PROBLEM 14.55 (Continued)
Conservation of energy.
PROBLEM 14.56
Three small identical spheres A, B, and C, which can slide on a horizontal, frictionless surface, are attached
to three strings of length l which are tied to a ring G. Initially, the spheres rotate clockwise about the ring
which moves along the x axis with a velocity
v
0
. Suddenly the ring breaks and the three spheres move freely in
the xy plane. Knowing that
(3.5 ft/s) ,
A
vj
(6.0 ft/s) ,
C
vi
16 in.a
and
9in.,d
determine (a) the initial
velocity of the ring, (b) the length l of the strings, (c) the rate in rad/s at which the spheres were rotating
about G.
SOLUTION
PROBLEM 14.56 (Continued)
Conservation of energy:
(c) Rate of rotation: 5.29 rad/s
PROBLEM 14.57
A stream of water with density
= 1000 kg/m
3
is
discharged from a nozzle at the rate of 0.06 m
3
/s. Using
Bernoulli’s equation the gage pressure P in the pipe just
upstream from the nozzle is
2
1
2
2
2vvP
. Knowing
the nozzle is held to the pipe by six flange bolts, determine
the tension in each bolt, neglecting the initial tension
caused by the tightening of the nuts.
SOLUTION
3
6
b
b
PROBLEM 14.58
A jet ski is placed in a channel and
is tethered so that it is stationary.
Water enters the jet ski with
velocity
v
1
and exits with velocity
v
2
. Knowing the inlet area is A
1
and the exit area is A
2
, determine
the tension in the tether.
SOLUTION
22 11
PROBLEM 14.59
The nozzle shown discharges a stream of water at a flow rate
Q 475 gal/min with a velocity v of magnitude 60 ft/s. The stream is split
into two streams with equal flow rates by a wedge that is kept in a fixed
position. Determine the components (drag and lift) of the force exerted by the
stream on the wedge. (1 ft3 7.48 gal.)
SOLUTION
PROBLEM 14.60
The nozzle shown discharges a stream of water at a flow rate
Q 500 gal/min with a velocity v of magnitude 48 ft/s. The stream is split
into two streams of equal flow rates by a wedge that is moving to the left at a
constant speed of 12 ft/s. Determine the components (drag and lift) of the
force exerted by the stream on the wedge. (1ft3 = 7.48 gal.)
SOLUTION
PROBLEM 14.61
A rotary power plow is used to remove snow
from a level section of railroad track. The plow
car is placed ahead of an engine that propels it
at a constant speed of 20 km/h. The plow car
clears 160 Mg of snow per minute, projecting it
in the direction shown with a velocity of 12 m/s
relative to the plow car. Neglecting friction,
determine (a) the force exerted by the engine
on the plow car, (b) the lateral force exerted by
the track on the plow.
s
(2.666.7)(12cos 30 12sin 30 5.5556 )
dt
x
PROBLEM 14.62
Tree limbs and branches are being fed at A at the rate of
5 kg/s into a shredder which spews the resulting wood
chips at C with a velocity of 20 m/s. Determine the
horizontal component of the force exerted by the shredder
on the truck hitch at D.
x
PROBLEM 14.63
Sand falls from three hoppers onto
a conveyor belt at a rate of 90 lb/s
for each hopper. The sand hits the
belt with a vertical velocity 1
v 10
ft/s and is discharged at A with a
horizontal velocity 2
v 13 ft/s.
Knowing that the combined mass
of the beam, belt system, and the
sand it supports is 1300 lb with a
mass center at G, determine the
reaction at E.
PROBLEM 14.64
The stream of water shown flows at a rate of 550 liters/min and moves
with a velocity of magnitude 18 m/s at both A and B. The vane is
supported by a pin and bracket at C and by a load cell at D that can exert
only a horizontal force. Neglecting the weight of the vane, determine the
components of the reactions at C and D.
SOLUTION
3
(1000 kg/m )(550 liters/min)(1 min)
dm Q
PROBLEM 14.65
The nozzle shown discharges water at the rate of 3
40 ft /min. Knowing
that at both A and B the stream of water moves with a velocity of
magnitude 75 ft/s and neglecting the weight of the vane, determine the
components of the reactions at C and D.
SOLUTION
33
62.4 lb/ft 40 ft /min 1.29193 lb s/ft
dm Q
y
PROBLEM 14.66
A stream of water flowing at a rate of 1.2 3
m/min
and moving
with a velocity of magnitude 30 m/s at both A and B is deflected
by a vane welded to a hinged plate. Knowing that the combined
mass of the vane and plate is 20 kg with the mass center at point
G, determine (a) the angle ,
(b) the reaction at C.
SOLUTION
3
kg m 1 min
dm Q
PROBLEM 14.67
A stream of water flowing at a rate of 1.2 3
m/min
and
moving with a velocity of magnitude v at both A and B is
deflected by a vane welded to a hinged plate. The combined
mass of the vane and plate is 20 kg with the mass center at
point G. Knowing that 45 ,
determine (a) the speed v of
the flow, (b) the reaction at C.
SOLUTION
3
kg m 1 min
dm Q
PROBLEM 14.68
Coal is being discharged from a first conveyor
belt at the rate of 120 kg/s. It is received at A by
a second belt that discharges it again at B.
Knowing that
1
3 m/sv
and
2
4.25 m/sv
and that the second belt assembly and the coal
it supports have a total mass of 472 kg,
determine the components of the reactions at C
and D.
SOLUTION
Velocity before impact at A:
PROBLEM 14.68 (Continued)
x
y
PROBLEM 14.69
The total drag due to air friction on a jet airplane traveling at 900 km/h is 35 kN. Knowing that the exhaust
velocity is 600 m/s relative to the airplane, determine the mass of air that must pass through the engine per
second to maintain the speed of 900 km/h in level flight.
SOLUTION
dm
PROBLEM 14.70
While cruising in level flight at a speed of 600 mi/h, a jet plane scoops in air at the rate of 200 lb/s and
discharges it with a velocity of 2100 ft/s relative to the airplane. Determine the total drag due to air friction on
the airplane.
PROBLEM 14.71
In order to shorten the distance required for landing, a jet airplane is equipped
with movable vanes, that partially reverse the direction of the air discharged by
each of its engines. Each engine scoops in the air at a rate of 120 kg/s and
discharges it with a velocity of 600 m/s relative to the engine. At an instant when
the speed of the airplane is 270 km/h, determine the reverse thrust provided by
each of the engines.
PROBLEM 14.72
The helicopter shown can produce a maximum downward air
speed of 80 ft/s in a 30-ft-diameter slipstream. Knowing that
the weight of the helicopter and its crew is 3500 lb and
assuming
3
0.076 lb/ft
for air, determine the maximum
load that the helicopter can lift while hovering in midair.
SOLUTION
dm
PH