Basics of Fluid Mechanics, 2014a

126 CHAPTER 4. FLUIDS STATICS
The Archimedes balance on the body is
ρ l V a = ρ w V w
(4.XXVI.b)
d = f −1 ρ lV a
ρ w
(4.XXVI.c)
End Solution
Example 4.27:
In example 4.26 a general solution was provided. Find the reverse function, f −1 for
cone with 30 ◦ when the tip is in the bottom.
Solution
First the function has to built for d (depth).
Thus, the depth is
( ) 2 d√3
πd
V w =
3
d = 3 √ 9 πρw
ρ l V a
End Solution
= πd3
9
(4.XXVII.a)
(4.XXVII.b)
4.6.1 Stability
Figure 4.37 shows a body made of
hollow balloon and a heavy sphere
connected by a thin and light rod.
This arrangement has mass centroid
close to the middle of the sphere.
The buoyant center is below the middle
of the balloon. If this arrangement
is inserted into liquid and will
be floating, the balloon will be on the
top and sphere on the bottom. Tilting
the body with a small angle from
Empty
buoyancy
center
gravity
center
Full
a b c
Fig. -4.37. Schematic of floating bodies.
its resting position creates a shift in the forces direction (examine Figure 4.37b). These
forces create a moment which wants to return the body to the resting (original) position.
When the body is at the position shown in Figure 4.37c ,the body is unstable
and any tilt from the original position creates moment that will further continue to
move the body from its original position. This analysis doesn’t violate the second
law of thermodynamics. Moving bodies from an unstable position is in essence like a
potential.