Problem 3.50
[Difficulty: 3]
F
A
H
= 25 ft
y
R
= 10 ft
h
A
B
z
x
y
Given:
Geometry of gate
Find:
Force F
A
for equilibrium
Solution:
Basic equation
F
R
A
p
⌠
⎮
⎮
⌡
d
=
dp
dh
ρ
g
⋅
=
Σ
M
z
0
=
or, use computing equations
F
R
p
c
A
⋅
=
y'
y
c
I
xx
Ay
c
⋅
+
=
where y would be measured
from the free surface
Assumptions: static fluid;
ρ
= constant; p
atm
on other side; door is in equilibrium
Instead of using either of these approaches, we note the following, using y as in the sketch
Σ
M
z
0
=
F
A
R
⋅
A
yp
⋅
⌠
⎮
⎮
⌡
d
=
with
p
ρ
g
⋅
h
⋅
=
(Gage pressure, since p =
p
atm
on other side)
F
A
1
R
A
y
ρ
⋅
g
⋅
h
⋅
⌠
⎮
⎮
⌡
d
⋅
=
with
dA
r dr
⋅
d
θ
⋅
=
and
y
r sin
θ
()
⋅
=
hHy
−
=
Hence
F
A
1
R
0
π
θ
0
R
r
ρ
g
⋅
r
⋅
sin
θ
⋅
H
r sin
θ
⋅
−
⋅
r
⋅
⌠
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 Fall '07
 Lear
 Fluid Mechanics, Gate

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