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.

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, $\tan \theta = \frac{y}{l}$

Referring to Fig. 21, the moment caused by W about M equals the moment about M caused by moving m to m’.  Or, $W \times GM \tan \theta = mx$

from which $GM = \frac{mx}{W \tan\theta}$

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. It should be noted that the weight W includes the movable weight m.  That’s not really clear from the example above but it is the case.