Basics of Fluid Mechanics, 2014a

4.6. BUOYANCY AND STABILITY 117
In going over these calculations, the calculations of the center of the area were not
carried out. This omission saves considerable time. In fact, trying to find the center of
the area will double the work. This author find this method to be simpler for complicated
geometries while the indirect method has advantage for very simple geometries.
End Solution
4.6 Buoyancy and Stability
One of the oldest known scientific research
on fluid mechanics relates to buoyancy
due to question of money was car-
a
b
Fig. -4.34. Schematic of Immersed Cylinder.
ried by Archimedes. Archimedes principle
is related to question of density and
volume. While Archimedes did not know
much about integrals, he was able to capture
the essence. Here, because this material
is presented in a different era, more advance mathematics will be used. While the
question of the stability was not scientifically examined in the past, the floating vessels
structure (more than 150 years ago) show some understanding 15 .
The total forces the liquid exacts on a body are considered as a buoyancy issue.
To understand this issue, consider a cubical and a cylindrical body that is immersed
in liquid and center in a depth of, h 0 as shown in Figure 4.34. The force to hold the
cylinder at the place must be made of integration of the pressure around the surface
of the square and cylinder bodies. The forces on square geometry body are made only
of vertical forces because the two sides cancel each other. However, on the vertical
direction, the pressure on the two surfaces are different. On the upper surface the
pressure is ρg(h 0 − a/2). On the lower surface the pressure is ρg(h 0 + a/2). The
force due to the liquid pressure per unit depth (into the page) is
F = ρg ((h 0 − a/2) − (h 0 + a/2)) lb= −ρgabl= −ρgV (4.147)
In this case the l represents a depth (into the page). Rearranging equation (4.147) to
be
F
= ρg (4.148)
V
The force on the immersed body is equal to the weight of the displaced liquid. This
analysis can be generalized by noticing two things. All the horizontal forces are canceled.
Any body that has a projected area that has two sides, those will cancel each other.
Another way to look at this point is by approximation. For any two rectangle bodies,
the horizontal forces are canceling each other. Thus even these bodies are in contact
with each other, the imaginary pressure make it so that they cancel each other.
15 This topic was the author’s high school name. It was taught by people like these, 150 years ago
and more, ship builders who knew how to calculate GM but weren’t aware of scientific principles behind
it. If the reader wonders why such a class is taught in a high school, perhaps the name can explain it:
Sea Officers High School.
r0
h0