What would happen to an object if the centripetal force acting on it stops?

Picture a ball on a string being whirled about the head of an experimenter. If the string breaks, the centripetal force disappears. The ball leaves on a tangent path form its (previous) circular path. Yes, it's that simple. The string provided centripetal force, by virtue of its tensile strength, to the ball to keep that ball moving in a circle. When the string broke, there was no force left to accelerate the ball "in" and keep it moving in an arc.