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1100 Solutions Manual
A test is conducted in which an
80 kg
person sitting in a
15 kg
cart is propelled by the jets emitted by
two household fire extinguishers with a combined initial mass of
18 kg
. The cross section of the exhaust
nozzles is
3cm
in diameter, and the density of the exhaust is
⇢D1:98 kg=m3
. The vehicle starts from rest,
and it is determined that the initial acceleration of the “jet cart” is
1:8 m=s2
. Recalling that the mass flow
rate out of the nozzle is given by
PmoD⇢Svo
, where
S
is the area of the nozzle cross section and
vo
is the
exhaust speed, determine voat the initial time. Ignore any resistance to the horizontal motion of the cart.
Solution
voDs2max
⇢⇡d2:(3)
Dynamics 2e 1101
Problem 5.165
Consider a rocket in space so that it can be assumed that no external forces act on the rocket. Let
vo
be the
constant speed of the exhaust gases relative to the rocket. In addition, let
mbCm
f
be the total mass of the
rocket and its fuel at the initial time, and let
mb
be the mass of the body after all the fuel is burned. If the
rocket is fired from rest, determine an expression for the maximum speed that the rocket can achieve.
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permission of McGraw-Hill, is prohibited.
1102 Solutions Manual
A stationary
4cm
diameter nozzle emits a water jet with a speed
of
30 m=s
. The water jet impinges on a vane with a mass of
15 kg
.
Recalling that water has a mass density of
1000 kg=m3
, determine
the minimum static friction coefficient with the ground such that the
vane does not move if
D20ı
and
✓D30ı
. Neglect the weight of
the water layer in contact with the vane, as well as friction between
the water and the vane.
Solution
RxD1
4⇢⇡d2v2
w.cos cos ✓/and RyD1
4⇢⇡d2v2
w.sin ✓Csin /: (5)
Dynamics 2e 1103
Substituting Eqs. (5) into Eq. (8), we obtain an equation for swhose solution is
w.cos cos ✓/
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permission of McGraw-Hill, is prohibited.
Problem 5.167
A diffuser is attached to a structure whose rigidity in the horizontal
direction can be modeled by a linear spring with constant
k
. The
diffuser is hit by a water jet issued with a speed
vwD55 ft=s
from
a
2in:
diameter nozzle. Assume that the friction between the jet and
the diffuser is negligible and that the diffuser’s motion in the vertical
direction can be neglected. Recalling that the specific weight of
water is
D62:4 lb=ft3
, if the opening angle of the diffuser is
✓D40ı
, determine
k
such that the horizontal displacement of the
diffuser does not exceed
0:25 in:
from the diffuser’s rest position.
Assume that the water jet splits symmetrically over the diffuser.
Solution
Because of the symmetry we can analyze one of the two jets in which the main jet splits.
Referring to the figure on the right, we model the chosen half-jet as a steady flow system
whose control volume is defined by the cross sections at
A
and
B
. Then, applying the
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1105
Problem 5.168
A water jet with a mass flow rate
Pm
f
at the nozzle impinges with a
speed
vw
on a fixed flat vane inclined at an angle
✓
with respect to the
horizontal. Assuming that there is no friction between the water jet and
the vane, the jet will split into two flows with mass flow rates
Pm
f1
and
Pm
f2
. Neglecting the weight of the water, determine how
Pm
f1
and
Pm
f2
depend on
Pm
f
,
vw
, and
✓
.Hint: Due to the no-friction assumption,
there is no force that slows down the water in the direction tangent
to the vane, and this implies that the momentum in that direction is
conserved.
v1Dv2Dvw:(2)
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permission of McGraw-Hill, is prohibited.
Problem 5.169
A person wearing a jet pack lifts off from rest and ascends along a straight vertical trajectory. Let
M
denote the initial combined mass of the pilot and the equipment, including the fuel in the pack. Assume
that the mass flow rate
Pmo
and exhaust gas speed
vo
are known constants and that the pilot can take off as
soon as the rocket engine is started. If the engine exhaust is completely directed in the direction of gravity,
determine the expression of the pilot’s speed as a function of time,
M
,
Pmo
,
vo
, and
g
(the acceleration due
to gravity) while the pack is providing a thrust. Neglect air resistance and assume that gravity is constant.
Photo credit: Courtesy of JetPack International, LLC
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1107
Problem 5.170
A
28;000 lb
A-10 Thunderbolt is flying at a constant speed of
375 mph
when it fires a
4
s burst from its forward-facing seven-
is
3250 ft=s
. Assume that each of the plane’s two jet engines
maintains a constant thrust of
9000 lb
, that the plane is subject
the
4
s burst, modeling the airplane’s change of mass due to firing
as a continuous mass loss.
Hence, using the (full precision value of the) above expressions, we have that the (continuous) mass outflow
Pmois given by
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permission of McGraw-Hill, is prohibited.
1108 Solutions Manual
A faucet is letting out water at a rate of
15 L=min
. Assume that the
internal diameter
d
of the faucet is uniform and equal to
1:5 cm
, the
distance
`D20 cm
, and the static water pressure at the wall is
0:30 MPa
.
Neglecting the weight of the water inside the faucet, as well as the weight
of the faucet itself, determine the forces and the moment that the wall
exerts on the faucet. Recall that the density of water is
⇢D1000 kg=m3
,
and neglect the atmospheric pressure at the spout. Hint: Define your
control volume using a section along the wall.
Solution
follows:
vAx DQ
Ay D0; vBx D0; vBy DQ
and Mwhose solution is
ND1
4⇡PAd2C4⇢Q2
⇡d2;VD4⇢Q2
⇡d2;MD4⇢Q2`
⇡d2;
ND53:37 N;VD0:3537 N;MD0:07074 Nm:
Dynamics 2e 1109
Problem 5.172
Consider a wind turbine with a diameter
dD110
m and the airflow
streamlines shown, which are symmetric relative to the axis of
the turbine. Since the airflow is tangent to the streamlines (by
definition), these lines can be taken to define the top and bottom
surfaces of a control volume. Suppose that pressure measurements
indicate that the flow experiences atmospheric pressure at the cross
sections
A
and
B
(as well as outside the control volume) where
the wind speed is
vAD7m=s
and
vBD2:5 m=s
, respectively.
Furthermore, assume that the average pressure along the streamlines
defining the control volume is also atmospheric. Finally, assume
that
O
is on the line of action of the overall weight of the turbine
and that the diameter of the flow cross section at
A
is
85
% of the
rotor diameter and that the rotor hub is at a distance
hD75
m
above the ground. If the density of air is constant and equal to
⇢D1:25 kg=m3
, determine the force exerted by the air on the wind
turbine and the reaction moment at the base of the support.
Substituting Eqs. (4)–(6) into Eqs. (1)–(3), we have, ND0,MRD0, and
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permission of McGraw-Hill, is prohibited.
