If pulling the string down, and shortening the radius to r/2, increases the tangential velocity of the object, I suppose releasing the string, so the radius increases to 2r, will diminish its tangential velocity of the object. Is this correct? How does the velocity diminish? Thank you.
Make a drawing as the one I gave. Release the string. The object continues in a tangential straight trajectory. Wait until its distance to the center is 2R. Look at the speed: decompose it in a component parallel to the circle of radius 2R and a radial component. Hold again the string. The radial component is annulled and only rests the tangential component which is smaller than the original one.
Is it clear?