Question

a ball at the end of a string is moving in a horizontal circle of radius R. IF the string is shortened until the ball moves in a new circle of radius R/2, by what factor does the angular velocity change. Net external torque is zero. answer is by 4So . . . I know that since the radius goes down the moment of inertia goes down and the angular vel goes up. But why by 4? How do I know which formula to use? for example v = (r) (w) so if the radius goes down by 1/2 the w (ang velocity) should go up by 2 -- not by 4. I guess you have to use centripetal acc = rw^2 but I don't understand why.