Integrating centrifugal force

1. The problem statement, all variables and given/known data



This is not homework. Suppose I am rotating a ball on a string (m=1, r = .2, v = 10 m/s)

2. Relevant equations



F_c = (m) v²/r = 500 N*m

If I reduce the length of the string to .1, v becomes 20, so

F_c = 4000 N*m



what is the work I have done, what kind of integration do I need,

does the energy spent depend on the speed/time I use to pull the string?



Thanks for your help