# Newton, gravitation and second law

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I bring to the forum the query made by a student at the school.

- Newtonian gravitation.
- Simple case of two equal spheres that gravitate to each other (equal masses).

Let's write the Newtonian equation of gravitational force.
$F = G \ \dfrac{m \ m}{r^2}$
Both masses are the same. Then there is the following.
$F = G \ \dfrac{m^2}{r^2} \ \ \ \ \ \ \ \ \ (1)$
We clear the mass in the equation of Newton's second law.
$m = \dfrac{F}{a}$
We raise both members to the square.
$m^2 = \dfrac{F^2}{a^2} \ \ \ \ \ \ \ \ \ (2)$
In (1) we replace $m^2$ as indicated by (2).
$F = G \ \dfrac{\dfrac{F^2}{a^2}}{r^2}$
We simplify and cleared F.
$F = \dfrac{a^2 \ r^2}{G} \ \ \ \ \ \ \ \ \ (3)$

Equation (3) reports that the force is directly proportional to the square of the distance. This is not what Newton's gravitational equation informs.

The gravitational equation and the second law, applied to the same system, result in something that we have not seen in school.

How should we understand equation (3)?

Edited by quiet

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You should understand it by realizing that the force is proportional to the square of the distance only if acceleration is being held constant, which is generally an unnatural assumption.

Mass being held constant is a somewhat natural idea; acceleration being held constant is not.

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12 minutes ago, uncool said:

You should understand ...

I guess the explanation would be more obvious if you could include some math. Thanks in advance.

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You brought simplified version of Newtonian equations. The more appropriate is the one which takes r as input parameter, to create gradient of forces/accelerations etc.

$F(r)=\frac{G m_1 m_2}{r^2}$

$a(r)=\frac{G m}{r^2}$

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30 minutes ago, quiet said:

I guess the explanation would be more obvious if you could include some math. Thanks in advance.

...the math is what you provided.

You asked how we should understand the equation. What math are you asking for?

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53 minutes ago, quiet said:

(3) reports that the force is directly proportional to the square of the distance. This is not what Newton's gravitational equation informs.

Newton’s gravitational law does not include the variable a, which depends on r. So you can’t make this comparison.

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59 minutes ago, uncool said:

You should understand it by ...

21 minutes ago, Sensei said:

You brought simplified version ...

20 minutes ago, uncool said:

...the math is what you provided...

19 minutes ago, swansont said:

Newton’s gravitational law does not ...

All the answers help, because in the classroom they stimulate reflection. Thanks for answering.

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ok-- I see it

Edited by OldChemE

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There seems to be an error in the derivation of (3). You go from a2/r2 to a2r2 (sorry for lack of superscript- im on my phone!)

But uncool’s answer is spot on.

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1 hour ago, Strange said:
1 hour ago, Strange said:

There seems to be an error in the derivation of (3).

Hello strange. In what kind of error are you thinking?

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@strange

Hello strange. In what kind of error are you thinking?

---------

I am thinking more in a methodological abuse than a mathematical error.

Is there some reason for suppose that the same physical property is responsible of the inertia and the gravitation?

If there is no reason, and if we anyway suppose yes, we are commiting a serious abuse. See. If you, instead $m$ use another symbol in the equation of gravitatory force, without suppossing the new symbol is the same that the mass, neither supposing that have the same physical meaning, then, in the gravitatory equation you can't apply the second Newton's law.

Something that requires to take much care in physics is to avoid confusions between different properties that have the same dimentions.

Example. The action and the angular momentum are dimentionally identical. Can someone admit that these two terms have the same physical nature and the same physical meaning?

Like I see the problem, inertia and gravitation are not related with the same cause. For this reason, I guess that there is an abuse in to state that both, inertia and gravitation, are referred to the same property.

I know that, in case we can not stablish the Einstein's equivalence principle, we will have serious trobles with General Relativity.

Today, we prefer GR over Newton's physics. Anyway, the inertial mass is a property that rules moving objects, and the property with same dimentions used in gravitational formula, is someting capable to rule the force between objects in mutual relative rest.

Edited by quiet

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6 hours ago, Strange said:

There seems to be an error in the derivation of (3)

Is the formatting a little off? Strange seems right but then next line (3) seems ok above.

Should we read it like this (ugly parenthesis added by me)? Line between a2 and r2 should be longer?

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47 minutes ago, Ghideon said:

There is no mathematical error in this equation. If you smplify respect to F remains F at the first power. Then you clear F and obtain the equation (3).

