# Newton´s third law: a justification

1. Mar 10, 2010

### jpas

Hi,

I´m doing a school project about the history of physics and I think I could use your help. When we study Newton´s laws in school there´s often no justification as to why they are the way they are.

Particularly, the third law just seems Newton´s wild guess. There seem to be no good alternatives to it, since they lead to wrong predictions (if body 1 exerts a force on body 2 and body 2 doesn´t exert any force on body 1, then we could have the reverse if we started thinking first about body 2: then, body 1 would be the one that exerts no force - hope this isn´t confusing.)

So, can you cook up any explanation on why action equals reaction?

P.S.: I think the second law can be expected from the first, which in turn is well justified, if you think of a frictionless floor.

2. Mar 10, 2010

### dacruick

its just the way it is. I'm not sure what type of explanation you are looking for.

3. Mar 10, 2010

### jpas

Hi dacruick,

I´m looking for ideas about how it was derived, or, at least, about why it was proposed. Why did Newton say that action equals reaction? Why couldn´t an action be double of its reaction? Why is there even a reaction? Is the third law just a random hypothesis that is confirmed by experience or is there some thought behind it?

As an example, you can find a perfectly good justification about why the law of inertia is the way it is. If you imagine different balls rolling down different floors, each with less friction than the previous, then the balls would roll through more and space until they stopped. If you imagine a frictionless floor, you´ll verify the law of inertia.

4. Mar 10, 2010

### SpectraCat

Good! Now try the same thing for the third law ... imagine that you are on an ice rink, and you push on a marble. Now try the same thing with a bowling ball, then with a steel ball of equal mass to your own, and then with a granite sphere the mass of a large dump truck. Think about which body will move in each case if you push on each of them with the same amount of force, using exactly the same motion. That is the phenomenology behind the third law of motion. The relative masses in each of the examples I mentioned just serve to emphasize that the force is being applied equally to both bodies, even though to you it may feel like you are doing the pushing. So, if you push on a less massive body, most of the motion will be imparted to it, but it you push on a more massive body, most of the motion will be imparted to you, according to the law of conservation of momentum.

HTH.

5. Mar 10, 2010

### dacruick

Think about punching something hard like brick. And then think about punching something, with the same force, like Styrofoam. The reason it hurts more when you punch brick is because the force that you've applied is not dispersed over a long period of time because the brick doesn't budge. And now the Styrofoam. Since it indents when you punch it, the Force you are applying is gradually being reduced because of the "give" that the Styrofoam has. So to really open up this concept think about why something is soft. Something is soft when it does not have the structural bond strength to react with the same force you are putting on it. And what happens when it does not have that force to counter-act your force? Of course the styrofoam gives in. and it will keep giving until it gives enough that it has the force to match your fist, or you break through it. So you could see this as a property of hardness. hardness isrepresentative a material's ability to counteract applied force.

6. Mar 10, 2010

### tiny-tim

Hi jpas!

The first and second laws only deal with the forces on one body.

Newtons needed a law to relate the forces on different bodies, particularly so that he could deal with rigid bodies.

And the action and reaction need to be equal so that internal forces cancel … if they weren't equal, internal forces could move a body.

7. Mar 10, 2010

### Staff: Mentor

That is actually the correct way that science is done. ANY theory starts out with some set of unproven statements called postulates. They are the scientific equivalent of axioms. From those postulates you make a set of experimental predictions and you carry out your experiments and see if the results match the predictions. If so, then you say that the postulates are "verified", but they are never "proven". The entire justification for accepting the postulates is simply that they work and lead to accurate predictions.

In Newton's theory the 3 laws and the law of gravitation are postulates. They are accepted simply because they work. In other theories (e.g. Hamiltonian or Lagrangian mechanics) Newton's 3rd law can be derived from spatial symmetry (conservation of momentum per Noether's theorem), but those theories have their own postulates that are equivalent and are accepted simply because they have been experimentally verified.

8. Mar 11, 2010

### fusedestiny

Re: Jpas; Newton´s third law

To Jpas;

The theory of Newton's third law explained "action = reaction". By looking at the formula of action = reaction I can simply said that {m1u1 + m2u2 = m1v1 + m2v2}. On the left hand of the formula is the velocity before collision and on the right hand is the velocity after collision. Hence; {m1u1 + m2u2 = m1v1 + m2v2} = {action = reaction}.

In my own research and finding of Newton's laws, there must always be a continuum within Newon's laws. Example when I stated [F = M x A] = Newton's second law. But Newton's second law never be possible without Newton's first law [no force means no acceleration, and hence the body will maintain its velocity].

Now back to Newton's third law; I can said that Newton's third law is the contiuum of Newton's first law -> Newton's second law -> Newton's third law
or from Newton's third law <- Newton's second law <- Newton's law.

This phenomenon is demonstrated by Newton's cradle.

9. Mar 11, 2010

### jpas

Hi everybody,

I had never thought of the third law as equivalent to the law of the conservation of momentum. The conservation law is much more intuitive (internal forces can´t move a body!). Problem solved.

Let me just show how the law of the conservation of momentum implies the third law (you guys already know it, but I think it would be good to leave it here for the history)

Consider two bodies, A and B, on an isolated system. By the law of the conservation of momentum:

$$\frac {d \vec p_{system}}{dt}=0$$

then, $$\frac {d \vec p_{A}}{dt} + \frac {d \vec p_{B}}{dt}=0$$

by the second law,

$$\vec F_B=-\vec F_A$$

10. Mar 11, 2010

### epenguin

That is really a circular argument if you think about it. If you did not have the law of inertia you'd have no basis to say the balls would behave in the way you describe.

