A paradox inside Newtonian world

[Tomaz Kristan]

I think the problem lies in that you cannot have 'infinite balls'

Of course. Only that Newtonism forgets to tell you that.

Everybody knows, that you can't have the infinite number of (even ever smaller) balls in the real life.

Here, we are talking about an abstract theory, which is apparently inconsistent. What is still officially unknown and unheard of.

 

[radou]

Of course. Only that Newtonism forgets to tell you that.

So what?

P.S. I don't mean to sound rude, I'm just asking a simple question.

 

[ObsessiveMathsFreak]

1 - do you agree that the force from the RIGHT side is always finite, for every ball?

2 - do you agree that the force from the LEFT side is always SMALLER than TWICE the immediate left neighbor's gravity force? [As the immediate left neighbor contains half of the mass on the left side, for every sphere.]

I'm finding it difficult to follow in words.

What means, that we have a finite force everywhere?

But there are an infinite number of forces.

Do you agree then, that those forces are all balanced by the surface reaction force of every ball?

Are they? You haven't proven that yet. The reaction forces may or may not be cancelling everything out.

(Now, is it normal, that the tex preview doesn't work? So I can't write down this as a bunch of formulae!)

Please, do try. Otherwise I'm going to have to write them out instead.

 

[Gelsamel Epsilon]

Hmm, well I've been told numerous times that infinity amounts of things screws everything up.

Not only that but you cannot get "infinity" amount of things so the problem is irrelevant.

 

[Tomaz Kristan]

So what?

P.S. I don't mean to sound rude, I'm just asking a simple question.

It's still possible, that I am wrong. I don't see how, not do anybody else, but it could be, after all.

The other possibility is, that we will be forced to abandon those infinities, and everything will be under control, again.

I would like, that the infinity concept was illogical. Most people hate this idea, though.

Unwanted side effect could also be, that people will become more agnostic, scientifically. They will say: For more than 300 years, you had an error just before your noses, and you haven't seen it! How one can believe science?

That would be a bad thing to happen. In fact, science only harbored the magic (of infinity) for too long. Once we clean it, the science will be better than ever before.

 

[Tomaz Kristan]

Not only that but you cannot get "infinity" amount of things so the problem is irrelevant.

I repeat. Not in reality, that is quite common view lately. But in well established axiomatic systems, like Newtonian mechanics, the infinity is included as a vital part.

 

[Tomaz Kristan]

Left side forces:

2*F(x,x+1)>F(x,x+1)+F(x,x+2)+F(x,x+3)+...+F(x,x+y)+...

Is that true, ObsessiveMathsFreak?

 

[Tomaz Kristan]

F(x,x+1)=G*m(x)*m(x+1)/d(x,x+1)^2

D(x,x+1)=(10^-x+10^-(x+1))/2

 

[Tomaz Kristan]

m(x)=2^-x

So, is that true, that the force on every ball is finite, ObsessiveMathsFreak?

 

[ObsessiveMathsFreak]

What about the reactive forces? You're not solving for them.

 

[Tomaz Kristan]

What about the reactive forces? You're not solving for them.

The reactive force of the surface of any ball, is exactly opposite to the (finite) net gravity force. Fg=-Fr.

That is by the definition. It is not forbidden anywhere. So, why not?

That holds for the complex. Everything perfectly balanced and resting.

Jupiter however, far right away, is forced to drifts closer. No ball in the complex can go there to meet Jupiter, since it has a far more dominant left ball. Going there, would mean not behaving as F=M*a says.

So, we have a pathetic mass of 2 tones (even 2 grams, or 0.6 micrograms ...), which is quite anchored to the left point, and the incoming Jupiter on the right side.

The complex is a bizarre star dragger, with lots of balls. Strange as hell, isn't it?

 

[ObsessiveMathsFreak]

The reactive force of the surface of any ball, is exactly opposite to the (finite) net gravity force. Fg=-Fr.

That is by the definition. It is not forbidden anywhere. So, why not?

That holds for the complex. Everything perfectly balanced and resting.

No it isn't, or at least, you cannot simply say it is. You have assummed that the reactive forces are cancelling each net gravity force but that may or may not be the case. Essentially here you have assummed that the accelleration of all masses is zero to begin with. You don't yet know that. You must calculate. Talk will not do.

