Stopped
In this situation no forces are involved and the ship has its centre of gravity approximately amidships when the ship is stopped.
Making Headway
Two forces now come into play, firstly the forward momentum of the ship and secondly, longitudinal resistance to forward momentum, created by the water ahead of the ship. These two forces must ultimately strike a balance and the pivot point moves forward. As a rough guide it can be assumed that 25% of the ships forward momentum, at constant speed, is spent in overcoming longitudinal resistance and the pivot point will be approximately 1/4L from forward.
Making Sternway
The situation is now totally reversed, the momentum of sternway must balance longitudinal resistance, this time created by the water astern of the ship. The pivot point moves aft and establishes itself approximately 1/4L from the stern.