Two forces are acting on the ball when it is hanging. One is the weight of the ball equal to ‘mg’ where ‘m’ is the mass of the ball. This force is acting vertically downward. The other force is tension in the string which acts vertically upward. When the ball is just supported by the string, the downward acting weight is just equal to the upward tension in the string. We know that there is an elastic limit of the string and in the given problem; the string is quite at the verge of its elastic limit. If the force is further increased in any direction, the string will break. When the ball is set in swinging, the path of motion is a circle. (The length of the string is no doubt equal to the radius). Hence, a centripetal force will come to play in the circular motion of the ball-string system. Since the string cannot bear any further force, it will break.