Basics of Fluid Mechanics, 2014a

4.6. BUOYANCY AND STABILITY 125
on the ratio of t/R which similar analysis to the above. For a given ratio of t/R, the
weight displaced by the sphere has to be same as the sphere weight. The volume of a
sphere cap (segment) is given by
V cap = πh2 (3R − h)
3
Where h is the sphere height above the water. The volume in the water is
V water = 4 πR3
3
− πh2 (3R − h)
3
= 4 π ( R 3 − 3 Rh 2 + h 3)
3
(4.XXV.d)
(4.XXV.e)
When V water denotes the volume of the sphere in the water. Thus the Archimedes law
is
ρ w 4 π ( R 3 − 3 Rh 2 + h 3)
= ρ s 4 π ( 3 tR 2 − 3 t 2 R + t 3)
(4.XXV.f)
3
3
or
(
R 3 − 3 Rh 2 + h 3) = ρ w
(
3 tR 2 − 3 t 2 R + t 3)
(4.XXV.g)
ρ s
The solution of (4.XXV.g) is
(√
−fR (4 R3 − fR)
h =
−
2
) 1
fR− 2 R3 3
2
+
(√ ) 1
−fR (4 R3 − fR) fR− 2 R3 3
−
2
2
(4.XXV.h)
Where −fR = R 3 − ρ w
(3 tR 2 − 3 t 2 R + t 3 ) There are two more solutions which
ρ s
contains the imaginary component. These solutions are rejected.
End Solution
Example 4.26:
One of the common questions in buoyancy is the weight with variable cross section and
fix load. For example, a wood wedge of wood with a fix weight/load. The general
question is at what the depth of the object (i.e. wedge) will be located. For simplicity,
assume that the body is of a solid material.
Solution
It is assumed that the volume can be written as a function of the depth. As it was
shown in the previous example, the relationship between the depth and the displaced
liquid volume of the sphere. Here it is assumed that this relationship can be written as
R 2
V w = f(d, other geometrical parameters)
(4.XXVI.a)