A paradox inside Newtonian world

[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?

 

[nealh149]

Yes because the force on the center of mass will be equal to the sum of the forces on all particles in the system. Thus the forces add to be infinite. This is a problem.

 

[Tomaz Kristan]

First of all, the force to the mass center is NOT the sum of all forces. Not at all.

Do you know that?

 

[ObsessiveMathsFreak]

First of all, the force to the mass center is NOT the sum of all forces. Not at all.

Do you know that?

It hadn't occurred to us.

$$M\mathbf{R} = \sum_i m_i \mathbf{r}_i$$
$$M\dot{\mathbf{R}} = \sum_i m_i \dot{\mathbf{r}_i}$$
$$M\ddot{\mathbf{R}} = \sum_i m_i \ddot{\mathbf{r}_i}$$

 

[Tomaz Kristan]

Do you know, that this is not the case?

You and I could be forced to accelerate in the opposite directions, yet the center of the mass of the you&me system, would not move at all. Let alone to be accelerated.

Just one example.

 

[ZapperZ]

This thread has gone long enough, and I believe that nothing has been achieved. This topic is now closed, and no new ones should be opened related to it.

Zz.