[Coda]

Pulling the rope perpendicular to the ball's motion will just make it fall in towards the center. Pulling the ball up in the first place is also not putting any force in the direction of the orbit.

I'll give you a hint: It's a little bit of a trick question. It's a question that does indeed have a real answer, but the process of trying to find it will reveal the shortcomings in your model.

The angle symbol notation is used by some graphing calculators. I've not ever seen it used in a textbook. The parametric notation is mostly used when it's not obvious what coordinate system is in use or when you're defining functions in the coordinate space instead of individual points. (In the latter case, you'll often see "r(t) = ..., θ(t) = ..." to specify the coordinate in terms of a parameter.)