Problem 5.155
One option when traveling to Mars from the Earth is to use a Hohmann transfer
orbit like that described in Probs. 5.148–5.152. Assuming that the Sun is the
primary gravitational influence and ignoring the gravitational influence of
Earth and Mars (since the Sun accounts for 99.8% of the mass of the solar
system), determine the change in speed required at the Earth
ve
(perihelion
in the transfer orbit) and the required change in speed at Mars
vm
(aphelion
in the transfer orbit) to accomplish the mission to Mars using a Hohmann
transfer. In addition, determine the amount of time
⌧
it would take for orbital
transfer. Use
1:989⇥1030 kg
for the mass of the Sun, assume that the orbits
of Earth and Mars are circular, and assume that the changes in speed are
impulsive, that is, they occur instantaneously. In addition, use
150⇥106km
for the radius of Earth’s orbit and
228⇥106km
for the radius of Mars’s orbit.
Solution
We start with determining the circular orbit speed (see Eq. (5.82) on p. 388 of the textbook) and the elliptical
orbit speed (see Eq. (5.105) on p. 391 of the textbook) at aphelion corresponding to a distance from the Sun
Finally, we determine the time needed for the orbital transfer. This time is half of the full elliptical orbit
period. Hence, using Eq. (5.97) on p. 390 of the textbook, we have
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Dynamics 2e 1091
Problem 5.156
A fluid is in steady motion in the conduit shown. The lines depicted are tangent to the velocity of the fluid
particles in the conduit (these lines are called streamlines). Explain whether or not the control volume
defined by the cross sections
A
and
B
in the figure is consistent with the assumptions laid out in this
section.
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Problem 5.157
A hydraulic system is being used to actuate the control surfaces of a plane. Suppose that there is a time
interval during which (a) the speed of the hydraulic fluid within a particular line is constant relative to
the line itself and (b) the plane is performing a turn. Explain whether or not the force balance for control
volumes presented in this section is applicable to the analysis of the hydraulic fluid in question.
3
3
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Dynamics 2e 1093
Problem 5.158
The cross sections labeled
A
and
B
in case (a) are identical to the corresponding cross sections in case
(b). Assume that, in both (a) and (b), a fluid in steady motion flows through
A
with speed
v1
and exits
the system at
B
with a speed
v2
. If the pipe sections are to remain stationary and if the mass flow rate is
identical in the two cases, determine whether the magnitude of the horizontal force acting on the pipes due
to the water flow in case (a) is smaller than, equal to, or larger than that in case (b). In addition, for both (a)
and (b), establish the direction of the force.
Solution
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Problem 5.159
Experience tells us that when a steady water jet comes out of a nozzle
B
, the hose line
A
attached to the
nozzle is in tension, that is, the nozzle exerts a force on the hose that is in the direction of the flow. If the
end of the nozzle at
B
were capped to stop the water flow, would the force exerted by the nozzle on the
hose decrease, stay the same, or increase?
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Dynamics 2e 1095
Problem 5.160
Revisit Example 5.17 and use the numerical result in Eq. (13) of the example, along with the fact that the
specific weight of water is
62:4 lb=ft3
, to determine the volumetric flow rate at the nozzle and the nozzle
diameter.
Solution
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permission of McGraw-Hill, is prohibited.
Problem 5.161
The rocket shown has
7lb
of propellant with a burnout time (time required to
burn all the fuel) of
7
s. Assume that the mass flow rate is constant and that
the speed of the exhaust relative to the rocket is also constant and equal to
6500 ft=s
. If the rocket is fired from rest, determine the initial weight of the
rocket’s body if the rocket is to experience an initial acceleration of 6g.
Substituting Eqs. (1) and (3) into the last of Eqs. (2) and solving for for Wb, we have
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Dynamics 2e 1097
Problem 5.162
The tip
B
of a nozzle is
1:5 in:
in diameter, whereas the diameter at
A
where the hose is attached is
3in:
If water flows through the nozzle at
200 gpm
(“gpm” stands for gallons per minute; 1 U.S. gallon is defined
as
231 in:3
) and the water static pressure in the line is
300 psi
, determine
the force necessary to hold the nozzle stationary. Recall that the specific
weight of water is
D62:4 lb=ft3
, and neglect the atmospheric pressure
at B.
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1098 Solutions Manual
An intubed fan is mounted on a cart connected to a fixed wall by a linear elastic spring with constant
kD50 lb=ft
. Assume that in a test the fan draws air at
A
with negligible speed and that the outgoing flow
causes the cart to displace to the left so that the spring is stretched by
0:5 ft
from its unstretched position.
Assuming that the specific weight of the air
D7:5⇥102lb=ft3
is constant, and letting the diameter of
the tube at Bbe dD4ft (the cross section is assumed to be circular), determine the airspeed at B.
Solution
Dynamics 2e 1099
where
ı
is the stretch of the spring, and we have expressed the spring force as
FsDkı
. Substituting the
expression for Rxin Eq. (3) into the last of Eqs. (4) we obtain an equation in vBx whose solution is
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