Are we looking at an impulsive force in pulling the string from the center?
Then neither momentum or energy is conserved in the process for the system being the particle in rotation.
We would look at the velocity vector for the particle moving in the first circle and the new velocity vector for the particle moving in the inner circle as a function of the pull.
The motion of the object would be that of a spiral arc from the original orbit to the new orbit.
Imagine two concentric circles and a spiral arc that is tangent to both circles as we take snap shots in time.
What do we see, if we do this experiment…the object takes on a spiral trajectory where both the tangential and perpendicular velocity are changing.
We see this example for an electron moving through a constant magnetic field where the Electric charges on a plate capacitor are varied.
These spirals have been studied as ideals: