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I came across a good website that explains how Newton derived his formula of gravitational force click here

Why is $\frac{k}{4\pi ^2}=\frac{c}{M}$ and $f=(kmM)/r^2$ not $(4\pi^2)/r^2\sqrt{MmcC}$? Which I tried myself and it seems that I can't see how they came to that conclusion.

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So, the first equation is wrong. It is not $\frac{k }{4 \pi r^2} = \frac{c}{M}$, but $$\frac{k }{4 \pi^2} = \frac{c}{m}.$$ This relation simply is a definition of k, nothing more. Being it a constant, you can call it as you wish.

Second, I don't know how you did the math, but the starting equation is $f =M 4 \pi^2 \frac{c}{r^2}$. Using the previous relation, $c = \frac{m k }{4 \pi^2}$, you find $$f = k \frac{mM}{r^2}.$$

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  • $\begingroup$ Yes defining $k$ does make sense in Mathematics. Well it is defined so that in end there be $mM$ included in gravitational force equation. Right? $\endgroup$
    – Not Tfue
    Sep 20, 2020 at 22:47
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    $\begingroup$ Yes, you're perfectly right!!! $\endgroup$
    – SoterX
    Sep 22, 2020 at 6:21

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