are equivalent is true.

imply that F = ma.

In my example where I add unit pulls and the same number of unit bodies, if a = (iF)^2/iM = i(constant) then the relationship between Δv and Δt would not be linear. It would be proportional to the number of unit forces or unit bodies that I have added.

The result would be that I can increase the change in velocity of a unit body per unit of time by i+1 times simply by being in the reference frame of i other non-interacting unit bodies each being independently pulled by the same unit of force. The conclusion would be that the change in velocity where the same pull is applied to the same body for the same period of time is not the same in all inertial reference frames, which negates the premise.

AM
 
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  • #24
AlonsoMcLaren said:
1. How do you define mass?
In my example I didn't define mass. I defined a unit body. I think it is possible only to logically deduce from the first law and Galilean relativity that the acceleration for the same force will vary inversely as the number unit bodies to which the force is applied.

You would have to do some real experiments to determine the relationship between mass and the number of unit bodies. If it was known that there was a relationship between mass and the number of fundamental particles (protons+electrons or neutrons) a body contains, one could have deduced a general relationship between mass and the number of unit bodies.

2. "You will not have used Newton's second law in order to produce your data" But you have to use 2nd law to show that force is additive. That is to say, the force provided by two springs stretched dx each is twice the force provided by one spring stretched dx.

If F=ma^2 (or some other weird stuff), force will not be additive.
If F = ma^2 (or anything but F=ma) the premises will be contradicted - see my immediately previous post. I don't think that involves any assumptions about adding forces.

AM
 
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  • #25
AlonsoMcLaren said:
1. How do you define mass?
Mass is defined differently in different theories. Newton defined mass as the amount of matter, such as (later) the standard kg.
2. "You will not have used Newton's second law in order to produce your data" But you have to use 2nd law to show that force is additive. That is to say, the force provided by two springs stretched dx each is twice the force provided by one spring stretched dx.

If F=ma^2 (or some other weird stuff), force will not be additive.
As discussed earlier (see posts #13, #17), Newton likely used a force definition similar to the later kgf. Force defined like that must be additive just as mass is additive in classical physics. Consequently, if one would find that F=ma^2, then that would disprove Newton's second law.
 
  • #26
harrylin said:
Mass is defined differently in different theories. Newton defined mass as the amount of matter, such as (later) the standard kg.
At first this might seem a bit circular. But Newton may have had in mind a primitive concept of atoms - that all bodies consisted of finite numbers of indivisible elements of matter. In that sense his definition is not circular at all. "Quantity of matter" (or however Newton expressed it) is really the essence of what mass is. We know this from our knowledge of atomic structure. The mass of a body is, to a very close approximation, measured by the number of neutrons or protons+electrons contained in the body ie. its "quantity of matter" or "quantity of the individual elements of matter".

As discussed earlier (see posts #13, #17), Newton likely used a force definition similar to the later kgf. Force defined like that must be additive just as mass is additive in classical physics.
Unit forces are necessarily additive the same way that unit masses are if you start with the simple premise that all inertial frames are equivalent.

If I apply one unit of force to each of two equal bodies for the same unit of time, I get the same change in motion of each body. I also get the same result if I applied the same unit force to just one unit body for the same unit of time. Since the two bodies are in the same reference frame at all times, if I join them together to form a 2 unit body I have not changed anything about the physics (ie. I am not applying a net force to the two body system simply by joining them). But now I have two units of force being applied to a two-unit body for a unit of time and I get the same change in motion. Since nothing has changed in the physics, I have to conclude that forces necessarily add together just as the bodies add together eg.. 1 + 1 = 2.

Consequently, if one would find that F=ma^2, then that would disprove Newton's second law.
Certainly. But it would also be inconsistent with the premise that all inertial frames of reference (as defined in accordance with the first law, being the frame of reference of a body on which no net forces act) are equivalent in the sense that no local experiment can be done to differentiate between two inertial frames of reference.

