A ball of a certain mass is attached to a string of certain length and the other end of the string is attached to a swivel at the top of a pole. (string does NOT wrap around pole).

If the ball is wacked horizontally, and spins around the pole at a constant velocity, the angle made by the pole and the string will increase as the velocity increases. At rest, 0 deg. Angle. At theoretical max. velocity, it appears to aproach 90 deg.'s

What is the formula to figure out the angle at a given velocity ?
Or vice versa?

Please respond if you know the answer.

I am going to restate your question giving the symbols I will use in solving this problem before continuing with the answer. The ball of weight mg, where m is mass and g is the acceleration due to gravity, is on a string of length l that is swung around the pole at a constant velocity v sweeping out a circle of radius R about the pole. The string makes an angle φ with the pole. Tension, F, is acting in the direction of the string and can be broken down into its components Fx and Fy. There is a centripetal force keeping the ball going around in a circle which is equal to mv^2/R. This centripetal force is provided by the horizontal component of the tension, F. In other words, Fx = mv^2/R. We also know that the horizontal component of tension is Fx = F sin φ since sin φ = Fx/F. Now according to Newton’s Second Law, ∑F = ma. For the vertical direction, since it is not moving vertically, this equation becomes ∑Fy = F cos φ + (-mg) = 0 (F cos φ is the vertical component of tension which exactly balances the weight, mg, of the ball). From this equation we can calculate the force of tension to be F = mg/ cos φ. Now that we know what the tension is, we can use it in our summation for the forces in the horizontal direction which can be stated as follows:
∑Fx = F sin φ = mv^2/R
Now plugging in the formula for tension we calculated,
(mg/ cos φ )*sin φ = mv^2/R
Remembering that tan φ = sin φ/ cos φ, and canceling out the mass we get,
g tan φ = v^2/R
This can then be solved for velocity in terms of the angle or for the angle in terms of velocity. Both are given below.
φ = tan-1(v^2/(gR))
v = sqrt.(gR tan φ )
So there is the formula to figure out the angle at a given velocity. Let me know if this answer is what you were looking for or if you were looking for something different.

The Apprentice