Problem *3.106
[Difficulty: 4]
F
B
W
F
L
F
U
x
y
Given:
Data on sphere and tank bottom
Find:
Expression for SG of sphere at which it will float to surface;
minimum SG to remain in position
Assumptions:
(1) Water is static and incompressible
(2) Sphere is much larger than the hole at the bottom of the tank
Solution:
Basic equations
F
B
ρ
g
⋅
V
⋅
=
and
Σ
F
y
F
L
F
U
−
F
B
+
W
−
=
where
F
L
p
atm
π
⋅
a
2
⋅
=
F
U
p
atm
ρ
g
⋅
H2
R
⋅
−
()
⋅
+
⎣
⎦
π
⋅
a
2
⋅
=
F
B
ρ
g
⋅
V
net
⋅
=
V
net
4
3
π
⋅
R
3
⋅
π
a
2
⋅
2
⋅
R
⋅
−
=
WS
G
ρ
⋅
g
⋅
V
⋅
=
with
V
4
3
π
⋅
R
3
⋅
=
Now if the sum of the vertical forces is positive, the sphere will float away, while if the sum is zero or negative the sphere will stay
at the bottom of the tank (its weight and the hydrostatic force are greater than the buoyant force).
Hence
Σ
F
y
p
atm
π
⋅
a
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 Fall '07
 Lear
 Fluid Mechanics, Buoyancy, Force, Archimedes, ΣFy

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