Ok so first of all I'm much more of a math guy than a physics guy there might be something wrong in here that I don't know because I don't know much about physics/
I tried to find how much the string wraps around the circle if it moves a certain angle. This next part is hard to put into words, but I drew a diagram and did some geometry and found that if the angle changes a certain amount, the string wraps around an arc that is that angle so the length of the string decreases by the radius multiplied by the difference in angle (I'm using radians because it makes this way smoother).
Next I had to convert the difference in angle to a difference in time. Assuming that the circle wrapping thing doesn't happen (and we can do this if we take a limit as delta t approaches 0 because calculus), the particle moves an arc with length delta t times v0. Divide this by the length of the string to get the change in angle. You can then plug this into r0 times delta theta, divide by delta t on both sides, and then you can get the change in length with respect to time. Solve the differential equation and get the length at a certain time. Then you just have to plug this into the classic mv^2/r for circular motion to get the tension on the string at any given time.
Sorry if this comes off as schizo rambling, I'm really really bad at putting math into words.
