Mechanical Engineering Chapter 16 Problem The Coefcient Restitution The Same For Both Impacts Show That The

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subject Authors Anthony M. Bedford, Wallace Fowler

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Problem 16.82 If the coefcient of restitution is the
same for both impacts, show that the cue ball’s path
after two banks is parallel to its original path.
Solution: The strategy is to treat the two banks as two successive
oblique central impacts. Denote the path from the cue ball to the rst
294
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Problem 16.83 The velocity of the 170-g hockey puck
is vP=10i4j(m/s). If you neglect the change in the
velocity vS=vSjof the stick resulting from the impact,
and if the coefcient of restitution is e=0.6, what should
vSbe to send the puck toward the goal? 20°
Direction
of goal
x
y
vP
vS
Solution: The strategy is to treat the collision as an oblique cen-
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Problem 16.84 In Problem 16.83, if the stick responds
to the impact the way an object with the same mass as
the puck would, and if the coefcient of restitution is
e=0.6, what should vSbe to send the puck toward
the goal?
Problem 16.85 At the instant shown (t1=0), the posi-
tion of the 2-kg object’s center of mass is r=6i+4j+
2k(m) and its velocity is v=−16i+8j12k(m/s).
No external forces act on the object. What is the object’s
angular momentum about the origin Oat t2=1s?
O
r
x
y
z
Problem 16.86 Suppose that the total external force
(a) Use Newton’s second law to determine the object’s
Solution:
v=t2
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Problem 16.87 A satellite is in the elliptic earth orbit
shown. Its velocity at perigee Ais 8640 m/s. The radius
of the earth is 6370 km.
(a) Use conservation of angular momentum to deter-
mine the magnitude of the satellite’s velocity at
apogee C.
(b) Use conservation of energy to determine the mag-
nitude of the velocity at C.
(See Example 16.8.)
B
A
C
13,900 km
Solution:
Problem 16.88 For the satellite in Problem 16.87,
determine the magnitudes of the radial velocity vrand
transverse velocity vθat B. (See Example 16.8.)
Solution: Use conservation of energy to nd the velocity
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Problem 16.89 The bar rotates in the horizontal plane
about a smooth pin at the origin. The 2-kg sleeve A
slides on the smooth bar, and the mass of the bar is
negligible in comparison to the mass of the sleeve.
The spring constant k=40 N/m, and the spring is
unstretched when r=0. At t=0, the radial position
of the sleeve is r=0.2 m and the angular velocity of
the bar is w0=6 rad/s. What is the angular velocity of
the bar when r=0.25 m?
r
A
v0
k
Problem 16.90 At t=0, the radial position of the
sleeve Ain Problem 16.89 is r=0.2 m, the radial
velocity of the sleeve is vr=0 and the angular velocity
of the bar is w0=6 rad/s. What are the angular velocity
of the bar and the radial velocity of the sleeve when
r=0.25 m?
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Problem 16.91 A 2-kg disk slides on a smooth
horizontal table and is connected to an elastic cord whose
tension is T=6rN, where ris the radial position of
the disk in meters. If the disk is at r=1 m and is given
an initial velocity of 4 m/s in the transverse direction,
what are the magnitudes of the radial and transverse
components of its velocity when r=2 m? (See Active
Example 16.7.)
r
Solution: The strategy is to (a) use the principle of conservation
Problem 16.92 In Problem 16.91, determine the max-
imum value of rreached by the disk.
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Problem 16.93 A 1-kg disk slides on a smooth
horizontal table and is attached to a string that passes
through a hole in the table.
(a) If the mass moves in a circular path of constant
radius r=1 m with a velocity of 2 m/s, what is
the tension T?
(b) Starting from the initial condition described in
part (a), the tension Tis increased in such a way
that the string is pulled through the hole at a
constant rate until r=0.5 m. Determine the value
of Tas a function of rwhile this is taking place.
r
T
Solution:
T=−mv2/rer
|T|=(1)(2)2/1=4N
(b) By conservation of angular momentum,
mr0v0=mrvT,vT=r0v0
r
and from Newton’s second law
|T|=mv2
T/r =mr0v0
r2
/r
T=(1)(1)(2)
r2#r=4/r3N
r
V
er
m
Problem 16.94 In Problem 16.93, how much work is
done on the mass in pulling the string through the hole
as described in part (b)?
