Basics of Fluid Mechanics, 2014a

118 CHAPTER 4. FLUIDS STATICS
On the other hand, any shape is made of many small rectangles. The force on
every rectangular shape is made of its weight of the volume. Thus, the total force is
made of the sum of all the small rectangles which is the weight of the sum of all volume.
In illustration of this concept, consider the cylindrical
shape in Figure 4.34. The force per area (see Figure 4.35)
is
P
dA vertical
{ }} { { }} {
dF = ρg (h 0 − r sin θ) sin θrdθ (4.149)
h 0
The total force will be the integral of the equation (4.149)
F =
∫ 2π
0
ρg (h 0 − r sin θ) rdθ sin θ (4.150)
r
θ
Rearranging equation (4.149) transforms it to
F = rgρ
∫ 2π
The solution of equation (4.151) is
0
(h 0 − r sin θ) sin θdθ (4.151)
Fig. -4.35. The floating
forces on Immersed Cylinder.
F = −πr 2 ρg (4.152)
The negative sign indicate that the force acting upwards. While the horizontal force is
F v =
∫ 2 π
0
(h 0 − r sin θ) cos θdθ=0 (4.153)
Example 4.19:
To what depth will a long log with radius, r, a length, l and density, ρ w in liquid with
density, ρ l . Assume that ρ l >ρ w . You can provide that the angle or the depth.
Typical examples to explain the buoyancy are
of the vessel with thin walls put upside down into
liquid. The second example of the speed of the
floating bodies. Since there are no better examples,
these examples are a must.
h 1
t
h in
w
h
Example 4.20:
A cylindrical body, shown in Figure 4.36 ,is floating
in liquid with density, ρ l . The body was inserted
into liquid in a such a way that the air had remained
in it. Express the maximum wall thickness, t, asa
function of the density of the wall, ρ s liquid density,
Fig. -4.36. Schematic of a thin wall
floating body.