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Problem *3.103
[Difficulty:4]
Given:
Sphere partially immersed in a liquid of specific gravity SG.
Find:
(a) Formula for buoyancy force as a function of the submersion depth d
(b) Plot of results over range of liquid depth
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
F
buoy
ρ
g
⋅
V
d
⋅
=
(Buoyant force is equal to weight of displaced fluid)
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts everywhere
d
R
sin
θ
R
d
max
h
We need an expression for the displaced volume of fluid at an arbitrary
depth d. From the diagram we see that:
d
R 1
cos
θ
max
()
−
=
at an arbitrary depth h:
h
d
R 1
cos
θ
−
⋅
−
=
r
R sin
θ
⋅
=
So if we want to find the volume of the submerged portion of the sphere we calculate:
V
d
0
θ
max
h
π
r
2
⌠
⎮
⌡
d
=
π
0
θ
max
θ
R
2
sin
θ
2
⋅
R
⋅
sin
θ
⋅
⌠
⎮
⌡
d
⋅
=
π
R
3
⋅
0
θ
max
θ
sin
θ
3
⌠
⎮
⌡
d
⋅
=
Evaluating the integral we get:
V
d
π
R
3
⋅
cos
θ
max
3
3
cos
θ
max
−
2
3
+
⎡
⎢
⎣
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 Fall '07
 Lear
 Fluid Mechanics

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