The complex is a bizarre star dragger, with lots of balls. Strange as hell, isn't it?

Things will become at lot less strange once you do those caluclations.

 

[Tomaz Kristan]

Don't expect too much from those surface reaction forces! They will do nothing.

Will they?

 

[ObsessiveMathsFreak]

Don't expect too much from those surface reaction forces! They will do nothing.

Will they?

If they fail to balance out the gravitational forces and one another, then the massess will begin accellerating either to the left or to the right. You'll have to calculate the answer to find out.

 

[Tomaz Kristan]

How could they do that? If they are solid enough, the reaction force of the surface is equal, but opposite to the gravity force.

What else?

 

[greedangerfoolishego]

So what you are saying, is that after all these masses fall in on each other, assuming collision is inelastic and that it forms one big mass eventually (ideally anyway), is that there is a net force still acting on the system and the resultant mass will continue to accelarate to the left?

Have you considerred your frame of reference for this problem? If you take x=o at the smallest mass (albeit infinite mass density), then this means that your frame of reference will be accelerating once that particle at x=0 is acted upon by gravitational force from the other masses. Maybe that's the real problem? If I remember correctly, Newton's Laws don't hold for non-inertial frames of reference?

Or is it the case that there is no mass at x=0, therefore won't be affected by gravitational forces, albeit infinite mass density?

 

[Tomaz Kristan]

So what you are saying, is that after all these masses fall in on each other

At this point, I have long ago lost my interest. I am gazing them only until Jupiter is still far away. The conservation of momentum has been broken, I don't care anymore, what allegedly happened after that.

Even more. Don't want to speculate about an impossible case further. It would be like to ask, if there are integers inside the set of all sets, which do not contain themselves. Game already over.

assuming collision is inelastic and that it forms one big mass eventually (ideally anyway), is that there is a net force still acting on the system and the resultant mass will continue to accelarate to the left?

Yes, as a matter of fact, it does seems so.

Have you considerred your frame of reference for this problem?

Well, it is not. I am safely parked at T(10,10,10) watching the show.

If you take x=o at the smallest mass (albeit infinite mass density)

No mass at all there at zero. But the infinite density, yes, and only at this point.

Or is it the case that there is no mass at x=0, therefore won't be affected by gravitational forces, albeit infinite mass density?

This one! Nothing at 0, but something at every point right of the zero to the rightmost point of the 1 tone ball. [Until Jupiter comes, at least.]

 

[ObsessiveMathsFreak]

How could they do that? If they are solid enough, the reaction force of the surface is equal, but opposite to the gravity force.

If the mass was "solid", i.e. unmoving. But if you assume that then you've assumed what you were trying to prove. You haven't proven what you set out to prove.

You still haven't proved that the collective mass doesn't move anywhere.

 

[Tomaz Kristan]

If the mass was "solid", i.e. unmoving. But if you assume that then you've assumed what you were trying to prove. You haven't proven what you set out to prove.

You still haven't proved that the collective mass doesn't move anywhere.

I proved for every ball on the left side. Therefore the complex also can't move.

 

[greedangerfoolishego]

At this point, I have long ago lost my interest. I am gazing them only until Jupiter is still far away. The conservation of momentum has been broken, I don't care anymore, what allegedly happened after that.

I thought the whole point of conservation of momentum was that total momentum beforehand = total momentum afterwards. How can you ignore further events if there is that time factor involved?

Why are you saying that there is a violation of conservation laws here? Are you saying that because all of the forces in your system don't cancel out and give zero, there is a brakedown of momentum conservation, and thus Newtonian mechanics?

I am stuggling with what exactly the paradox is here. What SHOULD happen?

 

[ObsessiveMathsFreak]

I proved for every ball on the left side. Therefore the complex also can't move.

No you didn't. You assummed that the reactive forces were exactly cancelling the gravity + previous reactive forces. That assumption forced the accelleration of each ball to be zero, but that's not a valid proof.

 

[Tomaz Kristan]

I am stuggling with what exactly the paradox is here. What SHOULD happen?

The mass center should not move without an external force. But it does. Jupiter comes more to the left, but complex stays put, instead of traveling toward Jupiter.

So the gravity center moves, and no external force acted.

