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

1. ### titaniumpen

45
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 newton.JPG

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...

2. ### Simon Bridge

15,280
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. ### titaniumpen

45
Thanks, what you just said is exactly what I was thinking, especially the spring part!

4. ### technician

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. ### titaniumpen

45
^ 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. ### Simon Bridge

15,280
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. ### technician

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. ### Simon Bridge

15,280
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:
... 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. ### harrylin

That's a good question!
I agree, that experiment makes use of Newton's second law and does not demonstrate it.

Newton's second law, according to one translation:
"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!].

Last edited: Jan 5, 2012
10. ### Andrew Mason

6,895
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. ### Simon Bridge

15,280
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. ### harrylin

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. ### harrylin

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.

Last edited: Jan 5, 2012
14. ### Andrew Mason

6,895
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 travelling 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 $F=Ma$ and $F\propto Ma$, 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

Last edited: Jan 5, 2012
15. ### Simon Bridge

15,280
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. ### titaniumpen

45
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. ### harrylin

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

Last edited: Jan 6, 2012
18. ### Andrew Mason

6,895
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. ### physwizard

153
thats right. its not really a law, it is the definition of 'force'

20. ### Andrew Mason

6,895
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