Basics of Fluid Mechanics, 2014a

4.6. BUOYANCY AND STABILITY 133
The moment of inertia is triangle (see explanation
in example (3.7) is
I xx = ah3
2
And the volume is
V body = a 2 √
√
h 2 − a2
4 = a2 h
1 − 1 4
The point B is a function of the density ratio of the solid and liquid. Denote the liquid
density as ρ l and solid density as ρ s . The point B can be expressed as
B = aρ s
2 ρ l
a 2
h 2
And thus the distance BG is
BG = a 2
(
1 − ρ s
ρ l
)
The limiting condition requires that GM =0so that
Or explicitly
ah 3
ρ l
√ 2
ρ s a 2 h 1 − 1 4
ρ l I xx
ρ s V body
= BG
= a
a 2 2
h 2
(
1 − ρ s
ρ l
)
After rearrangement and using the definitions of ξ = h/a ¯ρρ l /ρ s results in
¯ρξ 2 (
√ = 1 − 1¯ρ
)
1 − ξ2
4
The solution of the above solution is obtained by squaring both sides and defining a
new variable such as x = ξ 2 . After the above manipulation and selecting the positive
value and to keep stability as
√ √64
¯ρ 4 −64 ¯ρ 3 +¯ρ 2 −2¯ρ+1
¯ρ
+ 1¯ρ − 1
x<
2 √ 2¯ρ
End Solution
4.6.1.1 Stability of Body with Shifting Mass Centroid