Basics of Fluid Mechanics, 2014a

4.6. BUOYANCY AND STABILITY 127
A wooden cubic (made of pine, for example)
is inserted into water. Part of the block
floats above water line. The cubic mass (gravity)
centroid is in the middle of the cubic. How-
G
ever the buoyant center is the middle of the volume
under the water (see Figure 4.38). This
B
situation is similar to Figure 4.37c. However,
any experiment of this cubic wood shows that
Fig. -4.38. Schematic of floating cubic.
it is stable locally. Small amount of tilting of
the cubic results in returning to the original position.
When tilting a larger amount than π/4
, it results in a flipping into the next stable position. The cubic is stable in six positions
(every cubic has six faces). In fact, in any of these six positions, the body is in situation
like in 4.37c. The reason for this local stability of the cubic is that other positions are
less stable. If one draws the stability (later about this criterion) as a function of the
rotation angle will show a sinusoidal function with four picks in a whole rotation.
β
M
ΔF
δF
δF
ΔF
G
GM
dA
B
B’
Fig. -4.39. Stability analysis of floating body.
So, the body stability must be based on the difference between the body’s local
positions rather than the “absolute” stability. That is, the body is “stable” in some
points more than others in their vicinity. These points are raised from the buoyant force
analysis. When the body is tilted at a small angle, β, the immersed part of the body
center changes to a new location, B’ as shown in Figure 4.39. The center of the mass
(gravity) is still in the old location since the body did not change. The stability of the
body is divided into three categories. If the new immerse volume creates a new center in
such way that couple forces (gravity and buoyancy) try to return the body, the original
state is referred as the stable body and vice versa. The third state is when the couple
forces do have zero moment, it is referred to as the neutral stable.
The body, shown in Figure 4.39, when given a tilted position, move to a new
buoyant center, B’. This deviation of the buoyant center from the old buoyant center