The Metacentric Diagram

Naval Architecture


When a ship is heeled through a small angle, the centre of buoyancy moves to a new position B1 and the buoyancy force acts through a point M on the centreline. M is known as the transverse metacentre.

For angles of up to about 10 degrees, M is a fixed point. The positions of B and M depend only upon the geometry of the ship and are fixed for the draught at which it is floating. The designer can provide information on B and M for each mean draught, assuming the waterline is parallel to the design waterline. Small departures from this state are unlikely to be significant. With such a plot, known as a metacentric diagram, a master can find M and B for the condition the ship is in.

metacentric diagram

We have seen how the centre of buoyancy, the centroid of the volume of the displaced water, can be found. BM is given by:

BM = I/V, where I is the transverse second moment, or inertia, of the waterplane and V is the volume of displacement.

metacentric diagram2

There is a corresponding longitudinal metacentre for small changes in angle about a transverse axis. It will be shown later that the two metacentres are critical to a study of a ship’s stability. The position of the longitudinal metacentre is defined by:

BHL = IL/V, where IL is the longitudinal inertia of the waterplane about a transverse axis through the centroid of the area of the waterplane.


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