Problem *3.125
[Difficulty: 3]
V
= 5 m/s
d
d
R
= 1 m
r
T
Given:
Pail is swung in a vertical circle. Water moves as a rigid body.
Point of interest is the top of the trajectory.
Find:
(a) Tension in the string
(b) Pressure on pail bottom from the water.
Solution:
We will apply the hydrostatics equations to this system.
a
g
p
G
G
ρ
=
+
∇
−
Governing Equations:
(Hydrostatic equation)
r
r
a
g
r
p
=
+
∂
∂
−
(Hydrostatic equation radial component)
Assumptions:
(1) Incompressible fluid
(2) Rigid body motion
(3) Center of mass of bucket and water are located at a radius
of 1 m where V = r
ω
= 5 m/s
Summing the forces in the radial direction:
T
−
M
b
M
w
+
()
g
⋅
−
M
b
M
w
+
a
r
=
where
a
r
V
2
r
−
=
Thus the tension is:
TM
b
M
w
+
V
2
r
g
−
⎛
⎜
⎝
⎞
⎠
⋅
=
where:
M
b
15 N
⋅
s
2
9.81 m
⋅
×
kg m
⋅
Ns
2
⋅
×
=
M
b
1.529 kg
⋅
=
and:
M
w
ρ
V
⋅
=
ρ
π
4
⋅
d
2
⋅
h
⋅
=
M
w
999
kg
m
3
⋅
π
4
×
0.4 m
⋅
2
×
0.2
×
m
⋅
=
M
w
25.11 kg
⋅
=
Now we find T:
T
1.529
25.11
+
k
g
⋅
5
m
s
⋅
⎛
⎝
⎞
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 Fall '07
 Lear
 Fluid Dynamics, Fluid Mechanics, 1 m, 2 m, 2 m, 3 M, 2 kg

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