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 differential 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 differential.
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
radicalbig
V
0
−
V ,
where
k
=
A
h
√
A
0
radicalBigg
2
g
1
−
(
A
h
/A
0
)
2
.
c) Solve the differential equation
dV
dt
=
A
h
v
h
=
k
radicalbig
V
0
−
V
with the initial condition
V
= 0 at
t
= 0: i.e., find
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
?
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 Fall '06
 Buchler
 Physics, Escapement, Astronomical clock, water clock, Ctesibius, Su Sung

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