a spring with a spring constant k= 2 n/m is used to project a mass around a loop of radius R= 2m. To what minimum distance Xmin must the spring be compressed for the object to go around the loop without falling off ?

Hint: you need to find the minimum speed required for the ball at the top of the loop such that it stays on the track and doesn't fall inward. Then it's a matter of using energy principles to find the speed it must have when it first starts up the loop, and from that the energy that must be imparted from the spring. Try it, post your attempt, and we'll check it for you.