ES 123
Problem Set 6 Solution
Lihua Jin
Mar 26, 2010
1. According to the mass conservation, we have
0
0
V
V
ρ
ρ
=
(1)
where
ρ
and
V
are the density and volume at any time. Since the cross-section area
keeps a constant, we can cancel area from Eq. (1), i.e.
0
0
L
L
ρ
ρ
=
(2)
where we know
0
L
L
vt
=
−
(3)
so the dependence of the density on time is
0
0
0
L
L
vt
ρ
ρ
=
−
(4)
You can also use the differential equation of mass conservation to solve this problem.
Actually, Eq. (1) is the ‘integral’ form, which is similar to the integral momentum
conservation law we talked about in class. Eq. (1) can be derived from the differential
equation of mass conservation.
Concept question:
Assume the air in the cylinder is an ideal gas, which satisfies
p
R
T
M
ρ
=
(5)
so the force is proportional to the pressure and can be calculated as
(
)
0
0
0
L R
A
M
T
F
pA
L
vt
ρ
=
=
−
(6)
which can be sketched as shown in Fig. 1. The force increases with the time.
Fig. 1

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