Yours is not a justification why it is but rather that it is - even that is not so obvious. It is more obvious to us than it was to Newton and his contemporaries because we are more used to massive objects like trains where friction is limited and which go on for quite long without any propulsion if they are not braked. Instead for the ancients their objects were mostly not massive and frictional forces, which we abstract out of the picture when laying out mechanics, usually rather dominating. So as everything soon stops it was natural for Aristotle to think everything moving had a force keeping it moving it all the time, e.g. the stone you threw had air pushing it after it left your hand. Actually 50% of people still think in Aristotelian fashion. Galileo did an experiment something like you say, sliding things down a trough of parchment that minimised friction and noted that once they had acquired a velocity they tended to maintain it as the trajectory was made flat at the end (from memory). I imagine Newton got his idea from projecting stones along frozen fens as a boy, well I like to think that.

Last edited: Mar 12, 2010
11. Mar 11, 2010

### fusedestiny

To jpas

Perhaps I should explain why the law of the conservation of momentum?? and I truely thanks epenguin for bringing moment of inertia to this problem.

My argument:

If an object A exerts a force on an object B, then B exerts an equal but opposite force on A.
The inertia of an object is its reluctance to change velocity. Inertia increase with increasing mass.

The momentum, p, of an object depends on its mass m, and velocity v: p = m*v.
The principle of conservation of momentum states that, whenever objects interact, their total momentum remains constant, provided that no external force acts on the objects, therefore:
Total momentum before the interaction = total momentum after the interaction like what I mention before.

12. Mar 11, 2010

### D H

Staff Emeritus
The reverse is also true: Newton's third law implies the conservation of momentum; the strong form of Newton's third law implies angular momentum as well.

Which is better? Lagrangian mechanics (100 years after Newton) starts from the conservation principles and is in a sense much more elegant than Newtonian mechanics. That said, there are some problems where it is very hard to come up with a Lagrangian formulation (e.g., non-conservative problems) but where the Newtonian formulation works just fine.

Which is deeper? Newton's third law and the momentum conservation laws are equivalent. Saying which one is deeper is a bit too much arguing about angels dancing on a pin. Besides, that argument is a bit moot. Noether's theorem is much deeper than either Newton's law or the conservation laws. That linear and angular momentum are conserved because space is translationally and rotationally symmetric -- that is deep.

13. Mar 12, 2010

I haven't read all the comments yet, but as I sure you have been told,
there is nothing to prov here, those are axioms, this is just the way our world is.
And proving it with conversation of momentum is meaningless, because conversation of momentum, is proved by the third law, and if not, how would you prov the conversation of momentum?

However it's clear that there will always be some axioms, that cannot be proved.
The aim is just to make a minimum of them.

The whole thing brings up the interesting question of what physics is.
Physics is the attempt to predict the result of one experiment, using the result of a previous one. This cause a creation of model of the world that wish to become as simple and aesthetic as possible.

When you think about it, there is no a real will (or way) to discover the real model of the universe, just to create a model that successfully predicts whatever we try.
It is possible that there is no forces or fields or energy whatsoever, and things are completely different, but as long as our model predicts perfectly the "real model" (a.k.a. the world we live in), it's a perfect model.

But our model is not yet so, though it has almost been thought as such, on the classical physics era, and once you discover it does not predicts correctly some things, you try to generalize it to a better model, that cover it all.

It is also possible, by the way, that even we have, say, a perfect model,
we as humans who loves aestheticism, which is a human thing anyway, will try to define things as aesthetically as possible, while it's not necessarily so.
For example, you have A that cause B and C, and B cause D, and C cause D. We say that A is the cause of D, because it's just a simpler definition, but I'm not sure this simplicity has meaning in non human concepts.
Anyway, as long as those are the only consequences of A, it philosophically equivalent whether it (A=>[B,C,D]) or (A=>[B,C], B=>[D], C=>[D]).
Things getting different if you discover some E which is consequence of A but does not cause D.
Here you'll have to give up about D being consequence of A, or grow wise and define some other common thing between C and B, that exist in A, but E doesn't share it (hope you follow).
Again, I'm not sure there is meaning to aesthetics beyond humans. We just like it, so we try to define our world so,
and I'm not opposite, I LOVE the aesthetic model as well - I am human.
But you start to see the aesthetics starts to breaks, (the discover of E in the above example), for example the model of the atom of Niels Bohr.
The electrons do not continuously lose energy as they travel in acceleration in the atom, (as they do in any other situation). Why not? because.
That might be a prov the universe itself is not aesthetic. Or it is and we just didn't yet discover it.

So about the 3rd law, maybe it's not true, but it works. it works for gad damn a lot of things. But you never know, maybe some day they'll find something that violets the 3rd law. In such case they probably try to blame something else at start, but if there will be no choice, they'll try to generalize the 3rd law even to freaking situations.

Anyway, I think that it's good opportunity to salute Newton, which wisdom has somewhat forgot in the shadow for modern heroes. I don't believe Einstein himself would do much better in the stead of Newton.
The decision that there is no reason for a body moves in constant velocity. The definition of force, and as you said the stubbornness and the impressive intuition of of the 3rd law, the intuition of the law of gravitational and much more. Not to mention the on-the-way invitation of calculus.
I'm sure he could have helped physics a lot if he lived today.

Last edited: Mar 12, 2010