Page 64 - Introduction to Naval Architecture

FLOTATION AND STABILITY 51

Figure 4,16 Fluid free surface

Effect ofUquidfree surfaces
A ship in service will usually have tanks which are partially filled with
liquids. These may be the fuel and water tanks the ship is using or may
be tanks carrying liquid cargoes. When such a ship is inclined slowly
through a small angle to the vertical the liquid surface will move so as
to remain horizontal. In this discussion a quasi-static condition is
considered so that slopping of the liquid is avoided. Different
considerations would apply to the dynamic conditions of a ship rolling.
For small angles, and assuming the liquid surface does not intersect the
top or bottom of the tank, the volume of the wedge that moves is:

2
11); <p dx, integrated over the length, I, of the tank.
Assuming the wedges can be treated as triangles, the moment of
transfer of volume is:

where I\ is the second moment of area of the liquid, or free, surface.
The moment of mass moved =p ff»/ 1, where p f is the density of the liquid
in the tank. The centre of gravity of the ship will move because of this
shift of mass to a position Gj and:

where p is the density of the water in which the ship is floating and V
is the volume of displacement.