Still doubts? Make a dimensional analisys and verfy that units are coherent.

Edited by quiet

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1 hour ago, quiet said:

There is no mathematical error in this equation.

I did not say there is an error. I said the formatting may look ambigous. I tried to show that by adding paranthesis without changing the mathematical meaning.

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1 hour ago, Ghideon said:

I did not say there is an error. I said the formatting may look ambigous. I tried to show that by adding paranthesis without changing the mathematical meaning.

Oh, sorry. I had a confusion.

The format is fine. Both lines of division are well located and well sized.

---------

• There is something important that deserves to be elucidated.

When the principle of equivalence of Einstein is introduced in the physics of Newton, the result is absurd and self-contradictory. I do not see how to avoid that problem.

Are Newton's physics and the principle of equivalence mutually incompatible? This question needs a well-founded answer.

In case of being mutually incompatible, another question appears. Does Special Relativity remain valid if we reject Newton's physics?

If it does not remain valid, another question appears. Can GR be valid if SR is discarded?

Edited by quiet

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2 hours ago, quiet said:

The format is fine. Both lines of division are well located and well sized.

As Ghideon has pointed out, your formatting was completely ambiguous. I assumed you meant $\frac{F^2}{(\frac{a^2}{r^2})} = F^2 \frac{r^2}{a^2}$ whereas it seems you meant $\frac{(\frac{F^2}{a^2})}{r^2} = \frac{F^2}{a^2} \frac{1}{r^2} = \frac{F^2}{a^2 r^2}$.

8 hours ago, quiet said:

Is there some reason for suppose that the same physical property is responsible of the inertia and the gravitation?

The equivalence principle says that acceleration is indistinguishable from a gravity. So inertia is indistinguishable from gravitational mass.

Of course, that is just a statement of what we observe, not the reason. But I think this is one of those cases where the "why" question falls outside science.

2 hours ago, quiet said:

When the principle of equivalence of Einstein is introduced in the physics of Newton, the result is absurd and self-contradictory. I do not see how to avoid that problem.

I'm not sure what problem you are referring to. There is no contradiction in anything you have said.

2 hours ago, quiet said:

Are Newton's physics and the principle of equivalence mutually incompatible?

Why would they be?

2 hours ago, quiet said:

Can GR be valid if SR is discarded?

Of course not. SR is just a special case of GR (the clue is in the name).

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2 hours ago, Strange said:

As Ghideon has pointed out, your formatting was completely ambiguous...

Hi, strange.

The equation written with LaTex shows the main line of division at the height of the sign equal and slightly longer than the other line, belonging to the numerator. Keeping this in mind, the notation looks unambiguous.
---------
The principle of equivalence assumes that physical property itself intervenes in inertia and gravitation. The physics of Newton does not allow to suppose that, because in case of supposing it you arrive at the equation (3), obtained in the initial post of this thread. In Newtonian physics you can not put the same m in the gravitational equation and in the formula of the second law, because by doing that you get to (3) irremediably.

I repeat what was expressed in a previous post. Two magnitudes that have very different physical natures can have the same dimensions. You can measure in Kg both physical properties, one that intervenes in inertia and another that intervenes in gravitation. That does not mean that the same physical property intervenes in both phenomena. Newton's physics requires accepting that they are different properties, because when you assume that it is the same property you inevitably arrive at (3).

A theory that demands to distinguish sharply between the property that intervenes in gravitation and the property that intervenes in the inertia, does not allow to accept the principle of equivalence. You must choose between Newton and Einstein regarding inertia and gravitation. You can not have both of them in the same kind of physics.

To the same type of physics belong Newton's theory, the Lagrangian and Hamiltonian theories, thermodynamics, electrodynamics, special relativity and quantum theory, because none of these theories postulates the principle of equivalence. General relativity belongs to another type of physics, whose principle of equivalence is incompatible with the kind of physics mentioned previously.

The mathematics of the initial post of the thread is simple and brief, but it brings out an essential detail. If you say that General Relativity is a physical theory, then Newton's theory, Lagrangian and Hamiltonian theories, thermodynamics, electrodynamics, special relativity and quantum theory are not physical theories. And if you say that the latter are physical theories, then general relativity is not a physical theory. Like it or not, that's inevitably so. For that reason the hope of linking general relativity with quantum theory is futile. Union attempts suffer from the same type of incompatibility that we see in equation (3).

