A paradox inside Newtonian world

[Tomaz Kristan]

"Every ball is finite" ... what does that mathematically mean? Do you mean "countable"?

No. Every ball is just one. With a finite density and mass. Countably many of them.

The sequence of particles MUST END at x=0.

No ball is there at x=0 in my example.

 

[ObsessiveMathsFreak]

I've told you. Almost twice as big, as the force to the rightmost ball. Which is almost twice as big, as between the two rightmost balls.

And what is the force on the rightmost ball? Give us an equation for it.

 

[ssd]

No ball is there at x=0 in my example.

WRONG:
LOL,
if you distribute a mass on x-axis at distances 1/(10^N), N=0,1,2,...ad inf (of course the term you use is not 'ad inf' but mathematically it is so as per your description) then where do you end ...
Alternatives:
1/ At some negative value of x
2/ At some positive value of x
3/ At x=0

In other way of saying this is : what is the total length on x-axis over which mass particles are distributed?
1/ length>1
2/ length <1
3/ length=1

Another way of saying this:
Take a rod of length 1mt. Just as you place your mass particles, cut the rod at 1/(10^N) meters, N= 1,2,3,...ad inf.
That is, cut a piece of length 1/10mt . From this piece make a cut of 1/100mt... from this cut 1/1000mt and so on. Consider N as of the identical nature of your described N (i.e. countable, finite, the maximum not known etc as you have described or thought of).

Now if you add the length of all the pieces what shall you get :
1/ Something less than 1 mt
2/ Something more than 1 mt
3/ 1mt.

BTW this is probably my penultimate post in this thread since nothing attrective is left in your problem and I shall say goodbye to it after possibly looking at your answers to this post.

 

[Tomaz Kristan]

OMF,

0.992/0.81 units. The unit is the force between two 1 kg balls 1 meter apart. From center to center.

ssd,

You are not very strong at real numbers, do you?

 

[ssd]

ssd,

You are not very strong at real numbers, do you?

No ball is there at x=0 in my example.

Certainly not as strong as you : I cannot find a real number between '1' and '0.99999...recurring' . You are stronger enough to find such a number (who knows, possibly many) as the total length (starting from x=1 and going towards left) of particle distribution = sum of 9/(10^N) over N=1,2,,...ad inf ends before x=0 according to your opinion .

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.

I am glad to know that a new Giant has emerged in my life time who proved all Giants from 300 or more years wrong (may I say, they were ignorant of an error) at least in one case...(consequently,may be, in many retrospective cases).

THE END from my side.
With best wishes,
Thanks and bye .

 

[ObsessiveMathsFreak]

OMF,

0.992/0.81 units. The unit is the force between two 1 kg balls 1 meter apart. From center to center.

And just to make things completely clear, how is this force on the rightmost ball calculated again?

 

[Tomaz Kristan]

OMF,

We have this situation:

Force = SUM(N=1...) 2^-N/(1-10^-N)^2

Every next ball is a little (1/10) more distant and a half as massive.

 

[Tomaz Kristan]

ssd,

The point at x=0, is NOT inside the structure described. Does not matter, if we have points or (ever smaller) balls at 10^-N, for every natural N. The 0 is outside.

 

[ObsessiveMathsFreak]

Force = SUM(N=1...) 2^-N/(1-10^-N)^2

Every next ball is a little (1/10) more distant and a half as massive.

But that's not the total force on each ball. That's only the gravitational force on each ball. You haven't taken into account the surface normal reaction force on each ball. One on the left side of the rightmost ball, and one on both sides for every other ball.

You must solve for these forces before you can calculate the force on the center of mass.

 

[Tomaz Kristan]

There is no normal reaction force at pdf example in the post #1. Balls are degenerated to points. They could be also just smaller balls, so that they don't touch.

 

[masudr]

By the way, has anyone calculated the total internal energy of the system?

 

[ObsessiveMathsFreak]

There is no normal reaction force at pdf example in the post #1. Balls are degenerated to points. They could be also just smaller balls, so that they don't touch.

They could be, but in that case the case becomes identical to the point ball case in which the force on the center of mass is a divergent sum as I explained in post 27. So the force is divergent and the center of mass cannot be said to move at all.

 

[Tomaz Kristan]

By the way, has anyone calculated the total internal energy of the system?

Yes. It's not finite. As is not the total internal energy of any two mass points system.

 

[Tomaz Kristan]

So the force is divergent and the center of mass cannot be said to move at all.

No, the force is convergent. Wana bet?

 

[masudr]

Yes. It's not finite. As is not the total internal energy of any two mass points system.

Surely the internal energy of two uncharged particles is simply

$$U= \frac{1}{2}m_1\dot{x}^2_1 + \frac{1}{2}m_2\dot{x}^2_2 -\frac{Gm_1 m_2}{r}$$

Was the "not" a typo?

 

[Tomaz Kristan]

Well, the r can become as small as you want, and the energy so defined as big as you want. For every two mass particle system.

The escape velocity arbitrary great. That's what would I call "the total internal energy".

 

[masudr]

Well, the r can become as small as you want, and the energy so defined as big as you want. For every two mass particle system.

The escape velocity arbitrary great. That's what would I call "the total internal energy".

So technically it's impossible to bring two uncharged particles together. That matches experimental evidence. Do you agree with this?

If your system has an internal energy that diverges, even when there aren't two particles in contact, then it means that it is impossible to bring particles into that configuration from particles at infinity. Do you agree with this?

 

[ObsessiveMathsFreak]

No, the force is convergent. Wana bet?