AM
 
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  • #27
Andrew Mason said:
At first this might seem a bit circular. But Newton may have had in mind a primitive concept of atoms - that all bodies consisted of finite numbers of indivisible elements of matter. In that sense his definition is not circular at all. "Quantity of matter" (or however Newton expressed it) is really the essence of what mass is. We know this from our knowledge of atomic structure. The mass of a body is, to a very close approximation, measured by the number of neutrons or protons+electrons contained in the body ie. its "quantity of matter" or "quantity of the individual elements of matter".
Yes indeed, thanks for the addition.:smile: Newton certainly was thinking along those lines:
The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

THUS air of double density, in a double space, is quadruple in quantity [..]. The same thing is to be understood of snow, and fine dust or powders [..]. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums [..]
Andrew Mason said:
Unit forces are necessarily additive the same way that unit masses are if you start with the simple premise that all inertial frames are equivalent.
[..]
the premise that all inertial frames of reference (as defined in accordance with the first law, being the frame of reference of a body on which no net forces act) are equivalent in the sense that no local experiment can be done to differentiate between two inertial frames of reference.
AM
Sure. My point was that in Newton's presentation Galilean relativity follows from the laws of nature; and those laws can be put to the test based on non-circular definitions.
 
  • #28
Why is F=ma^2 inconsistent with the fact that all inertial frames are equivalent?
 
  • #29
AlonsoMcLaren said:
Why is F=ma^2 inconsistent with the fact that all inertial frames are equivalent?

I think that Andrew clearly explained that... Did you follow his derivation in post #14 (+ his elaboration in post #23)?

Note: probably one has to add the assumption that measurements of time and distance are the same in all inertial frames.
 
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  • #30
harrylin said:
I think that Andrew clearly explained that... Did you follow his derivation in post #14 (+ his elaboration in post #23)?

Note: probably one has to add the assumption that measurements of time and distance are the same in all inertial frames.
Yes. That is important. Newton assumed it. The essential difference between Galilean relativity and Special Relativity is that measurements of time (and simultanaeity and, therefore, distance) do not translate equally between frames: t' ≠ t.

AM
 
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  • #31
AlonsoMcLaren said:
Why is F=ma^2 inconsistent with the fact that all inertial frames are equivalent?
In looking at this again, I did not specifically address the situation where F/m changes. So I will try to do that here. It can get a little tricky so it is important to identify the principles that follow from the premise (that all inertial reference frames are equivalent).

Since all inertial frames are equivalent then whether I apply a unit of pull to each of two equal bodies simultaneously or sequentially for a unit of time should not matter. It will result in the same change of each body's motion when the pulls end i.e. if they start in the same reference frame they will end up in the same reference frame. (The only difference will be a spatial separation depending on how long I wait between the sequential applications of pull). Let's call this principle Principle 1.

Second, if two bodies are physically equal and at rest in the same reference frame, then one body can be substituted for the other and the same result will be obtained when they are subjected to the same pull for the same amount of time. Let's call this principle, Principle 2.To make it simple, let the application of one unit of pull to a one unit body for one unit of time result in a speed that we will define as a unit of velocity, v1.

Exp. 1: I apply one unit pull to each of two single unit bodies simultaneously for one unit of time. The result will be a change of v1 for each body. Let's say that the bodies are initially at rest in reference frame i0. They end up at rest in reference frame i1 moving at velocity v1 with respect to i0.

Exp. 2: The same two single unit bodies are initially at rest in i0. I apply one unit of pull to each of the two bodies sequentially, each for one unit of time. By application of Principle 1, this should give the same results as Exp. 1. So the result will be a change of v1 for each body. They both end up in i1 (but separated by a distance).

Exp. 3: This is the same as Exp. 2 except that we start with the first body initially at rest in i0 and the second at rest in i1. Again, the result will be a change of v1 for each body. But in this case, the first body ends up in i1 and the second in i2 traveling at velocity v1 relative to i1 = 2v1 relative to i0.

