As the rope becomes shorter, the length of the rope is decreased and the centripetal force decreases according causing the ball to move in smaller circles.
The centripetal force is defined as the force that keeps an object moving in a circular path. Also, the centripetal force is equal to the tension in the rope and the tension in the rope depends on the length of the rope.
As the rope shortens, the tension in the rope decreases and the centripetal force on the ball decreases hence the ball moves in smaller and smaller circles, the ball goes around with a shorter and shorter period.
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