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33 minutes ago, quiet said:

A theory that demands to distinguish sharply between the property that intervenes in gravitation and the property that intervenes in the inertia, does not allow to accept the principle of equivalence. You must choose between Newton and Einstein regarding inertia and gravitation. You can not have both of them in the same kind of physics.

If you take GR in the weak gravity limit, you end up with Newton. How can that be, if they are in conflict?

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22 minutes ago, swansont said:

If you take GR in the weak gravity limit, you end up with Newton. How can that be, if they are in conflict?

In the limit you get Newton's gravitational formula. And that formula is incompatible with the principle of equivalence. Then GR is completely self-contradictory, since it contains a limit incompatible with its basic hypothesis.

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6 hours ago, quiet said:

The equation written with LaTex shows the main line of division at the height of the sign equal and slightly longer than the other line, belonging to the numerator. Keeping this in mind, the notation looks unambiguous.

So you are telling me that I couldn’t have misunderstood it and so I never said it appeared to be wrong.

6 hours ago, quiet said:

The principle of equivalence assumes that physical property itself intervenes in inertia and gravitation. The physics of Newton does not allow to suppose that, because in case of supposing it you arrive at the equation (3), obtained in the initial post of this thread. In Newtonian physics you can not put the same m in the gravitational equation and in the formula of the second law, because by doing that you get to (3) irremediably.

Good grief. Of course you can use the same mass in both. For example, you can use this to show why two different masses fall at the same rate.

You have derived a fairly meaningless equation (uncool explained how it can be interpreted physically). It does not show any contradiction. It can’t.

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On 12/18/2018 at 7:40 PM, quiet said:

Equation (3) reports that the force is directly proportional to the square of the distance.

Since acceleration in the formula is dependant on distance...it is simply incorrect to conclude that

On 12/18/2018 at 7:40 PM, quiet said:

How should we understand equation (3)?

With regard to force and distance?

Force is inversely proportional to distance squared, same as it was in equation (1), but now less obvious.

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A few years back we had another member that tried to follow the algebra route the quiet offered.

(remember he did not say it was correct physics, just that the algebra as presented followed the algebraic rules which I agree with)

The trouble is that the F and the a are not the same physically in all the equations.

This is made worse by the initial diagram which uses x for distance, but quiet's formula jumps to r.

In that older thread I went through several pages of step by step calculations to explain the proper route, but it was a couple of years ago adn I haven't yet found it.
If anyone can remember the thread I would be grateful because it is all explained in there.

Edited by studiot

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2 hours ago, J.C.MacSwell said:

...it is simply incorrect

Agree. And more, I have pointed out, in the first post of the thread, that is incorrect. The main question is where come from the incorrection? Have I made a wrong use of algebra? Have I altered Newton's equations? Have I commited an abuse?

Respect to an abuse, there is one. Wich? To  suppose that the same physical property is responsible for both, inertia and gravitation. This is simply incorrect. And this is the cause for what the same term $m$ appears in gravity formula and in the second law formula. This double appearing cause the absurde. And this double appearing come from the equivalence principle.

Edited by quiet

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On 12/19/2018 at 12:40 AM, quiet said:

Equation (3) reports that the force is directly proportional to the square of the distance. This is not what Newton's gravitational equation informs.

I am really wondering how you can conclude this. You use Newton's equations, do some legal mathematical operations with them, and then come to something that is not consistent with the equations you started with???  Wouldn't that make you suspicious of how to interpret the equations you derived?

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39 minutes ago, quiet said:

Agree. And more, I have pointed out, in the first post of the thread, that is incorrect. The main question is where come from the incorrection? Have I made a wrong use of algebra? Have I altered Newton's equations? Have I commited an abuse?

Respect to an abuse, there is one. Wich? To  suppose that the same physical property is responsible for both, inertia and gravitation. This is simply incorrect. And this is the cause for what the same term m appears in gravity formula and in the second law formula. This double appearing cause the absurde. And this double appearing come from the equivalence principle.

No the  principle of equivalence has nothing to do with this and introducing GR or even SR is surely off topic?

You are simply misinterpreting the Physics, not the algebra.

But since no one seems interested in my opinion, I will go back to my Christmas.

23 minutes ago, Eise said:

I am really wondering how you can conclude this. You use Newton's equations, do some legal mathematical operations with them, and then come to something that is not consistent with the equations you started with???  Wouldn't that make you suspicious of how to interpret the equations you derived?

Yes that a one (good) way to put it.

Edited by studiot