Exp. 4: This is the same as Exp. 3 except that now we have only one single unit body intially at rest in i0. I apply one unit of pull to the body for one unit of time and the change in velocity is v1 so it is now in frame i1. Then I apply one unit of pull to the SAME body for another unit of time. By application of Principle 2 this will give the same result as Exp. 3: it results in an additional change in velocity of v1 so the body ends up in i2 traveling at velocity v1 relative to i1. Each of the above four experiments involves the application of a force to a unit body for a unit of time twice. Each application results in the same change of motion of the unit body to which the pull is applied ie. v1. Since, the sequential application of the unit of pull to a unit body results in a change of 2v1, then, by principles 1 and 2, the simultaneous application of the same units of pull for the same unit of time (i.e. 2 units of pull applied to the same body simultaneously rather than sequentially each for a unit of time) will result in the same change of motion. So the unit body must end up in i2 traveling at velocity 2v1 relative to i0.Letting the standard unit of velocity be the velocity of a unit body after applying a unit of pull for one unit of time be v1 then (using U for a unit of Force, M for a unit body, and t1 for a unit of time):

(A) 1U to 1M for 2t1 → 2v1
(B) 2U to 1M for t1 → 2v1If [itex]F = ma^2 \text{ i.e. } \sqrt{F/m} = a[/itex] then the result in (B) would have to be: [itex]\sqrt{2}v_1[/itex]

AM
 
  • #32
Simon Bridge said:
The rubber band is being used as a Newtonmeter and has the same problem: you know 2 bands (in parallel) give twice the force because of Newton's laws. However, I'll agree it's a better demonstration ... you can pretend to define force in terms of how much a spring stretches and then show this is proportional to the rate of change of momentum: it'll hold em. (But involve a bit of a rework in how Newtonian physics is usually taught.)
You don't have to use Hooke's law explicitly. You just have to assume that the rubber band pulls with the same force given the same elongation. Two rubber bands pulling with twice the force also doesn't really require you assume Newton's laws. What if you pull two masses in parallel using two rubber bands at fixed elongation and confirm that the distances traveled per time are the same whether the masses are fixed together or allowed to move separately?
 
  • #33
As a proof? Or a demonstration?
Remember you need to prove that the force is equal to the rate of change of momentum - of, F=ma, for a fixed mass.

I like the way you've rigged pulling the masses side-by-side connected or no which makes sure the times are the same also. In fact, notice that you don't need to actually do the experiment - the proportionality logically cannot fail but follow - making a-priori knowledge of Newton's law possible (if this method is correct) which tells you that it is not a synthetic truth - eg. not of the World and therefore just a definition.

Looking more closely: Performing the experiment - which would be tricky to say the least - you are measuring F=nf (f= force due to one band/spring and n=1,2,3...) and nm (m is the mass that f pulls at a chosen constant acceleration) - plot F against nm, and get the slope f/m which is equal to independantly measured acceleration and proving f=ma... but wait: you don't know f! You need to know that to show the relation.

How do you find f?

You can show that f is proportional to m ... but you defined that to be the case when you defined force in terms of the extension of the rubber band - in the setup of the experiment... to show Newton's law you need to confirm the constant of proportionality
 
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  • #34
You use the two parallel bands to generate some arbitrary force F and then 2*F without assuming Hooke's law

You don't know a priori that pulling the masses together and separately will result in the same acceleration, which is why you do the experiment.

What you are trying to demonstrate in Newton's law is not the constant of proportionality (since that would be units-dependent) but the fact that a direct (linear) proportionality exists at all.
 
  • #35
Simon Bridge said:
As a proof? Or a demonstration?
Looking more closely: Performing the experiment - which would be tricky to say the least - you are measuring F=nf (f= force due to one band/spring and n=1,2,3...) and nm (m is the mass that f pulls at a chosen constant acceleration) - plot F against nm, and get the slope f/m which is equal to independantly measured acceleration and proving f=ma... but wait: you don't know f! You need to know that to show the relation.

How do you find f?

You can show that f is proportional to m ... but you defined that to be the case when you defined force in terms of the extension of the rubber band - in the setup of the experiment... to show Newton's law you need to confirm the constant of proportionality
Olivermsun is correct. f is just an arbitrary unit of "pull". By choosing units you can make the constant of proportionality to be 1.

