Page 43 - Introduction to Naval Architecture

FLOTATION AND STABILITY 31

Figure 4.1 Floating body

Figure 4.1. These act normal to the body's surface and can be resolved
into vertical and horizontal components. The sum of the vertical
components must equal the weight. The horizontal components must
cancel out otherwise the body would move sideways. The gravitational
force mg can be imagined as concentrated at a point G which is the
centre of mass, commonly known as the centre of gravity Similarly the
opposing force can be imagined to be concentrated at a point B.
Consider now the hydrostatic forces acting on a small element of the
surface, da, a depth y below the surface.

Pressure = density X gravitational acceleration X depth = pgy

The normal force on an element of area da = pyg da
If #> is the angle of inclination of the body's surface to the horizontal
then the vertical component of force is:
(pgyda)cos<p = pg(volume of vertical element)

Integrating over the whole volume the total vertical force is:
pgV where V is the immersed volume of the body.

This is also the weight of the displaced water. It is this vertical force
which 'buoys up' the body and it is known as the buoyancy farce or simply
buoyancy. The point, B, through which it acts is the centroid of volume
of the displaced water and is known as the centre of buoyancy.