You shoot an arrow into the air and keep your eye on it as it follows a parabolic flight path to the ground. You note that the arrow turns in flight so that it is always tangent to its flight path. What makes it do that?

When the arrow is hot in the air, the gravitational force acts on its center of gravity, and rotates it about the axis of rotation. The gravitational force in this case provides the centripetal force to the arrow which allows it to curve in a parabolic path.
It is important to note that action of gravitation force is such that the arrow always remains tangent to its flight path.