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PROBLEM 14.73
Prior to take-off the pilot of a 3000-kg twin-engine airplane
tests the reversible-pitch propellers by increasing the reverse
thrust with the brakes at point B locked. Knowing that point G
is the center of gravity of the airplane, determine the velocity
of the air in the two 2.2-meter-diameter slipstreams when
the nose wheel A begins to lift off the ground. Assume
1.21 3
kg/m and neglect the approach velocity of the air.
SOLUTION
PROBLEM 14.74
The jet engine shown scoops in air at A at a rate of 200 lb/s
and discharges it at B with a velocity of 2000 ft/s relative to
the airplane. Determine the magnitude and line of action of
the propulsive thrust developed by the engine when the speed
of the airplane is (a) 300 mi/h, (b) 600 mi/h.
SOLUTION
2000 880
PROBLEM 14.75
A jet airliner is cruising at a speed of 900 km/h with each of its
three engines discharging air with a velocity of 800 m/s relative
to the plane. Determine the speed of the airliner after it has lost
the use of (a) one of its engines, (b) two of its engines. Assume
that the drag due to air friction is proportional to the square of
the speed and that the remaining engines keep operating at the
same rate.
SOLUTION
Let v be the airliner speed and u be the discharge relative velocity.
PROBLEM 14.76
A 16-Mg jet airplane maintains a constant speed of 774 km/h while
climbing at an angle 18 .
The airplane scoops in air at a rate of
300 kg/s and discharges it with a velocity of 665 m/s relative to the
airplane. If the pilot changes to a horizontal flight while maintaining
the same engine setting, determine (a) the initial acceleration of the
plane, (b) the maximum horizontal speed that will be attained.
Assume that the drag due to air friction is proportional to the square
of the speed.
2
22
(215)
1.87228 N s /m
PROBLEM 14.76 (Continued)
(b) Maximum speed in horizontal flight:
PROBLEM 14.77
The propeller of a small airplane has a 2-m-diameter slipstream and produces a thrust of 3600 N when the
airplane is at rest on the ground. Assuming 3
1.225 kg/m
for air, determine (a) the speed of the air in the
slipstream, (b) the volume of air passing through the propeller per second, (c) the kinetic energy imparted per
second to the air in the slipstream.
SOLUTION
Calculation of
dm
dt
at a section in the airstream:
mass density volume
density area length
dt
PROBLEM 14.78
The wind turbine-generator shown has an output-power rating of 1.5 MW for a
wind speed of 36 km/h. For the given wind speed, determine (a) the kinetic energy
of the air particles entering the 82.5-m-diameter circle per second, (b) the
efficiency of this energy conversion system. Assume
1.21 kg/m3 for air.
SOLUTION
PROBLEM 14.79
A wind turbine-generator system having a diameter of 82.5 m produces 1.5 MW at
a wind speed of 12 m/s. Determine the diameter of blade necessary to produce
10 MW of power assuming the efficiency is the same for both designs and
1.21 kg/m3 for air.
11
1
22 2
21
1.5
PAd
PROBLEM 14.80
While cruising in level flight at a speed of 570 mi/h, a jet airplane scoops in air at a rate of 240 lb/s and
discharges it with a velocity of 2200 ft/s relative to the airplane. Determine (a) the power actually used to
propel the airplane, (b) the total power developed by the engine, (c) the mechanical efficiency of the airplane.
SOLUTION
dm
PROBLEM 14.81
In a Pelton-wheel turbine, a stream of water is deflected by a series of
blades so that the rate at which water is deflected by the blades is equal
to the rate at which water issues from the nozzle
(/ ).
A
mt Av
Using the same notation as in Sample Problem 14.8, (a) determine the
velocity
V
of the blades for which maximum power is developed,
(b) derive an expression for the maximum power, (c) derive an
expression for the mechanical efficiency.
SOLUTION
PROBLEM 14.81 (Continued)
Input power rate of supply of kinetic energy of the stream
2
in
2
11
()
2
1
A
P
mv
t
mv
PROBLEM 14.82
A circular reentrant orifice (also called Borda’s mouthpiece) of
diameter D is placed at a depth h below the surface of a tank.
Knowing that the speed of the issuing stream is 2vgh and
assuming that the speed of approach
1
v
is zero, show that the
diameter of the stream is
/2.dD
(Hint: Consider the section of
water indicated, and note that P is equal to the pressure at a depth h
multiplied by the area of the orifice).
44
2
PROBLEM 14.83
A railroad car of length L and mass 0
m when empty is moving
freely on a horizontal track while being loaded with sand from
a stationary chute at a rate /.dm dt q Knowing that the car
was approaching the chute at a speed 0,v determine (a) the
mass of the car and its load after the car has cleared the chute,
(b) the speed of the car at that time.
0
L
PROBLEM 14.84*
The depth of water flowing in a rectangular channel of width b
at a speed v
1
and a depth d
1
increases to a depth d
2
at a
hydraulic jump. Express the rate of flow Q in terms of b, d
1
,
and d
2
.
2
PROBLEM 14.85*
1
2
PROBLEM 14.84
The depth of water flowing in a rectangular
channel of width b at a speed v
1
and a depth d
1
increases to a
depth d
2
at a hydraulic jump. Express the rate of flow Q in
terms of b, d
1
, and d
2
.
12 2 1 2 1
12
2
21
1221
12
()
1()()
2
tbdbd
Qd d bd d d d
bd d
PROBLEM 14.85* (Continued)
PROBLEM 14.86
A chain of length l and mass m lies in a pile on the floor. If its end A is raised
vertically at a constant speed v, express in terms of the length y of chain that is off
the floor at any given instant (a) the magnitude of the force
P
applied at A, (b) the
reaction of the floor.
PROBLEM 14.87
Solve Problem 14.86, assuming that the chain is being lowered to the floor at a
constant speed v.
PROBLEM 14.86
A chain of length l and mass m lies in a pile on the floor. If its
end A is raised vertically at a constant speed v, express in terms of the length y of
chain that is off the floor at any given instant (a) the magnitude of the force
P
applied at A, (b) the reaction of the floor.
PROBLEM 14.88
The ends of a chain lie in piles at A and C. When released from rest at time
0,t
the chain moves over the pulley at B, which has a negligible mass.
Denoting by L the length of chain connecting the two piles and neglecting
friction, determine the speed v of the chain at time t.
PROBLEM 14.88 (Continued)
