Final answer:
As the axle rotates faster, the ball's tangential velocity and centripetal force increase, reducing the angle x between the rope and the vertical due to increased tension and conservation of angular momentum.
Step-by-step explanation:
When the axle on which the rope with ball at one end is attached rotates faster, the tangential velocity of the ball increases due to increased angular velocity. According to the principles of circular motion, the centripetal force which acts towards the center also increases with tangential velocity. As a result, the tension in the rope increases, which pulls the ball closer to the vertical axle, thus reducing the angle x between the rope and the vertical.
In another context provided by the conservation of angular momentum, if the angular velocity increases, the moment of inertia decreases because the ball moves closer to the axle. This reduction in moment of inertia is necessary to conserve angular momentum if no external torque is applied. Therefore, the angle x reduces as the angular velocity increases.