As we have seen above, because all inertial frames are equivalent (our premise) the application of one unit f to one unit m will result in a constant acceleration a1. Let's define f as the "pull" exerted by a certain stretched elastic such that the motion of an arbitrary unit body, m, will experience an acceleration a1 (ie. a unit change of velocity v1 per unit of time, t1). We can also deduce from the premise that we apply a pull of nf to n unit bodies, ie. nm, the acceleration must be a constant a1.

My previous post was an attempt to show that we can also deduce from the premises (the first law and the equivalence of inertial frames) that n unit pulls, ie nf, applied to i unit bodies ie. I am will result in an acceleration of those unit bodies at the rate [itex]a = \frac{n}{i}a1[/itex].

nf would just be a simultaneous application of n unit stretched rubber bands to an aggregation of i unit bodies. We don't have to know anything about Hooke's law or anything else in order to deduce Newton's second law.

If the unit body is one proton+electron or neutron then a gram could be defined as an aggregate of N (Avogadro's number) unit bodies.

AM
 
<h2>1. How does Newton's Second Law relate to this experiment?</h2><p>The Second Law of Motion, also known as the Law of Acceleration, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this experiment, we can observe how the acceleration of the object changes as we manipulate the net force and mass, thus proving the Second Law.</p><h2>2. What is the experimental setup for proving Newton's Second Law?</h2><p>The experimental setup will involve a cart or object with a known mass, a force sensor to measure the net force applied, and a motion sensor to measure the acceleration of the object. The experiment will involve changing the net force acting on the object while keeping its mass constant, and vice versa, to observe the relationship between force, mass, and acceleration.</p><h2>3. Can Newton's Second Law be proven through one experiment?</h2><p>No, Newton's Second Law cannot be proven through one experiment alone. It is a fundamental law of physics that has been extensively tested and proven through various experiments and observations. This experiment will provide evidence and support for the Second Law, but it cannot be proven solely through one experiment.</p><h2>4. How can we ensure the accuracy of the results in this experiment?</h2><p>To ensure the accuracy of the results, we must follow proper experimental procedures and use precise and calibrated equipment. It is also important to repeat the experiment multiple times and take an average of the results to minimize errors. Additionally, we must consider and control any external factors that could affect the results, such as friction or air resistance.</p><h2>5. What are the implications of proving Newton's Second Law?</h2><p>Proving Newton's Second Law has significant implications in the field of physics and engineering. It helps us understand the relationship between force, mass, and acceleration, which is crucial in designing and building structures and machines. It also forms the basis for other laws and principles, such as the Law of Conservation of Momentum and the Law of Universal Gravitation.</p>

1. How does Newton's Second Law relate to this experiment?

The Second Law of Motion, also known as the Law of Acceleration, states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. In this experiment, we can observe how the acceleration of the object changes as we manipulate the net force and mass, thus proving the Second Law.

2. What is the experimental setup for proving Newton's Second Law?

The experimental setup will involve a cart or object with a known mass, a force sensor to measure the net force applied, and a motion sensor to measure the acceleration of the object. The experiment will involve changing the net force acting on the object while keeping its mass constant, and vice versa, to observe the relationship between force, mass, and acceleration.

3. Can Newton's Second Law be proven through one experiment?

No, Newton's Second Law cannot be proven through one experiment alone. It is a fundamental law of physics that has been extensively tested and proven through various experiments and observations. This experiment will provide evidence and support for the Second Law, but it cannot be proven solely through one experiment.

4. How can we ensure the accuracy of the results in this experiment?

To ensure the accuracy of the results, we must follow proper experimental procedures and use precise and calibrated equipment. It is also important to repeat the experiment multiple times and take an average of the results to minimize errors. Additionally, we must consider and control any external factors that could affect the results, such as friction or air resistance.

5. What are the implications of proving Newton's Second Law?

Proving Newton's Second Law has significant implications in the field of physics and engineering. It helps us understand the relationship between force, mass, and acceleration, which is crucial in designing and building structures and machines. It also forms the basis for other laws and principles, such as the Law of Conservation of Momentum and the Law of Universal Gravitation.

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