Solution: The work done is
0r2
0
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Problem 16.95 Two gravity research satellites (mA=
250 kg, mB=50 kg)are tethered by a cable. The
satellites and cable rotate with angular velocity ω0=
0.25 revolution per minute. Ground controllers order
satellite Ato slowly unreel 6 m of additional cable. What
is the angular velocity afterward?
A
B
0
12 m
ω
Solution: The satellite may be rotating in (a) a vertical plane, or
from which WA=−mAgjand WB=−mBgj(or, alternatively).
The angular momentum impulse is
A repeat of the argument above for any additional length of cable
ω=H
32mA+152mB=0.0116 rad/s =0.1111 rpm
Case (b): Assume that the system rotates in the x-zplane, with
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Problem 16.96 The astronaut moves in the x–y plane
at the end of a 10-m tether attached to a large space
station at O. The total mass of the astronaut and his
equipment is 120 kg.
(a) What is the astronaut’s angular momentum about
Obefore the tether becomes taut?
(b) What is the magnitude of the component of his
velocity perpendicular to the tether immediately
after the tether becomes taut?
y
2i (m /s)
6 m
Ox
(120)200
=−1440k(kg-m2/s).
(b) From conservation of angular momentum,
(b) H=−1440k=−(10)(120)vk, from which
v=1440
(10)(120)=1.2 m/s
6 m
10 m
Problem 16.97 The astronaut moves in the xyplane
at the end of a 10-m tether attached to a large space
station at O. The total mass of the astronaut and his
equipment is 120 kg. The coefcient of restitution of the
“impact” that occurs when he comes to the end of the
tether is e=0.8. What are the xand ycomponents of
his velocity immediately after the tether becomes taut?
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Problem 16.98 A ball suspended from a string that
goes through a hole in the ceiling at Omoves with
velocity vAin a horizontal circular path of radius rA. The
string is then drawn through the hole until the ball moves
with velocity vBin a horizontal circular path of radius
rB. Use the principle of angular impulse and momentum
to show that rAvA=rBvB.
Strategy: Let ebe a unit vector that is perpendicular
to the ceiling. Although this is not a central-force
problem — the ball’s weight does not point toward
O— you can show that e·(r×(F)=0, so that e·HO
is conserved.
O
B
rB
rA
A
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Problem 16.99 The Cheverton re-ghting and rescue
boat can pump 3.8 kg/s of water from each of its two
pumps at a velocity of 44 m/s. If both pumps point in
the same direction, what total force do they exert on the
boat.
Problem 16.100 The mass ow rate of water through
the nozzle is 1.6 slugs/s. Determine the magnitude of the
horizontal force exerted on the truck by the ow of the
water.
20 ft
35 ft
12 ft
20°
Solution: We must determine the velocity with which the water
304
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Problem 16.101 The front-end loader moves at a
constant speed of 2 mi/h scooping up iron ore. The
constant horizontal force exerted on the loader by the
road is 400 lb. What weight of iron ore is scooped up
in 3 s?
Problem 16.102 The snowblower moves at 1 m/s and
scoops up 750 kg/s of snow. Determine the force exerted
by the entering ow of snow.
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Problem 16.103 The snowblower scoops up 750 kg/s
of snow. It blows the snow out the side at 45above
the horizontal from a port 2 m above the ground and the
snow lands 20 m away. What horizontal force is exerted
on the blower by the departing ow of snow?
vf=13.36 m/s.
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Problem 16.104 A nozzle ejects a stream of water hor-
izontally at 40 m/s with a mass ow rate of 30 kg/s, and
the stream is deected in the horizontal plane by a plate.