It's divergent. We've been over this before. But considering you're so adamant that it is converging, you'll be so good as to grace us all with a proof then?

 

[Tomaz Kristan]

It's divergent.

What's divergent? Where have anybody showed that?

Show me, where is shown, please.

 

[ObsessiveMathsFreak]

What's divergent? Where have anybody showed that?

Show me, where is shown, please.

Good, very good. Shifting the burden of proof. Clever.

But I'm afraid I'll have to decline your most generous invitation and insist that you, as the primary proponent of the finite sum leading to a paradox, must be the one to provide this, the final step in your argument, namely, the proof that the force on the center of mass is a finite sum.

 

[Tomaz Kristan]

All you need is in the post #1, to understand the whole problem. That all forces are finite and pointed to the left is almost trivial to see.

I am not forcing you to make any new calculations. You are free to think what you want.

 

[masudr]

Yes, but given that the energy diverges, it is impossible to bring particles into this configuration from particles at rest at infinity.

 

[Eli Botkin]

That "all forces" are "pointed to the left" is neither trivial nor true. For every force pointed to the left there is an equal and opposite force (pointed to the right). That is what is meant by the 3rd law, action and reaction.

 

[ObsessiveMathsFreak]

I am not forcing you to make any new calculations. You are free to think what you want.

Well, I think I need some solid calculations. So does this thread.

 

[Tomaz Kristan]

Yes, but given that the energy diverges, it is impossible to bring particles into this configuration from particles at rest at infinity.

Of course it's possible in the abstract Newtonian world. Not in the real life, sure.

 

[Tomaz Kristan]

That "all forces" are "pointed to the left" is neither trivial nor true. For every force pointed to the left there is an equal and opposite force (pointed to the right). That is what is meant by the 3rd law, action and reaction.

I know. The third law does not permit this. But the second law demands it.

That's why we have a paradox here.

 

[masudr]

Of course it's possible in the abstract Newtonian world. Not in the real life, sure.

It's not even possible there. It is not a point in the phase of space of possible states.

 

[Tomaz Kristan]

masdur,

You can have two mass points of 1 kg each, without spliting one 2 kg mass point into two. What would require an infinite amount of energy.

In the same way, one can just set my construction #1 as the initial situation.

 

[Eli Botkin]

Tomaz:
As you know, this is a long thread and I entered very late. Very possibly you’ve already replied to my following questions, in which case I ask your indulgence.

You say “But the second law demands it. That’s why we have a paradox here.”

But the second law states only that F = ma, so it seems to me that you are basing your argument on an assumption that for this infinite sequence of masses ALL accelerations are to the left, and consequently ALL forces must be to the left as well.

Suppose we were to truncate the sequence to the first googolplex of masses, would you still hold the same view? Maybe not. And if not, isn't it possible (and maybe likely) that an ‘infinite’ sequence of masses brings about a result that is beyond our ken if it isn't correct to extrapolate its behavior from an ever increasing ‘finite’ set?

So maybe the “paradox”, based on such an assumption, is of your own making and not nature’s.

 

[Tomaz Kristan]

Suppose we were to truncate the sequence to the first googolplex of masses, would you still hold the same view?

Not at all. With any finite number of masses, everything is just fine. No paradox there.

it isn't correct to extrapolate its behavior from an ever increasing ‘finite’ set?

No, it is not always correct just to extrapolate that way. For example, every finite set of naturals have a maximum element. But there is not so for an infinite set of them. Never.

For the reals, maximum may be there for the infinite set of real numbers. Sometimes. Sometimes not.

So maybe the “paradox”, based on such an assumption, is of your own making and not nature’s.

Nature does not have this problem. Newton's abstract world, our theory has it. That's my point.

 

[ObsessiveMathsFreak]

No, it is not always correct just to extrapolate that way. For example, every finite set of naturals have a maximum element. But there is not so for an infinite set of them. Never.

But extrapolating from finite sets is the only way we have of evalutating convergent sums. Without this extrapolation, all infinite sums are undefined.

 

[Eli Botkin]

Tomaz:
But why insist that Newton's theory has to apply to infinite sets of masses?
Newton proposed a physical theory. Infinite sets are not within that scope.
Your "paradox" may be entertaining but I don't see it as any challenge to the theory.

 

[Tomaz Kristan]

But extrapolating from finite sets is the only way we have of evalutating convergent sums. Without this extrapolation, all infinite sums are undefined.

I respect this. Gave you several examples, related to this case.

The infinite sum of forces, all converge to a finite value.

 

[Tomaz Kristan]

But why insist that Newton's theory has to apply to infinite sets of masses?

Be cause there is no rule to forbid that. It is allowed, and it's enough.

 

[K.J.Healey]

I skipped over like 15 pages here cause it was getting tedious, but did you ever show WHY you believe the collective masses on the left (of the ball +jupiter experiment) do not move toward jupiter? I don't understand that.
What would keep them from moving as a whole? The collective forces acting on one of the masses is all of the surrounding masses plus jupiter. While the near masses may keep it from moving, each one has some effect from Jupiter that I don't see how gets canceled out. Each one would have an effect, that would press it a little more on the one to its right, which in turn would result in a normal on the next one toward the left, which would move it to the right, all the way down the chain. What am i missing?

Also we're assuming an infinite number of balls? Because each ball would have a resultant force to the left EXCEPT the left-most one. It would have the resultant force to the right, and it would be great, correct? Basically the force of all balls to its right. By making the system infinite its almost like looking at the whole picture, minus the leftmost ball. Right?