Problem 2.51
[Difficulty: 4]
F
f
τ
A
⋅
=
x, V, a
Mg
⋅
Given:
Data on the block and incline
Find:
Initial acceleration; formula for speed of block; plot; find speed after 0.1 s.
Find oil viscosity if speed is 0.3 m/s after
0.1 s
Solution:
Given data
M5
k
g
⋅
=
A
0.1 m
⋅
()
2
=
d
0.2 mm
⋅
=
θ
30 deg
⋅
=
From Fig. A.2
μ
0.4
Ns
⋅
m
2
⋅
=
Applying Newton's 2nd law to initial instant (no friction)
Ma
⋅
⋅
sin
θ
⋅
F
f
−
=
⋅
sin
θ
⋅
=
so
a
init
g sin
θ
⋅
=
9.81
m
s
2
⋅
sin 30 deg
⋅
×
=
a
init
4.9
m
s
2
=
Applying Newton's 2nd law at any instant
⋅
⋅
sin
θ
⋅
F
f
−
=
and
F
f
τ
A
⋅
=
μ
du
dy
⋅
A
⋅
=
μ
V
d
⋅
A
⋅
=
so
⋅
M
dV
dt
⋅
=
⋅
sin
θ
⋅
μ
A
⋅
d
V
⋅
−
=
Separating variables
dV
g sin
θ
⋅
μ
A
⋅
Md
⋅
V
⋅
−
dt
=
Integrating and using limits
⋅
μ
A
⋅
−
ln 1
μ
A
⋅
⋅
d
⋅
sin
θ
⋅
V
⋅
−
⎛
⎝
⎞
⎠
⋅
t
=
or
Vt
⋅
d
⋅
sin
θ
⋅
μ
A
⋅
1e
μ
−
A
⋅
⋅
t
⋅
−
⎛
⎜
⎝
⎞
⎠
⋅
=
At t = 0.1 s
V5
k
g
⋅
9.81
×
m
s
2
⋅
0.0002
×
m
⋅
sin 30 deg
⋅
⋅
m
2
0.4 N
⋅
s
⋅
0.1 m
⋅
2
⋅
×
2
⋅
kg m
⋅
×
0.4 0.01
⋅
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 Fall '07
 Lear
 Fluid Mechanics

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