Applying newton's second law on circular motion In school, when we are studying circular motion, we analyse the forces acting on the body into two axis, the tangential one to the speed of the body and the perpendicular one and then we apply newton's second law separately on them. What I don't really get is why does the law still hold as it seems to me that we create a coordinate system on the body and then proceed to move it and rotate it with it. doesn't the coordinate system have to be still? 
 A: You are free to use a rotating coordinate system if you want to, as long as you pay the price. The price comes in the form of a force that is “built in” to the system. While in ordinary “still” coordinates a stationary body remains stationary and feels no force, in the rotating coordinate system a stationary body feels a built-in force pulling it away from a specific point which we call the centre of rotation. That force is called the centrifugal force. 
So you have a choice:


*

*“Still” coordinates: the body would go off in a straight line but is pulled into moving in a circle by the centripetal force (the tension in a string, perhaps).

*Rotating coordinates: the built-in centrifugal force is exactly balanced by the centripetal force, so the body remains stationary. 
For a bonus:


*If the body is not rotating, in “still” coordinates, but the entire universe, every star and every galaxy, is rotating round it, does the body feel a centrifugal force?


This takes you to Mach’s Principle and one of the philosophical inspirations for General Relativity. 
A: In that analysis that you did in school, the frame of reference was fixed for a moment.   The equations are applied to that moment.  The frame is fixed, at rest, and the object will move to a new position in that frame in the next instant.  You could continue the analysis in that frame, but the math would be difficult.   In that next instant, the tangential and radial force velocity, and acceleration would no longer be directed along coordinate axes as they were in the first moment.  The math will give the right answer, but it will be a mess.
To simplify things, we move to a new fixed coordinate system, one that at the next instant, the force, velocity, and acceleration are aligned with the axes of this new system.  We find that the mathematical description of the force is the same as it was in the first moment, and the mathematical description of the velocity is the same as it was in the first frame.  But, of course, the actual force, velocity, and acceleration are different.  But since the math description is the same, we see that the math will not change.  In fact, we've learned all there is to know in the first moment.  There's no need to take it any further.
Note that the two coordinate systems that I described are both fixed.  These are not coordinate systems that rotate with the object. It is possible to do the analysis in a moving coordinate system.  That's a different analysis altogether, and the interpretation of the results in that frame takes some scrutiny.
