Problem 4.143
[Difficulty: 4]
Given:
Data on vane/slider
Find:
Formula for acceleration and speed; plot
Solution:
The given data is
ρ
999
kg
m
3
⋅
=
M
30 kg
⋅
=
A
0.005 m
2
⋅
=
V
20
m
s
⋅
=
μ
k
0.3
=
The equation of motion, from Problem 4.141, is
dU
dt
ρ
V
U
−
(
)
2
⋅
A
⋅
M
g
μ
k
⋅
−
=
The acceleration is thus
a
ρ
V
U
−
(
)
2
⋅
A
⋅
M
g
μ
k
⋅
−
=
Separating variables
dU
ρ
V
U
−
(
)
2
⋅
A
⋅
M
g
μ
k
⋅
−
dt
=
Substitute
u
V
U
−
=
dU
du
−
=
du
ρ
A
⋅
u
2
⋅
M
g
μ
k
⋅
−
dt
−
=
But
u
1
ρ
A
⋅
u
2
⋅
M
g
μ
k
⋅
−
⎛
⎜
⎝
⎞
⎠
⌠
⎮
⎮
⎮
⎮
⌡
d
M
g
μ
k
⋅
ρ
⋅
A
⋅
−
atanh
ρ
A
⋅
g
μ
k
⋅
M
⋅
u
⋅
⎛
⎜
⎝
⎞
⎠
⋅
=
and u = V  U so
M
g
μ
k
⋅
ρ
⋅
A
⋅
−
atanh
ρ
A
⋅
g
μ
k
⋅
M
⋅
u
⋅
⎛
⎜
⎝
⎞
⎠
⋅
M
g
μ
k
⋅
ρ
⋅
A
⋅
−
atanh
ρ
A
⋅
g
μ
k
⋅
M
⋅
V
U
−
(
)
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
=
Using initial conditions
M
g
μ
k
⋅
ρ
⋅
A
⋅
−
atanh
ρ
A
⋅
g
μ
k
⋅
M
⋅
V
U
−
(
)
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
M
g
μ
k
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 Fall '07
 Lear
 Fluid Mechanics

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