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Problem 3.72
[Difficulty: 3]
Given:
Parabolic gate, hinged at O has a constant width.
b2
m
⋅
=
c
0.25 m
1
−
⋅
=
D2
m
⋅
=
H3
m
⋅
=
Find:
(a) Magnitude and line of action of the vertical force on the gate due to water
(b) Horizontal force applied at A required to maintain equilibrium
(c) Vertical force applied at A required to maintain equilibrium
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
dp
dh
ρ
g
⋅
=
(Hydrostatic Pressure  h is positive downwards)
Σ
M
z
0
=
(Rotational equilibrium)
F
v
A
y
p
⌠
⎮
⎮
⌡
d
=
(Vertical Hydrostatic Force)
x' F
v
⋅
F
v
x
⌠
⎮
⎮
⌡
d
=
(Location of line of action)
F
H
p
c
A
⋅
=
(Horizontal Hydrostatic Force)
h'
h
c
I
xx
Ah
c
⋅
+
=
(Location of line of action)
O
y
h’
B
x’
x
F
V
O
x
F
H
y
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts at free surface of water and
on
outside of gate
Integrating the hydrostatic pressure equation:
p
ρ
g
⋅
h
⋅
=
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 Fall '07
 Lear
 Fluid Mechanics, Gate

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