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.

Figure 21Let 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.

Example 2

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.