Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > sci.physics.relativity > #372806

Re: The Inverse Square Law explained

Show all headers | View raw

Odd Bodkin wrote:

> On 12/22/2015 10:00 AM, Alan Folmsbee wrote:
>> Inverse Square Law Revealed
>>
>> "Where does the 1/R^2 come from for the inverse square laws of force and
>> acceleration?"
>>
>> The answer...
> 
> Doesn't this come directly from Gauss' law?

It does not, indeed ;-)

<https://en.wikipedia.org/wiki/Gauss%27s_law>

Gauss’s law *for gravity* can be derived from Newton’s law instead.

However, it can be shown that Newton’s law follows from Gauss’s *theorem*, 
also known as [1] <https://en.wikipedia.org/wiki/Divergence_theorem> when 
*also* applying Gauss’s law to gravity.

Conversely, Newton’s law can be derived, *indirectly*, from Gauss’s law 
*for gravity* *with additional assumptions*, using Gauss’s/the divergence 
*theorem*:

<https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity#Deriving_Newton.27s_law_from_Gauss.27s_law_and_irrotationality>

But, of course, “[Gauss’s/the divergence] theorem was first discovered by 
Lagrange in 1762,[9] then later independently rediscovered by Gauss in 1813,
[…]” [↑1].  Newton posited his law of universal gravitation *about 100 years
earlier*, in Book 1, «De motu corporum» (“On the motions of bodies”), of 
the «Philosophiæ Naturalis Principia Mathematica» («Principia») of 1678:

<https://en.wikipedia.org/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica>

So neither Gauss’s theorem nor law could have been Newton’s foundation.
It is not known how Newton got the idea of the inverse square law, but the 
arguments in the «Principia» (Book Ⅰ, Section Ⅱ), suggest that it were 
Kepler’s laws of planetary motion (in «Epitome astronomiae Copernicanae» 
„Epitome of Copernican Astronomy”, 1617–1621) – this would not be too 
surprising given Newton’s interest in astronomical matters, and that it was 
Edmond Halley, later Astronomer Royal, whom he discussed his work with, and 
who funded the publication of the «Principia».

Susskind (see below) argues:

“The acceleration of circular motion is described by ω² R, where ω is the 
angular period of going round in an orbit of radius R.  Suppose Newton set 
that equal to some unknown force law [that depends on the radius, F(R)],

  ω² R = F(R),

and divided by R:

    ω² = F(R)∕R.

For the real case, inverse square law,

    ω² = F(R)∕R = (X∕R²)∕R = X∕R³

and in that form it is Kepler’s third law: ‘The square of the orbital 
period of a planet is directly proportional to the cube of the semi-major 
axis of its orbit.’”

> That's covered in calculus-based intro physics books for scientists and
> engineers, if I recall correctly.

It is also covered in physics lectures, such as lecture 1 of the series 
“Einstein’s General Theory of Relativity” as taught by Leonard Susskind, 
Ph.D., at Stanford University in 2008, from where I took most of the 
equations and the argument above.  Highly recommended:

<http://www.youtube.com/watch?v=hbmf0bB38h0>


PointedEars
-- 
Q: What did the female magnet say to the male magnet?  
A: From the back, I found you repulsive, but from the front
   I find myself very attracted to you.
(from: WolframAlpha)

Back to sci.physics.relativity | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread

Thread

The Inverse Square Law explained Alan Folmsbee <omnilobe@gmail.com> - 2015-12-22 08:00 -0800
  Re: The Inverse Square Law explained David Fuller <fuller.david@hotmail.com> - 2015-12-22 10:13 -0800
    Re: The Inverse Square Law explained David Fuller <fuller.david@hotmail.com> - 2015-12-22 10:17 -0800
    Re: The Inverse Square Law explained David Fuller <fuller.david@hotmail.com> - 2015-12-22 10:19 -0800
  Re: The Inverse Square Law explained Tom Roberts <tjroberts137@sbcglobal.net> - 2015-12-22 12:50 -0600
    Re: The Inverse Square Law explained alsor@interia.pl - 2015-12-22 12:14 -0800
      Re: The Inverse Square Law explained JanPB <filmart@gmail.com> - 2015-12-28 21:11 -0800
        Re: The Inverse Square Law explained alsor@interia.pl - 2015-12-29 09:18 -0800
  Re: The Inverse Square Law explained Odd Bodkin <bodkinodd@gmail.com> - 2015-12-22 16:06 -0600
    Re: The Inverse Square Law explained Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-12-27 00:42 +0100
      Re: The Inverse Square Law explained alsor@interia.pl - 2015-12-27 07:50 -0800
        Re: The Inverse Square Law explained Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-12-27 20:12 +0100
          Re: The Inverse Square Law explained alsor@interia.pl - 2015-12-27 14:30 -0800
            Re: The Inverse Square Law explained Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-12-28 00:23 +0100
              Re: The Inverse Square Law explained alsor@interia.pl - 2015-12-27 15:58 -0800
                Re: The Inverse Square Law explained Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2015-12-28 11:59 +0100
                Re: The Inverse Square Law explained Waye Kitchens <wscom@complang.net> - 2015-12-28 11:18 +0000
                Re: The Inverse Square Law explained The Starmaker <starmaker@ix.netcom.com> - 2015-12-29 10:11 -0800
                Re: The Inverse Square Law explained Gary Harnagel <hitlong@yahoo.com> - 2015-12-29 13:08 -0800
                Re: The Inverse Square Law explained alsor@interia.pl - 2015-12-29 15:19 -0800
                Re: The Inverse Square Law explained David Fuller <fuller.david@hotmail.com> - 2015-12-29 17:13 -0800
                Re: The Inverse Square Law explained David Fuller <fuller.david@hotmail.com> - 2015-12-29 17:16 -0800
                Re: The Inverse Square Law explained alsor@interia.pl - 2015-12-31 05:31 -0800
      Re: The Inverse Square Law explained John Gogo <jfgogo22@yahoo.com> - 2016-01-01 18:28 -0800
        Re: The Inverse Square Law explained Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-01-02 23:48 +0100

csiph-web