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144 Chapt. 15 Fluids
the water coming out of the primary. The volume of water
V
accumulated in the catchment
measures time:
V
0
−
V
is the volume of water remaining in the primary.
a) Show that the di±erential bit of water volume sent to the catchment is
dV
=
A
h
v
h
dt ,
where
v
h
is the water velocity at the hole and
dt
is a time di±erential.
HINT:
Actually,
this is one of those obviously it’s
...
things.
b) Using the continuity equation
Av
= Constant
and Bernoulli’s equation
P
+
1
2
ρv
2
+
ρgy
= Constant
show that
A
h
v
h
=
k
r
V
0
−
V ,
where
k
=
A
h
√
A
0
R
2
g
1
−
(
A
h
/A
0
)
2
.
c) Solve the di±erential equation
dV
dt
=
A
h
v
h
=
k
r
V
0
−
V
with the initial condition
V
= 0 at
t
= 0: i.e., ²nd
V
explicitly as a function of time. At
what time
t
max
does
V
reach a maximum? What is
V
max
? At what time
t
empty
is the
primary empty? If one drops the 2nd order term in the
V
(
t
) solution, what would
V
be at
time
t
empty
?
d) Why is the simple water clock
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This note was uploaded on 11/16/2011 for the course PHY 2053 taught by Professor Buchler during the Fall '06 term at University of Florida.
 Fall '06
 Buchler
 Physics

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