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Problem 3.77
[Difficulty: 3]
Given:
Tainter gate as shown
w3
5
m
⋅
=
Find:
Force of the water acting on the gate
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
p dA
⋅
=
(Hydrostatic Force)
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
⋅
=
ρ
g
⋅
R
⋅
sin
θ
()
⋅
=
Resolving the hydrostatic force into horizontal and vertical components:
dF
H
dF cos
θ
⋅
=
pdA
⋅
cos
θ
⋅
=
ρ
g
⋅
R
⋅
sin
θ
⋅
w
⋅
R
⋅
d
θ
⋅
cos
θ
⋅
=
since
dA
w R
⋅
d
θ
⋅
=
Integrating this expression:
F
H
0
θ
1
θ
ρ
g
⋅
R
2
⋅
w
⋅
sin
θ
⋅
cos
θ
⋅
⌠
⎮
⌡
d
=
where
θ
1
asin
10 m
⋅
20 m
⋅
⎛
⎝
⎞
⎠
=
30 deg
⋅
=
F
H
ρ
g
⋅
R
2
⋅
w
⋅
0
30 deg
⋅
θ
sin
θ
( ) cos
θ
⋅
⌠
⎮
⌡
d
⋅
=
ρ
g
⋅
R
2
⋅
w
⋅
sin 30 deg
⋅
2
2
⋅
=
ρ
g
⋅
R
2
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This note was uploaded on 10/19/2011 for the course EGN 3353C taught by Professor Lear during the Fall '07 term at University of Florida.
 Fall '07
 Lear
 Statics, Fluid Mechanics, Gate

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