A paradox inside Newtonian world

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    Newtonian Paradox
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The discussion revolves around a paradox in Newtonian mechanics concerning gravitational forces and the behavior of masses. Participants debate the calculations and assumptions regarding how gravitational forces act on a system of masses arranged in a specific configuration. Key points include the assertion that the net force on certain masses is directed leftward, while others argue that the center of mass may not actually move left due to the dynamics of the system over time. The conversation highlights the complexities of infinite mass distributions and the implications for conservation laws in physics. Ultimately, the paradox challenges traditional interpretations of gravitational interactions in a Newtonian framework.
  • #361
masudr said:
We never consider an infinite number of particles. Remember that a rigid body is an infinite number of infinitesimal particles.


Wait a minute! A rigid body is a complex of infinite number of rigid bodies. In an ideal Euclidean/Newtonian world, of course.

I thought, everybody knows that.
 
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  • #362
Tomaz Kristan said:
Wait a minute! A rigid body is a complex of infinite number of rigid bodies. In an ideal Euclidean/Newtonian world, of course.
For example,

m=\int_V \rho(x,y,z)\,dx\,dy\,dz

Each infinitesimal volume of the object dx\,dy\,dz at the point (x,y,z) has density given by the function \rho(x,y,z).

In all calculations with rigid bodies, we can usually integrate over the volume and obtain results like the above.

Until you give me an example where I need to consider a rigid body as a complex of infinite number of rigid bodies, in which case I'd happily switch to that definition, I'd rather stick to my simpler, non-circular definition.

Tomaz Kristan said:
I thought, everybody knows that.

I obviously didn't. Why patronise?
 
  • #363
masudr said:
Each infinitesimal volume of the object dx\,dy\,dz at the point (x,y,z) has density given by the function \rho(x,y,z).

It is just another way of looking at the rigid bodies.

In all calculations with rigid bodies, we can usually integrate over the volume and obtain results like the above.

You can do the same for my (ball) example.

I'd rather stick to my simpler, non-circular definition.

It's not a definition, it's a property of a rigid body.
 
  • #364
Tomaz Kristan said:
You can do the same for my (ball) example.

I must admit I haven't followed the thread post-by-post. So what is the density function for your ball example?
 
  • #365
Every ball is 500 times denser than the previus one. And twice smaller by mass.
 
  • #366
Tomaz Kristan said:
Every ball is 500 times denser than the previus one. And twice smaller by mass.
But does the sum of forces converge?
 
  • #367
ObsessiveMathsFreak said:
But does the sum of forces converge?

No it doesn't. Why?

But the sum of all forces to any ball - it does!
 
  • #368
Tomaz Kristan said:
Every ball is 500 times denser than the previus one. And twice smaller by mass.

Where is the "previous" ball located compared to the first ball?
 
  • #369
Tomaz Kristan said:
But the sum of all forces to any ball - it does!

But it's not any old ball: it's a ball designed so that the sum of forces don't converge, isn't it?
 
  • #370
masudr said:
Where is the "previous" ball located compared to the first ball?

At 1/10^N.
 
  • #371
masudr said:
But it's not any old ball: it's a ball designed so that the sum of forces don't converge, isn't it?

Oh, no. They do converge!
 
  • #372
It is a finite force to every ball. An infinite number of forces sums to a finite number for every ball.

Those numbers go over every finite limit, but all are finite themselves.
 
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  • #373
Tomaz Kristan said:
Oh, no. They do converge!
You just said they didn't!
 
  • #374
No. Listen carefully!

The sum of all forces acting to every ball is finite. An infinite number of forces adds up to a finite number.

For every ball.

Do you agree?
 
  • #375
Tomaz Kristan said:
The sum of all forces acting to every ball is finite. An infinite number of forces adds up to a finite number.

For every ball.

Do you agree?

No. An infinite number of forces adds up to infinity. Of course in the limit as the number of forces grows without bound, then sum may be convergent. But we're not talking about such a case. Our scenario here is more akin to Hilbert's hotel than Zeno's paradox.
 
  • #376
ObsessiveMathsFreak said:
An infinite number of forces adds up to infinity

Not at all.

The sum of all left forces to any ball, is less than twice as the force the of immediate left ball.

Holds for every ball.

Why? Be cause the mass of the immediate left ball is already half of all the others on the left side.

See this now?
 
  • #377
Tomaz Kristan said:
See this now?
No. The sum of an infinite number of numbers is infinite. Seriously. Look it up.
 
  • #378
ObsessiveMathsFreak said:
The sum of an infinite number of numbers is infinite.

1/2+1/4+1/8+...+... = 1.

Don't you know that?
 
