[generuso]

It is a common and understandable error to assume that the linear velocity should remain constant.

If the object goes in a circle, the radial force always acts perpendicular to the direction of motion, and cannot change the linear velocity of the object. The centripetal force does not do any work, the energy of the rotating body remains fixed.

But there is a subtle difference when one does pull the object closer to the axis. First, it is easy to notice that one does expend work. This energy must go somewhere, and there is nowhere else for it to go except into the kinetic energy of the moving object.

But how exactly is this energy transferred? If you think about it, when the object is pulled in, it no longer goes in a circle, but follows a spiral. Its velocity is no longer strictly perpendicular to the direction of the force. If you integrate this seemingly small effect, this is precisely what makes pulling on the sting to increase the velocity of the object.

Ignoring the mechanism, a formal calculation, from conservation of angular momentum or conservation of energy would immediately tell how much the linear velocity of the object would increase when it is pulled closer to the axis.