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Problem 3.74
[Difficulty: 2]
Given:
Open tank as shown.
Width of curved surface
b1
0
f
t
⋅
=
Find:
(a) Magnitude of the vertical force component on the curved surface
(b) Line of action of the vertical component of the force
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
dp
dh
γ
=
(Hydrostatic Pressure  h is positive downwards)
L
x’
x
F
Ry
y
F
v
A
y
p
⌠
⎮
⎮
⌡
d
−
=
(Vertical Hydrostatic Force)
x' F
v
⋅
F
v
x
⌠
⎮
⎮
⌡
d
=
(Moment of vertical force)
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts at free surface of water
and
on outside of wall
Integrating the hydrostatic pressure equation:
p
γ
h
⋅
=
We can define along the surface
hLR
2
x
2
−
()
1
2
−
=
We also define the incremental area on the curved surface as:
dA
y
bdx
⋅
=
Substituting these into the force equation we get:
F
v
A
y
p
⌠
⎮
⎮
⌡
d
−
=
0
R
x
γ
LR
2
x
2
−
1
2
−
⎡
⎢
⎣
⎤
⎥
⎦
⋅
b
⋅
⌠
⎮
⎮
⎮
⌡
d
−
=
γ
−
b
⋅
0
R
x
2
x
2
−
−
⌠
⎮
⌡
d
⋅
=
γ
−
b
⋅
R
⋅
π
4
⋅
−
⎛
⎝
⎞
⎠
⋅
=
F
v
62.4
lbf
ft
3
⋅
10
×
ft
⋅
4
×
ft
⋅
10 ft
⋅
4f
t
⋅
π
4
×
−
⎛
⎝
⎞
⎠
×
⎡
⎢
⎣
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 Fall '07
 Lear
 Fluid Mechanics

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