In the earlier post on buoyancy and stability, methods for the determination of the metastatic height are discussed.  This is yet another method of doing this, taken (with a few modifications) from E.H. Lewitt, Hydraulics and Fluid Mechanics (Sir Isaac Pitman and Sons, 1923.)   This text also has a treatment on the “moment of inertia” method of computing metastatic height.  The U.S. Navy also uses this method and it’s described on pp. 2-1 and 2-2 in the document Naval Ships’ Technical Manual Chapter 096: Weights and Stability. Refer to the figure above to understand this narrative.

The metastatic height of a ship or pontoon may be found experimentally whilst the vessel is floating, if the position of the centre of gravity is known.

Let W be the weight of the ship, which is known, and let G be the centre of gravity. Let a known movable weight m be placed on one side of the ship.  A pendulum consisting of a weight suspended by a long cord is placed in the ship, and the position of the bob when at rest is marked. Let l be the length of the pendulum. The weight m is then moved across the deck through the distance x, the new position of m being denoted by m’. This will cause the ship to swing through a small angle \theta about its metacentre M. Then, as the pendulum inside the ship still remains vertical, the angle \theta may be measured by the apparent deflection of the pendulum.

Let the apparent horizontal displacement of the pendulum weight = y. Then,

\tan \theta = \frac{y}{l} (1)

Referring to Fig. 21, the moment caused by W about M equals the moment about M caused by moving m to m’.  Or,

W \times GM \tan \theta = mx (2)

from which

GM = \frac{mx}{W \tan\theta} (3)

and, as all the quantities on the right of this equation are known, the metacentric height can be calculated.

This experiment is often carried out on a ship in order to determine the exact position of G which is difficult to estimate from the distribution of the ship’s weight.

As an example, consider the case of a ship which displaces 3,000 tons of sea water. A load of 15 tons is shifted from the port (left) side 15′ to the starboard (right) side of the deck; the vessel changes its angle relative to the vertical from in a ratio of 1/30 (1 horizontal to 30 vertical.) Determine GM for this ship.

To use Equation (3), we equate W = 3000 tons and w = 15 tons. The distance x is the distance the weight is moved symmetrically to the centreline of the ship (see above,) in this case x = 15′. The tricky part is the angle; angle ratios such as this are common in real life, and the angle \theta = arctan(\frac{1}{30}) = 1.9^\circ . If we take the tangent of both sides of this expression, however, we realise that \tan\theta = \frac{1}{30} . Direct substitution of these quantities into Equation (3) yields GM = \frac{mx}{W \tan\theta} = \frac{15\times15}{3000\times\frac{1}{30}} = 4.5\,ft.

Some important notes:

  • It should be noted that the weight W includes the movable weight w.  That’s not really clear from the example above but it is the case.
  • The weight shifting is often done in two stages, i.e., from the port (left side) to the centreline (amidships) measuring the angle, and then from amidships to the starboard (right side.) The angles are measured at each stage and added together. This is described in detail in the lab procedure, which can be found at Fluid Mechanics Laboratory Video: Buoyancy and Stability. It is important to insure that the ship is perfectly vertical at the start with the shifted weight amidships (along the centreline.)
  • It is also often done a multiple points along the deck. The example above was done from -7.5′ from the centreline to 7.5′ from the centreline. If we took readings at different distances, say from -6 to 6, -4.5 to 4.5, -3 to 3, etc., theoretically all of these should give the same result for GM. In real life there will be variations, and statistical analysis will prove helpful in sorting out the actual result and dealing with outliers. This is also discussed in the lab procedure linked above.
  • The method as described here used a plumb bob to determine the angle. These days an inclinometer would be much easier.