then i must agree it works by reaction but i think my concept still is valid
you shoot the ball you aply a tension minor than v*v*m/r untill it reaches the horizontal now when it goes to the vertical you make the ball go to the original radius sou you aply a tension bigger than v*v*m/r so the tension or force is bigger in the second quarter than in the first but the time for each quarter is the same since they are simetrical trajectories
i dont think spiral trajectories centripetal force have been studied well enough since ive asked about this in an advanced mechanics forum and they have no idea