[Coda]

Regarding the orange force line: The tangential force line is 100% real -- it has a (net) magnitude of zero if it's not accelerating, but it's real. If you try to swing the object faster, then the magnitude will be nonzero until it reaches the new speed. That said, you and Espy are both right in that the force isn't necessary to be there for it to keep spinning, only for it to start spinning.

An interesting exercise is to think about how that force GETS there. When you're spinning the object around on the rope, you're not pushing it from the side in order to get it moving (unless you're playing tetherball) so how is it that you can accelerate it along that axis?

I personally prefer to notate polar coordinates either parametrically -- that is, r=1 θ=π/2 -- or using an angle symbol like 1∠π/2. That said, the standard notation is indeed (r, θ) in 2D and (r, θ, ϕ) in 3D.