The Pivoting Point Formula


Posted by John Wilde Crosbie on March 05, 2002 at 17:14:09:

In Reply to: Re: ship handling (specifics on pivot pt) posted by Capt. Timothy

McGill on January 25, 2000, at 16:43:02:

: I am a St. Lawrence Seaway Pilot engaged in studies on the effects of the

position of pivot point on shiphandling. I have come across several

contradictory sources and I am searching for clarification. Any input



In reply to the above may I refer Captain Timothy McGill to the following formula.

When a disturbing force is applied to a ship at a distance y metres from the

centre of gravity, she will pivot about a point which is x metres in the far

side of the centre of gravity.

The formula is simple and elegant and may be written thus:

-x = I/yM

where x is the distance of the pivoting point from the centre of gravity;

where - is the minus sign indicating that x is on the far side of the centre of

gravity to y;

where = means equals;

where y is the distance of the disturbing force such as the rudder or a tug from

the centre of gravity;

where M is the mass or tonnage of the ship and cargo;

where / is the division or divide sign;

where I is the Moment of Inertia of the ship and cargo about her centre of


For a homogeneous box shaped vessel:

I = M(lsquared + bsquared)/12

where l is the length of the vessel; and,

where b is the breadth of the vessel.

where lsquared means l multiplied by itself; and,

where bsquared means b multiplied by itself.

where + means plus.

Thus for such a box shaped vessel the formula becomes:

-x = (lsquared + bsquared)/12y


I have had to derive the formula myself as I have been unable to find the formula in the works on dynamics. I think this may be because engineers and mathematicians always refer motion

to the centre of gravity whereas the the concept of the pivoting point is only

of special interest to the pilot of a craft as it is the point about which he

sees his craft turning in relation to other objects.

For a vessel under way it is the point about which she turns relative to her

initial straight line course. For such a vessel the pivoting point may be

defined as the point on the ships centre line which does not sway to port or

starboard when the turn is commenced but is the point which is continuous from the

straight line course to the turning circle. When the ship is moving at speed the

pivoting point may not have much relevance as it may not be too material whether

the ship is turning about the pivoting point or her centre of gravity. It would

be interesting to have pilots views on this. A further complication arises in the case of a ship under way because while she is turning her historical  momentum increases the lateral velocity thereby driving the pivoting point forward while at the same time the phenomenon known as bow lift creates a turning force forward of the centre of gravity which on its own would drive the pivoting point aft. Both of these factors mean that the position of the pivoting point is not constant though they do tend to cancel out each other. For this reason when the ship has headway the formula can only be relied upon to calculate the initial pivoting point. 


The formula, of course, is derived by reference to the centre of gravity using the standard equations of linear and angular motion. When a disturbing force is applied to a ship, say by the rudder or a tug, both lateral linear motion and angular motion will be imparted to the hull. The pivoting point, by definition, is the point on the centre line of the ship where this lateral linear velocity exactly cancels out this angular velocity. The resulting formula is both simple and elegant. It will be seen that the position of the pivoting point is independent of the magnitude of the disturbing force. It is purely a matter of geometry. It depends on the distribution of weight in the ship but is independent of the actual weight. 


John Wilde Crosbie,

Dublin, Ireland.