  • #379
Tomaz Kristan said:
1/2+1/4+1/8+...+... = 1.

Don't you know that?

\lim_{n \to \infty} \sum_{k=1}^{n} \frac{1}{2^k} = 1

But you're not talking about a limiting case. You're talking about infinite terms directly.
 
  • #380
No. Not at all. The sum of forces to EACH ball is a finite one.
 
  • #381
Tomaz Kristan said:
No. Not at all. The sum of forces to EACH ball is a finite one.
Only in the limiting case. You said you weren't talking about the limiting case. You're talking directly about the case with an infinite amount of particles. The direct sum of an infinite number of numbers is not defined.
 
  • #382
What's your point OMF?

Do we have a finite force to every ball or we haven't?
 
  • #383
Tomaz Kristan said:
What's your point OMF?

Do we have a finite force to every ball or we haven't?

No there are an infinite amount of particles.
 
  • #384
ObsessiveMathsFreak said:
No there are an infinite amount of particles.

So what? You may divide ANY Euclidean body into an infinite number of smaller bodies.

Is it always a problem, or isn't?
 
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  • #385
Tomaz Kristan said:
So what? You may divide ANY Euclidean body into an infinite number of smaller bodies.

Is it always a problem, or isn't?
Not really. You can take the limit of subdivision as the divisions become infinite. From this, one can validly speak of densities, etc per unit volume or whatever.

But to say that anybody consists of an infinite number of smaller bodies is wrong. If that were the case, then the smallest amount of matter would have infinite mass.
 
  • #386
ObsessiveMathsFreak said:
to say that anybody consists of an infinite number of smaller bodies is wrong.

In the real world yes, it's wrong. NOT so in the Newtonian abstract world.

ObsessiveMathsFreak said:
If that were the case, then the smallest amount of matter would have infinite mass.

This is nonsense.
 
  • #387
I think there is confusion between infinitesimal volumes and actual particles. Do you think they are the same thing, Tomaz Kristan?
 
  • #388
masudr said:
I think there is confusion between infinitesimal volumes and actual particles.

I don't do infinitesimal volumes. I do only finite volumes in my case. For each real epsilon, there are only a finite number of grater than epsilon volumes.

But each volume is a finite one, non the less.

For the half, quarter, 1/8 and so on parts of any Euclidean body (like cube) is the same. Only those do not grow by density. But they may be squeezed to my balls and do just that.

In the abstract Newtonian realm.
 
  • #389
:smile: Hi Tomaz / All,

Thanks to hear for the following, maybe it will end up the long debate.

Let's not get into deep mathematical discussions. The main debated problem here is the physical rule. By the way, no scientist until these days said that Newtonian mechanics are wrong in structure. But, they agreed upon it is non relativistic in addition of being classical.

The very first Rule is only for rigid bodies. What we are talking about here are bodies of non-constant vector lengths (displacement).

Besides, this is not the body of the Newtonian Mechanics. But,of course, it is important.

The rule says that a rigid body in the vacuum can be having an angular velocity (directional acceleration) around his centre of gravity, which may be moving with constant velocity. This can be easily deduced by the general laws of motions:
1. Forces' Resultant to applied to the centre of "mass" to get acceleration of the centre of "mass".
2. Torques' Resultant to be applied to the body to get angular acceleration.

Hint: This is applicable in the 3-D Motion (and also to Relativistic Mechanics after some simple modifications).

Amr Morsi.
 
  • #390
Happy New Years. The situation this year is just as it was last year: there is no paradox. the problem is not well set, as mathematicians say, and thus has no solution. Let's look again.

First, let's look at the limit of the system as N - > infinity, after the equations of motion have been solved. That is, first solve for the system's behavior with a finite number of masses. Then, of course, there will be no net force acting on the CM. So, the sequence of forces acting on the CM is 0,0,0,...0,...
Under most circumstances, this sequence converges to, guess what, 0.

Now, let's take the limit of N-> infinity, before solving the equations of motion. If I've read things correctly, the gravitational potential of the system at any point r, along the line, will be V = - G*SUM(over n){ (1/2)^n/|r - (1/10)^n|}.

It's a fair bet to claim that this series does not converge. So, we get a different answer, non-answer in fact, than above.

QED -- the problem has no solution. (Even if one is clever enough to note that the potential has poles for r= (1/10)^n, one can finesse such poles as is usually done in most 1/r potential problems. See most any text on Maxwell+Newton+Einstein, Jackson, for example.)

In this case, one road leads to Berkeley, another leads to Palo Alto, another leads to Cambridge,MA, and so on. "Can't get there from here."

No paradox, just a badly stated problem, one that makes no sense. Sorry 'bout that.
Regards,
Reilly Atkinson
 
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