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
→⎯
p dA
→
⎯
⋅
=
(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
⋅
=
p w
⋅
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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '07
 Lear
 Statics, Fluid Mechanics

Click to edit the document details