A paradox inside Newtonian world

[Tomaz Kristan]

The honesty is the only tool required, as that lady said.

And a little bit of knowledge about limits also. To see that the left side force is always finite here.

 

[Tomaz Kristan]

The force, to every ball, from its left side is smaller, than if it was, if all the masses on the left were concentrated at the point, where only the nearest left ball is.

This way the mass of the nearest left ball would be only doubled and the left force would be double as much, as it is now, from this nearest ball alone.

The way it is described at the post #1, it's only 992/1000 as much.

It's just trivial to see this.

 

[ObsessiveMathsFreak]

Wakalixes.

Would you mind please explaining what in my last calculation was wrong exactly?

 

[Tomaz Kristan]

Where is your error?

The every next ball in the row is not ever closer. Their distancies go as 1.0d, 1.1d, 1.11d, 1.111d ... and so on.

I can't imagine, you didn't know this.

 

[CPL.Luke]

there is also another way of approximating this, that should become more accurate as n goes to infinity.

if you merely define a desnity function that represents the distribution continuously (would the op be opposed to this?)

you can write the gravitational field as being equal to -G int[1,0] den(x) dx/x^2

which is found by finding the contribution to the total field by any infinitesimal segment of mass.

I tried doing this with the density function as e^-x (because the calculations involving this exponential function are easier) and quickly found that the integral is divergent, as you end up with ln(0) in a number of the terms, as the divergent natural log appeared due to the division by x and not the exponential I think that its safe to infer that any exponential distribution will result in a divergent gravitational field.

thus the situation is unphysical and mathematically undefinable, and so is not worth talking about.

but things like this happen quite often when working in Newtonian gravity, just try and calculate the amount of energy that a particle gains when it falls into a point mass.

 

[Mentz114]

This is ridiculous. Almost "how many angels" stuff.

Tomaz, if you start with a physically imposible situation, it doesn't matter what physical model you use to evolve it, it will always remain a physically impossible situation.

My reaction to your 'paradox' - very amusing, so what ? It means nothing.

Stop wasting your time with these fantasies and study the real thing - GR cosmology.

 

[Tomaz Kristan]

Luke,

I don't give any guaranties to my example changed in any way. A modification still may be no good (what's the whole intent), but I stand only behind my construction.

Here, it's not the infinite amount of energy, what is the problem. It's the rebellion against the Third Law of Newton.

The bastard between Cantor's Infinity hotel and the Newtonian mechanics is the ugly one. The paradox. No matter of good advices I get.

 

[ObsessiveMathsFreak]

The every next ball in the row is not ever closer. Their distancies go as 1.0d, 1.1d, 1.11d, 1.111d ... and so on.

I can't imagine, you didn't know this.

My calculation considered the left force on every ball due mearly to the ball to its immediate left. If this sum is divergent, then the overall left force sum, which you yourself "proved" is "greater", is also a divergent sum. PLease read the argument again.

 

[Tomaz Kristan]

My calculation considered the left force on every ball due mearly to the ball to its immediate left.

This is one reason, why your calculation is wrong.

If this sum is divergent

Means nothing. The sum of all forces between various parts of a rigid body can easily be divergent. So what?

 

[Hurkyl]

Means nothing. The sum of all forces between various parts of a rigid body can easily be divergent. So what?

So you've been claiming for dozens of pages that the sum of all of the forces converges.

 

[Tomaz Kristan]

The sum of all the forces, to every ball is finite, yes. Negative, but finite.

 

[ObsessiveMathsFreak]

The sum of all the forces, to every ball is finite, yes. Negative, but finite.

And the sum of all the forces on all the balls is infinite. This has been proven twice now. Frankly, you're trolling at this point.

 

[Tomaz Kristan]

> And the sum of all the forces on all the balls is infinite.

The sum of all forces on all balls doesn't count. You can tile a cube to have the same "effect" of the "infinite sum of all forces".

It is not I who trolls here, OMF.

What only matters is:

The sum of all the forces, to every ball is finite, yes. Negative, but finite.

Now, if that's not true, is the only question.

 

[ObsessiveMathsFreak]

The sum of all the forces, to every ball is finite, yes. Negative, but finite.