 

[Tomaz Kristan]

No you didn't. You assummed that the reactive forces were exactly cancelling the gravity + previous reactive forces. That assumption forced the accelleration of each ball to be zero, but that's not a valid proof.

How do you expect, a surface reaction can move something? It could just be a 'passive' response to other force acting there.

The desk reactive force will not throw you pencil up. Not by itself. It can transfer some other force from bellow.

Do we have some here?

 

[greedangerfoolishego]

Having read the question again I think this is just incorrect application of Newtons laws, with terminology being misunderstood / things being mislabelled in the supposed paradox.

A collective group of body's is not itself one body.

The centre of mass reffered to in Newtons laws therefore isn't the centre of mass for the whole system, it is the centre of mass for one body i.e. one mass, not the net mass. So we have many centre of masses in this system.

Subsequently, gravitational force isn't an internal force at all, it is an external force acting on each of the body's.

Correct me if I am wrong!

 

[Tomaz Kristan]

A collective group of body's is not itself one body.

Nobody talks about that, at all.

The centre of mass reffered to in Newtons laws therefore isn't the centre of mass for the whole system

Wrong.

it is the centre of mass for one body i.e. one mass, not the net mass. So we have many centre of masses in this system.

Wrong.

Subsequently, gravitational force isn't an internal force at all, it is an external force acting on each of the body's.

Wrong.

Correct me if I am wrong!

Done.

 

[greedangerfoolishego]

Nobody talks about that, at all.
Wrong.
Wrong.
Wrong.
Done.

Fair enough!

In case I have been a bit hasty in thinking that Newton wasn't refferring to the centre of mass of a system of particles but rather the centre of mass of a rigid body,...

I think your mistake is setting x = 0 as a fixed reference point in your system.
This point interacts with the positions of the other particles in a way that gives rise to a non-inertial frame of reference, in which Newtons Laws don't hold.

Instead, how about using the centre of mass as a frame of reference?

http://en.wikipedia.org/wiki/Center_of_mass_frame

 

[ObsessiveMathsFreak]

How do you expect, a surface reaction can move something? It could just be a 'passive' response to other force acting there.

If you don't believe me, ride an elevator! Reactive forces can contribute to overall motion in their direction.

The desk reactive force will not throw you pencil up. Not by itself. It can transfer some other force from bellow.

Do we have some here?

We might. There's an infinite number of particles involved here after all. Who's to say.

Anyway, you need to actually write down some mathematical equations and solve for the accelleration of each particle. Otherwise this is all just philosophy.

P.S.

 

[Tomaz Kristan]

If you don't believe me, ride an elevator! Reactive forces can contribute to overall motion in their direction..

Well, there is a motor which runs the elevator somewhere, isn't it? And the force is translated from there.

Anyway, you need to actually write down some mathematical equations and solve for the accelleration of each particle. Otherwise this is all just philosophy.

What equations do you need? Fg=-Fr?

What forces are present here, except those two? Gravity and surface reaction?

None.

 

[greedangerfoolishego]

'A non inertial frame of reference is one in which a body violates Newton's Laws of Motion, mainly the First Law. In such a frame, despite no real force acting on a body at rest, it might move; or one that was already moving come at rest or change it's direction of motion.

Newton's first and second laws of motion do not hold in non-inertial reference frames. Specifically, masses in non-inertial reference frames appear to feel fictitious forces (such as the Coriolis force or the centrifugal force) that derive from the acceleration of the reference frame itself. Fictitious forces cause apparent accelerations in objects without any physical force causing the acceleration. Fictitious forces are proportional to the mass upon which they act; if such forces are observed, scientists will recognize that they are in a non-inertial reference frame. For example, the rotation of the Earth can be observed from the Coriolis force acting on a Foucault pendulum.'

I'm pretty sure by that by not using the centre of mass frame, you are in fact using a non inertial frame of reference. Incidentally, an elevator is a non-inertial frame of reference too, you feel imaginary forces as it comes to rest / starts moving because it is accelerating.

 

[Tomaz Kristan]

I'm pretty sure by that by not using the centre of mass frame, you are in fact using a non inertial frame of reference. Incidentally, an elevator is a non-inertial frame of reference too, you feel imaginary forces as it comes to rest / starts moving because it is accelerating.

As I've already said. I am sitting at X=10, Y=10, Z=10, watching down the show.