Satellite revolving problem gives two different answer Assume there's a satellite revolving about the Earth.  If I would like to decrease its radius, should I increase or decrease its velocity?
I know the answer apparently should be decreasing its speed, but the following two formulas give different answers. Can someone explain why two formulas give two different answers?
$r = mv² / F$, where r and v are directly proportional
$v = √(GM / r)$, where v and r are inversely proportional
 A: You cannot simply pluck equations out of your textbook and apply them to any situation.  Just because they both contain radius $r$ and speed $v$ does not mean that they are necessarily valid for your situation.  You have to think about what you are doing.
The 1st equation applies for circular motion. It tells you that $r$ is proportional to $v^2$ providing that centripetal force $F$ and mass $m$ are held constant.  If you change orbital radius, ie the distance between the satellite and the Earth, the gravitational force $F=GMm/r^2$ (which provides the centripetal force) also changes. So you cannot apply this equation.
However, substituting $F=GMm/r^2$ into the 1st equation gives you $v^2=GM/r$, which is the same as your 2nd equation.  This is the equation you need to use. It tells you how the speed and radius of a satellite or planet are related for a circular orbit.
A: Your question is puzzling but I think i have understood the answer. 
When the satellite undergoes slow down (due to friction or something) it comes closer to the earth and gain additional velocity. This velocity is same as in your equation and as in @sammy gerbil 's answer.
$v^2=GM/r$
Friction reduces the energy of the satellite and the energy of the satellite is 
E=KE+PE or
$E=\frac{1}{2}mv^2-\frac{GM}{r}= -\frac{GM}{2r}$
Hence after slow down the satellite comes closer to the earth and gain velocity, although the final equilibrium velocity (after the slow down) is higher than the initial velocity (before the slow down) the final energy is lower than the initial energy. I think this will solve the contradiction.