To every individual ball. Look this would go a lot faster if you bothered to do some calculations.

 

[Tomaz Kristan]

Calculation

The right side force to mass point m is:

(G*m*m/2)/(d*d)+(G*m*m/4)/(d*d*1.1*1.1)+...+... = (0.992*G*m*m)/(d*d)

What else do you need?

 

[K.J.Healey]

The right side force to mass point m is:

(G*m*m/2)/(d*d)+(G*m*m/4)/(d*d*1.1*1.1)+...+... = (0.992*G*m*m)/(d*d)

What else do you need?

What you have written there is what, this?:

$$\sum_{i=0}^{\infty} \frac{Gm^2}{2^{i+1} d^{2} \sum_{j=0}^{i} 10^{-j} }= \frac{0.922 Gm^2}{d^2}$$

Im just turning your ...'s into sums.

 

[Tomaz Kristan]

Looks quite good to me.

 

[K.J.Healey]

but its not right. Left side approaches zero.

 

[Tomaz Kristan]

The left side is quite weak, but I don't care. At least not more than for the OMF's calculation somewhere above.

But the right side is correct and that's enough.

 

[K.J.Healey]

how is it correct? i was just putting what you said into calculable terms. How did you get that number, please show the math.

 

[Tomaz Kristan]

I repeat myself:

>> (G*m*m/2)/(d*d)+(G*m*m/4)/(d*d*1.1*1.1)+(G*m*m/8)/(d*d*1.11*1.11)+...+... = (0.992*G*m*m)/(d*d)

It's trivial to see that.

 

[Tomaz Kristan]

Even

(G*m*m/2)/(d*d)+(G*m*m/4)/(d*d*1.1*1.1)+(G*m*m/8)/(d*d*1.11*1.11)+...+... < G*m*m/(d*d)

would be quite enough. The left side force is finite. Also always exceeds the right side force.

No more is needed, no fancy math can change this strange fact.

 

[BillJx]

Am I wrong or not? What's your say?

- Thomas

Thomaz. The mathematians, physicists and engineers have answered your question. They have answered it more than fully, in many different ways, in a 34 page thread. You choose not to listen. So what do you actually want??

 

[Tomaz Kristan]

Thomaz. The mathematians, physicists and engineers have answered your question.

Good! What was the answer?

Was it ... that a finite force is affecting every ball? All pointed to the left? At least at t=0?

 

[ObsessiveMathsFreak]

Was it ... that a finite force is affecting every ball? All pointed to the left? At least at t=0?

We knew that from the very beginning. Your paradox concerns the force on the center of mass. Could you give your proof that this force is also finite.

 

[Tomaz Kristan]

I don't care for the mass center. I care only for the mass particles.

I am glad that you agree with me about those.

> Could you give your proof that this force is also finite.

I could, after this is settled with the majority here. That all balls are forced to the left hand side.

 

[ObsessiveMathsFreak]

I don't care for the mass center. I care only for the mass particles.

Your paradox revolved around the fact that the center of mass of a closed system was supposedly moving. If your not going to aruge that anymore then there's not much else to discuss.

I could, after this is settled with the majority here. That all balls are forced to the left hand side.

Well I'm settled on the finite force on every individual ball part. Care to move on to the center of mass bit?

 

[Tomaz Kristan]

> Well I'm settled on the finite force on every individual ball part.

Fine. Everybody else also?

> Your paradox revolved around the fact that the center of mass of a closed system was supposedly moving.

It can also revolves around the strange fact you admit. Only left pointed forces at t=0. A matter of choice.

 

[nealh149]

This is absolutely ridiculous. OMP has provided a considerable amount of mathematic proof that this mathematic problem is unsolvable, results a divergency. Yet, Tomaz, you still continue to provide an equation that is set up, but not your process for solving it. OMP, thus far, has constructed the only VALID argument between the two of you. And you are still unable to provide full and complete calculation of the force on the center of mass, which is where the core of this seeminly fake paradox lies.

 

[Tomaz Kristan]

What is your point?

That it is all OK, if all the net forces, to every ball, are finite and left pointing, as long as the force to the mass center is divergent?

Is that your point?