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How did Newton derive his law of gravity?

  1. Sep 20, 2009 #1
    For some reason I have been extremely curious of how Newton derived his mathematical law for gravitation. You know

    [tex] F= \frac{GMm}{r^2}[/tex]

    So the first thing I did before i posted this was search Google and I foundhttp://www.relativitycalculator.com/Newton_Universal_Gravity_Law.shtml" [Broken]. It talks about a possible derivation of how Newton could have derived his law of gravity. It looks pretty good and I understand most of it BUT there is one part I don't understand. There is one part in the derivation that looks like this.

    [tex] \frac{C}{M}=\frac{c}{m}=\frac{k}{4\pi} [/tex]

    In this situation k is the gravitational constant and 4pi is.... 4pi. My question is WHERE did this k/(4pi) come from? Why set these two equations equal to k/(4pi)? Can someone tell me where this k/(4pi) came from??

    Thanks for your responce :D
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Sep 20, 2009 #2

    alxm

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    Science Advisor

    Well, get rid of it and see what happens!

    [tex]\frac{C}{M}=\frac{c}{m}=k[/tex]
    then
    [tex]C = kM, c=km[/tex]
    [tex]f^2 = f*f' = m\frac{4\pi^2C}{r^2}M\frac{4\pi^2c}{r^2}[/tex]
    [tex]= m\frac{4\pi^2kM}{r^2}M\frac{4\pi^2km}{r^2}[/tex]
    [tex]= 16\pi^4k^2\frac{M^2m^2}{r^4}[/tex]

    [tex]f = 4\pi^2k\frac{Mm}{r^2}[/tex]

    So you get the same result but with the unnecessary constant of [tex]4\pi^2[/tex] in front, so they defined k in a way that eliminates that constant..

    Of course, if you want to, you could also just say [tex]G = 4\pi^2k[/tex].
     
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