Determine the force exerted on the plate by the stream
in cases (a), (b), and (c). (See Example 16.11.)
x
45°
y
(a)
x
y
(b)
x
y
(c)
Solution: Apply the strategy used in Example 16.7. The exit veloc-
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Problem 16.105* A stream of water with velocity
80i(m/s)and a mass ow of 6 kg/s strikes a turbine
blade moving with constant velocity 20i(m/s).
(a) What force is exerted on the blade by the water?
(b) What is the magnitude of the velocity of the water
as it leaves the blade?
y
x
70°
20 m/s
80 m/s
Solution: Denote the xed reference frame as the nozzle frame,
and the moving blade frame as the blade rest frame. Assume that the
in the blade rest frame is not modied by the velocity of the blade.
Denote the velocity of the blade by vB. The inlet velocity of the water
vB
θ′
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Problem 16.106 At the instant shown, the nozzle Aof
the lawn sprinkler is located at (0.1,0,0)m. Water exits
each nozzle at 8 m/s relative to the nozzle with a mass
ow rate of 0.22 kg/s. At the instant shown, the ow
relative to the nozzle at Ais in the direction of the unit
vector
e=1
3i1
3j+1
3k.
Determine the total moment about the zaxis exerted on
the sprinkler by the ows from all four nozzles. x
y
A
Solution:
f
Problem 16.107 A 45-kg/s ow of gravel exits the
chute at 2 m/s and falls onto a conveyor moving at
0.3 m/s. Determine the components of the force exerted
θ
x
0.3 m/s
Solution: The horizontal component of the velocity of the gravel
ow is vx=2 cos 45=2 m/s. From Newton’s second law, (using
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Problem 16.108 Solve Problem 16.107 if θ=30.
Solution: Use the solution to Problem 16.107 as appropriate. From
The angle of impact:
Problem 16.109 Suppose that you are designing a toy
car that will be propelled by water that squirts from
an internal tank at 10 ft/s relative to the car. The total
weight of the car and its water “fuel” is to be 2 lb. If
you want the car to achieve a maximum speed of 12 ft/s,
what part of the total weight must be water?
20°
Problem 16.110 The rocket consists of a 1000-kg
payload and a 9000-kg booster. Eighty percent of the
booster’s mass is fuel, and its exhaust velocity is
1200 m/s. If the rocket starts from rest and external
forces are neglected, what velocity will it attain? (See
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Problem 16.111* The rocket consists of a 1000-kg
payload and a booster. The booster has two stages whose
total mass is 9000 kg. Eighty percent of the mass of
each stage is fuel, and the exhaust velocity of each stage
is 1200 m/s. When the fuel of stage 1 is expended,
it is discarded and the motor of stage 2 is ignited.
Assume that the rocket starts from rest and neglect
external forces. Determine the velocity attained by the
rocket if the masses of the stages are m1=6000 kg and
m2=3000 kg. Compare your result to the answer to
Problem 16.110.
Payload
12
Solution:
Problem 16.112 A rocket of initial mass m0takes off
straight up. Its exhaust velocity vfand the mass ow rate
of its engine mf=dm
f/dt are constant. Show that, dur-
ing the initial part of the ight, when aerodynamic drag
is negligible, the rocket’s upward velocity as a function
of time is
v=vfln m0
m0−˙mftgt.
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Problem 16.113 The mass of the rocket sled in
Active Example 16.9 is 440 kg. Assuming that the only
signicant force acting on the sled in the direction of its
motion is the force exerted by the ow of water entering
it, what distance is required for the sled to decelerate
from 300 m/s to 100 m/s?
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Problem 16.114* Suppose that you grasp the end of a
chain that weighs 3 lb/ft and lift it straight up off the
oor at a constant speed of 2 ft/s.
(a) Determine the upward force Fyou must exert as
a function of the height s.
(b) How much work do you do in lifting the top of the
chain to s=4 ft?
Strategy: Treat the part of the chain you have lifted
as an object that is gaining mass.
s
Solution: The force is the sum of the “mass ow” reaction and
Problem 16.115* Solve Problem 16.114, assuming
that you lift the end of the chain straight up off the
oor with a constant acceleration of 2 ft/s2.
Solution: Assume that the velocity is zero at s=0. The mass
from which

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