Vector Mechanics for Engineers: Statics and Dynamics

Transcription

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13.9 A package is projected 10 m up a 15° incline so that it just reaches
Kinetics of Particles: Energy and Momentum
Methods
the top of the incline with zero velocity. Knowing that the coefficient of kinetic friction between the package and the incline is
0.12, determine (a) the initial velocity of the package at A, (b) the
velocity of the package as it returns to its original position.
10 m
d
B
C
15°
A
Fig. P13.9 and P13.10
13.10 A package is projected up a 15° incline at A with an initial velocity
of 8 m/s. Knowing that the coefficient of kinetic friction between
the package and the incline is 0.12, determine (a) the maximum
distance d that the package will move up the incline, (b) the velocity of the package as it returns to its original position.
13.11 Boxes are transported by a conveyor belt with a velocity v0 to a
fixed incline at A where they slide and eventually fall off at B.
Knowing that mk 5 0.40, determine the velocity of the conveyor
belt if the boxes leave the incline at B with a velocity of 8 ft/s.
v0
20 ft
A
B
15°
Fig. P13.11 and P13.12
13.12 Boxes are transported by a conveyor belt with a velocity v0 to a
fixed incline at A where they slide and eventually fall off at B.
Knowing that mk 5 0.40, determine the velocity of the conveyor
belt if the boxes are to have zero velocity at B.
13.13 Packages are thrown down an incline at A with a velocity of 1 m/s.
1 m/s
d
A
2 m/s
The packages slide along the surface ABC to a conveyor belt which
moves with a velocity of 2 m/s. Knowing that mk 5 0.25 between
the packages and the surface ABC, determine the distance d if the
packages are to arrive at C with a velocity of 2 m/s.
13.14 Packages are thrown down an incline at A with a velocity of 1 m/s.
C
30°
B
7m
Fig. P13.13 and P13.14
The packages slide along the surface ABC to a conveyor belt which
moves with a velocity of 2 m/s. Knowing that d 5 7.5 m and mk 5 0.25
between the packages and all surfaces, determine (a) the speed of
the package at C, (b) the distance a package will slide on the conveyor belt before it comes to rest relative to the belt.
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13.15 The subway train shown is traveling at a speed of 30 mi/h when
the brakes are fully applied on the wheels of cars B and C, causing
them to slide on the track, but are not applied on the wheels of
car A. Knowing that the coefficient of kinetic friction is 0.35
between the wheels and the track, determine (a) the distance
required to bring the train to a stop, (b) the force in each
coupling.
30 mi/h
A
40 tons
B
50 tons
C
40 tons
Fig. P13.15
13.16 Solve Prob. 13.15 assuming that the brakes are applied only on the
wheels of car A.
13.17 A trailer truck enters a 2 percent downhill grade traveling at
108 km/h and must slow down to 72 km/h in 300 m. The cab has
a mass of 1800 kg and the trailer 5400 kg. Determine (a) the average braking force that must be applied, (b) the average force
exerted on the coupling between cab and trailer if 70 percent
of the braking force is supplied by the trailer and 30 percent by
the cab.
108 km/h
72 km/h
2% down grade
CROSS COUNTRY MOVERS
300 m
Fig. P13.17
13.18 A trailer truck enters a 2 percent uphill grade traveling at 72 km/h
and reaches a speed of 108 km/h in 300 m. The cab has a mass
of 1800 kg and the trailer 5400 kg. Determine (a) the average force
at the wheels of the cab, (b) the average force in the coupling
between the cab and the trailer.
108 km/h
72 km/h
2% up grade
300 m
Fig. P13.18
Problems
773
774
Methods
13.19 Two identical blocks are released from rest. Neglecting the mass
of the pulleys and the effect of friction, determine (a) the velocity
of block B after it has moved 2 m, (b) the tension in the cable.
