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Problem 3.81
[Difficulty: 3]
Given:
Cylindrical log floating against dam
Find:
(a) Mass per unit length of the log (b) Contact force per unit length between log and dam
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
→⎯
pdA
→
⎯
⋅
=
(Hydrostatic Force)
dF
H
dF
h
θ
dF
V
R = D
/2
Assumptions:
(1) Static fluid
(2) Incompressible fluid
(3) Atmospheric pressure acts on free surfaces and on the
first quadrant of the log
Integrating the hydrostatic pressure equation:
p
ρ
g
⋅
h
⋅
=
ρ
g
⋅
R
⋅
1
cos
θ
()
−
⋅
=
Resolving the incremental force into horizontal and vertical components:
dF
p dA
⋅
=
pw
⋅
R
⋅
d
θ
⋅
=
ρ
g
⋅
R
⋅
1
cos
θ
−
⋅
w
⋅
R
⋅
d
θ
⋅
=
ρ
g
⋅
R
2
⋅
w
⋅
1
cos
θ
−
⋅
=
dF
H
dF sin
θ
⋅
=
ρ
g
⋅
R
2
⋅
w
⋅
1
cos
θ
−
⋅
d
θ
⋅
sin
θ
⋅
=
dF
v
dF cos
θ
⋅
=
ρ
g
⋅
R
2
⋅
w
⋅
1
cos
θ
−
⋅
d
θ
⋅
cos
θ
⋅
=
Integrating the expression for the horizontal force will provide us with the contact force per unit length:
F
H
0
3
π
⋅
2
θ
ρ
g
⋅
R
2
⋅
w
⋅
1
cos
θ
−
⋅
sin
θ
⋅
⌠
⎮
⎮
⌡
d
=
ρ
g
⋅
R
2
⋅
w
⋅
0
3
π
⋅
2
θ
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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

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