Problem 3.75
[Difficulty: 2]
Given:
Gate formed in the shape of a circular arc has width w.
Liquid is water;
depth h = R
Find:
(a) Magnitude of the net vertical force component due to fluids acting on the gate
(b) Line of action of the vertical component of the force
Solution:
We will apply the hydrostatics equations to this system.
Governing Equations:
dp
dy
ρ
g
⋅
=
(Hydrostatic Pressure  y is positive downwards)
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 gate
Integrating the hydrostatic pressure equation:
p
ρ
g
⋅
y
⋅
=
Instead of y, we use
θ
as our variable of integration:
y
R sin
θ
()
⋅
=
Therefore,
dy
R cos
θ
⋅
d
θ
⋅
=
In addition,
dA
y
wR
⋅
sin
θ
⋅
d
θ
⋅
=
Therefore,
F
v
0
π
2
θ
ρ
g
⋅
R
⋅
sin
θ
⋅
w
⋅
R
⋅
sin
θ
⋅
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 Fall '07
 Lear
 Fluid Mechanics, Gate

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