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Problem 3.86
[Difficulty: 4]
Given:
Cylinder of mass M, length L, and radius R is hinged along its length and immersed in an incompressilble liquid to depth
Find:
General expression for the cylinder specific gravity as a function of
α
=H/R needed to hold
the cylinder in equilibrium for
α
ranging from 0 to 1.
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
dp
dh
ρ
g
⋅
=
(Hydrostatic Pressure  h is positive downwards from free surface)
dF
H
h
θ
dF
V
H =
α
R
F
v
A
y
p
⌠
⎮
⎮
⌡
d
=
(Vertical Hydrostatic Force)
Σ
M0
=
(Rotational Equilibrium)
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts on free surface of the liquid.
The moments caused by the hydrostatic force and the weight of the cylinder about the hinge need to balance each other.
Integrating the hydrostatic pressure equation:
p
ρ
g
⋅
h
⋅
=
dF
v
dF cos
θ
()
⋅
=
pdA
⋅
cos
θ
⋅
=
ρ
g
⋅
h
⋅
w
⋅
R
⋅
d
θ
⋅
cos
θ
⋅
=
Now the depth to which the cylinder is submerged is
H
h
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 Fall '07
 Lear
 Fluid Mechanics

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