In the case that you describe, an individual swinging a mass horizontally on the end of a string, the string does not run directly to the centre of rotation. Instead, it runs to your hand, which in turn is moving in a circle about its centre of rotation . Sometimes the arm is involved, sometimes only a rotation at the wrist. ( Mime winding up a sling to throwing speed to see what I mean)
If everything is constant (and there's no drag on the mass), the line from mass to hand to centre of rotation is straight; the string tension exerts only the centripetal force needed to maintain the circle, as well as an upward component to keep the mass from dropping downward.
If you then speed up the circular motion of your hand to a new constant angular velocity, your hand's angular motion gets ahead of the mass's angular motion, and gets continuously farther ahead. So now the tension in the string is not in the line from the mass to the centre of rotation. There is a tangential component to the tension, constantly increasing, which serves to speed up the rotation of the mass.
This tangential force speeds up the mass's rotation, until the mass is rotating faster than your hand. Then the mass gets ahead, and your hand slows it down, and so on.
Picture a pendulum clock in a wheel-style space station, with the pendulum set to swing in the plane of the wheel, but currently at rest. Then speed up the space station slightly...
Best Answer
The object will move in a curved path whose center is not where I am pulling from. This center, my hand and the mass form a triangle whose lead angle might be positive or negative depending if the speed of the mass is increasing or decreasing.
Consider the body above at B moving along the indicated curved path (like a closing spiral). While pulling from A with a force $F$, some of the force goes into rotating the mass about C (the $m v^2/r$ part) and some into accelerating the mass (the $m \dot{v}$) part.