Buoyancy and Stability

Buoyancy and Stability
The same principles used to compute hydrostatic forces on surfaces can be applied to
the net pressure force on a completely submerged or floating body. The results are the
two laws of buoyancy discovered by Archimedes in the third century B.C.:
1. A body immersed in a fluid experiences a vertical buoyant force equal to the
weight of the fluid it displaces.
2. A floating body displaces its own weight in the fluid in which it floats.
These two laws are easily derived by referring to Fig. 2.16. In Fig. 2.16a, the body
lies between an upper curved surface 1 and a lower curved surface 2. From Eq. (2.45)
for vertical force, the body experiences a net upward force
FB  FV(2)  FV (1)
 (fluid weight above 2)  (fluid weight above 1)
 weight of fluid equivalent to body volume (2.48)
Alternatively, from Fig. 2.16b, we can sum the vertical forces on elemental vertical
slices through the immersed body:
FB  body
(p2  p1) dAH (z2  z1) dAH  ()(body volume) (2.49)
1,518,000

108,800
846(37.3)

70,200
(55.0 lbf/ft3)(298.7 ft4)

12,300 lbf
84 Chapter 2 Pressure Distribution in a Fluid
Part (b)
These are identical results and equivalent to law 1 above.
Equation (2.49) assumes that the fluid has uniform specific weight. The line of action
of the buoyant force passes through the center of volume of the displaced body;
i.e., its center of mass is computed as if it had uniform density. This point through
which FB acts is called the center of buoyancy, commonly labeled B or CB on a drawing.
Of course, the point B may or may not correspond to the actual center of mass of
the body’s own material, which may have variable density.
Equation (2.49) can be generalized to a layered fluid (LF) by summing the weights
of each layer of density i displaced by the immersed body:
(FB)LF   ig(displaced volume)i (2.50)
Each displaced layer would have its own center of volume, and one would have to sum
moments of the incremental buoyant forces to find the center of buoyancy of the immersed
body.
Since liquids are relatively heavy, we are conscious of their buoyant forces, but gases
also exert buoyancy on any body immersed in them. For example, human beings have
an average specific weight of about 60 lbf/ft3. We may record the weight of a person
as 180 lbf and thus estimate the person’s total volume as 3.0 ft3. However, in so doing
we are neglecting the buoyant force of the air surrounding the person. At standard conditions,
the specific weight of air is 0.0763 lbf/ft3; hence the buoyant force is approximately
0.23 lbf. If measured in vacuo, the person would weigh about 0.23 lbf more.
For balloons and blimps the buoyant force of air, instead of being negligible, is the
controlling factor in the design. Also, many flow phenomena, e.g., natural convection
of heat and vertical mixing in the ocean, are strongly dependent upon seemingly small
buoyant forces.
Floating bodies are a special case; only a portion of the body is submerged, with
the remainder poking up out of the free surface. This is illustrated in Fig. 2.17, where
the shaded portion is the displaced volume. Equation (2.49) is modified to apply to this
smaller volume
FB  ()(displaced volume)  floating-body weight (2.51)
2.8