Derive Newton's 2nd law from Newton's Law of Universal Gravitation?

Question

When Fg = G*m1*m2/r^2

How can i prove Fg = m*g by

deriving from upper equation?
Do i need to plug in mass and radius of earth?

Answer 1

Newton's Law of Universal Gravitation is derived from Newton's 2nd Law, not the other way around. You can arrive at values of g (which is a special designator for a) by dividing through by one of the m's:
F = ma = - GmM/r^2
a = F/m = - GM/r^2
g = a(R) = - GM/R^2