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Problem 3.85
[Difficulty: 3]
Given:
Model cross section of canoe as a parabola.
Assume constant width W over entire length L
ya
x
2
⋅
=
a
1.2 ft
1
−
⋅
=
W2
f
t
⋅
=
L1
8
f
t
⋅
=
Find:
Expression relating the total mass of canoe and contents to distance d.
Determine maximum
allowable total mass without swamping the canoe.
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)
F
v
A
y
p
⌠
⎮
⎮
⌡
d
=
(Vertical Hydrostatic Force)
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts on free surface of the water and inner
surface of the canoe.
At any value of d the weight of the canoe and its contents is balanced by the net vertical force of the water on the canoe.
Integrating the hydrostatic pressure equation:
p
ρ
g
⋅
h
⋅
=
F
v
A
y
p
⌠
⎮
⎮
⌡
d
=
x
ρ
g
⋅
h
⋅
L
⋅
⌠
⎮
⎮
⌡
d
=
where
hH
d
−
()
y
−
=
To determine the upper limit of integreation we remember that
x
2
⋅
=
At the surface
yHd
−
=
Therefore,
x
Hd
−
a
=
and so the vertical force is:
F
v
2
0
−
a
x
ρ
g
⋅
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 Fall '07
 Lear
 Fluid Mechanics

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