Question

How much work does an agent do that applies centripetal force to an object to obligate it turning into a circle whose ratio measures 1 m, after a full spin? Consider that keeps uniform movement and explain yourself

Answer 1

As far as I know, the agent doesn't do any work whilst maintaining circular motion. Work is force times distance moved in the direction of the force. In circular motion, the object is always moving at right angles - tangential speed - relative to the radius - and thus the direction of the centripetal force - and so has no work done on it.
This is a hard question, I think. One way of demonstrating it would be to find a circular motion system that could show that the agent is doing this work. That seems like a difficult challenge. The most popular example of circular motion may be gravitation. So, if the gravitational field/agent were doing work, then energy would be being dissipated and the "battery would eventually go flat". Well, the earth is pretty massive, so presumably it would take some very long time for it to "lose it's gravitational pull" - ie go flat.
You may gather from the above that I don't know the answer, but I think I can see the problem ...

Answer 2

Work is done when the force has a non zero component in the direction of motion of the point of application (in vector language the work is the dot product of the force and displacement vectors). When a body moves in a curved path the centripetal force is toward the center of curvature of the path which, at any instant, is perpendicular to the direction of motion along the path. Therefore the instantaneous rate of work is zero.