Problem 3.65
[Difficulty: 3]
Given:
Gate shown with fixed width, bass of gate is negligible.
Gate is in equilibrium.
b
3 m
⋅
=
Find:
Water depth, d
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
dp
dh
ρ
g
⋅
=
(Hydrostatic Pressure - h is positive downwards)
F
R
p
c
A
⋅
=
(Hydrostatic Force)
y'
y
c
I
xx
A y
c
⋅
+
=
(Location of line of action)
Σ
M
z
0
=
(Rotational equilibrium)
L
y
l
d
M
h
θ
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
⋅
=
F
R
p
c
A
⋅
=
ρ
g
⋅
h
c
⋅
A
⋅
=
h
c
d
2
=
A
b
d
sin
θ
( )
⋅
=
Therefore,
F
R
ρ
g
⋅
b
⋅
d
2
⋅
2 sin
θ
( )
⋅
=
To find the line of application of this force:
y'
y
c
I
xx
A y
c
⋅
+
=
Since
I
xx
b l
3
⋅
12
=
and
A
b l
⋅
=
it follows that
y'
y
c
b l
3
⋅
12 b
⋅
l
⋅
y
c
⋅
+
=
y
c
l
2
12 y
c
⋅
+
=
where l is the length of the gate in contact with the water (as seen in diagram)

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