Okay I hope I can write this so it makes sense what I am thinking. For a pendulum you have: Fres = -mgsinθ And Fres points along the circumference: So: mRθ'' = -mgsinθ I wonna discuss this property that you can just express the acceleration as Rθ''. In a cartesian coordinate frame your acceleration would depend on the characteristic length scale of x. How do you know that the characteristic length scale of Rθ is the same as x? I know it sounds weird, but I hope you understand what Im going at.