ME 309 Fall 2008
Section 2 (Merkle)
Homework # 12
Due Wed 24 Sep 2008
Problem 121
A cart with a scoop on its underside rides above a water channel. The forward velocity of
the cart causes the water to be scooped up and stored inside the cart.
A very clever mechanical device is
used to prevent any water from flowing back out through the scoop.
The cart mass is given by M
c
and the
cart initially has no water inside it.
The scoop width is W and it dips into the water a depth h.
Neglect air
drag and wheel friction.
a)
Starting with a CV fixed on the cart, with an observer on the ground, find an equation from which
the force necessary to keep the cart moving at constant velocity can be expressed as a function of the
amount of mass in the cart.
Take the initial velocity to be V, and the cart to contain no water at the
outset..
b). Repeat a) for an observer fixed on the cart.
c)
Find an equation for the rate of deceleration of the cart if it is initially empty and its initial velocity
is V.
Again, fix the CV on the cart and express the equations from the viewpoint of an observer fixed
on the ground.
d)
Repeat part c) for an observer sitting on the decelerating cart.
Solution
:
Assumptions:
Uniform 1D flow across inlet stream
No body forces
Choose CV fixed on cart and enclosing cart plus vane.
Conservation of mass:
(
)
∫
∫
=
•

CV
CS
dt
d
dA
dVol
ρ
ρ
n
V
No mass outflow, mass inflow given by:
c
in
hWu
m
ρ
=
&
Result for mass conservation:
dt
dM
hWu
m
c
in
=
=
ρ
&
, where
M
is mass inside CV.
Part a)
Observer on Ground
:
Conservation of Momentum:
u
cart
water
water
F
scoop
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(
)
∫
∫
∑
∫
+
=
+
•

CV
coor
CV
CS
dVol
a
dt
d
dA
ρ
ρ
ρ
dVol
V
F
n
V
V
For uniform flow across all areas of interest:
M
a
dt
dM
u
dt
du
M
F
u
m
u
m
coor
c
c
c
out
in
in
+
+
=
+

&
&
(Note:
The velocity crossing the control surface at the inlet, as seen by observer on the ground, is zero.)
Momentum simplifies to:
dt
dM
u
F
c
=
Incorporating momentum equation:
2
c
hWu
F
ρ
=
Part b)
Observer on Cart
:
Conservation of mass is unchanged;
Conservation of momentum:
M
a
dt
dM
u
dt
du
M
F
u
m
u
m
coor
cV
cV
c
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 Spring '08
 MERKLE
 Mass, ρ, cart velocity, dt dt Coordinate, uc dt dt

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