What Are the Limitations of the Law of Universal Gravitation?

[Wildcatfan]

Update: Thanks for the help! I'll work on it...

 

[Identity]

I'm not sure what your equation is, but this is how I would derive Newton's Law

T^2 = \frac{4\pi^2}{GM}R^3

In circular motion,

F = \frac{mv^2}{R}

The period in circular motion is

T = \frac{2\pi R}{v} \Rightarrow v = \frac{2\pi R}{T}

So, by substitution,

F = \frac{m\left(\frac{2\pi R}{T}\right)^2}{R}=\frac{4\pi^2 mR}{T^2}

\Rightarrow T^2 = \frac{4\pi^2 mR}{F}

Substituting into Kepler's 3rd,

\frac{4 \pi ^2 mR}{F} = \frac{4\pi^2}{GM}R^3

\therefore F = \frac{GMm}{R^2}