2 kg
B
2 kg
A
60°
60°
Fig. P13.19 and P13.20
13.20 Two identical blocks are released from rest. Neglecting the mass
of the pulleys and knowing that the coefficients of static and
kinetic friction are mS 5 0.30 and mk 5 0.20, determine (a) the
velocity of block B after it has moved 2 m, (b) the tension in
the cable.
13.21 The system shown is at rest when a constant 150-N force is applied
to collar B. (a) If the force acts through the entire motion, determine the speed of collar B as it strikes the support at C. (b) After
what distance d should the 150-N force be removed if the collar
is to reach support C with zero velocity?
600 mm
8 kg
B
C
150 N
3 kg
A
h
Fig. P13.22
B
A
Fig. P13.21
13.22 Blocks A and B have masses of 11 kg and 5 kg, respectively, and
they are both at a height h 5 2 m above the ground when the
system is released from rest. Just before hitting the ground block A
is moving at a speed of 3 m/s. Determine (a) the amount of energy
dissipated in friction by the pulley, (b) the tension in each portion
of the cord during the motion.
776
Methods
13.26 A 10-lb block is attached to an unstretched spring of constant
k 5 12 lb/in. The coefficients of static and kinetic friction between
the block and the plane are 0.60 and 0.40, respectively. If a force
F is slowly applied to the block until the tension in the spring
reaches 20 lb and then suddenly removed, determine (a) the speed
of the block as it returns to its initial position, (b) the maximum
speed achieved by the block.
k = 12 lb/in.
F
10 lb
Fig. P13.26 and P13.27
13.27 A 10-lb block is attached to an unstretched spring of constant k 5
3 kg
2 kg
12 lb/in. The coefficients of static and kinetic friction between the
block and the plane are 0.60 and 0.40, respectively. If a force F is
applied to the block until the tension in the spring reaches 20 lb
and then suddenly removed, determine (a) how far the block will
move to the left before coming to a stop, (b) whether the block
will then move back to the right.
13.28 A 3-kg block rests on top of a 2-kg block supported by but not
Fig. P13.28
attached to a spring of constant 40 N/m. The upper block is suddenly
removed. Determine (a) the maximum speed reached by the 2-kg
block, (b) the maximum height reached by the 2-kg block.
13.29 Solve Prob. 13.28, assuming that the 2-kg block is attached to the
spring.
13.30 An 8-lb collar C slides on a horizontal rod between springs
A and B. If the collar is pushed to the right until spring B is compressed 2 in. and released, determine the distance through which
the collar will travel assuming (a) no friction between the collar
and the rod, (b) a coefficient of friction mk 5 0.35.
16 in.
A
6 lb
k = 18 lb/in.
6 in.
C
B
k = 12 lb/in.
Fig. P13.30
13.31 A 6-lb block is attached to a cable and to a spring as shown.
Fig. P13.31
The constant of the spring is k 5 8 lb/in. and the tension in the
cable is 3 lb. If the cable is cut, determine (a) the maximum displacement of the block, (b) the maximum speed of the block.
/Users/user-s172/Desktop/Tempwork/03:12:2008/MHDQ077:Beer:204
Problems
13.39 The sphere at A is given a downward velocity v0 and swings in a
vertical circle of radius l and center O. Determine the smallest
velocity v0 for which the sphere will reach point B as it swings
about point O (a) if AO is a rope, (b) if AO is a slender rod of
negligible mass.
B
13.40 The sphere at A is given a downward velocity v0 of magnitude
5 m/s and swings in a vertical plane at the end of a rope of length
l 5 2 m attached to a support at O. Determine the angle u at
which the rope will break, knowing that it can withstand a maximum tension equal to twice the weight of the sphere.
l
A
O
q
v0
13.41 A section of track for a roller coaster consists of two circular arcs
AB and CD joined by a straight portion BC. The radius of AB is
90 ft and the radius of CD is 240 ft. The car and its occupants, of
total weight 560 lb reach point A with practically no velocity and
then drop freely along the track. Determine the normal force
exerted by the track on the car as the car reaches point B. Ignore
air resistance and rolling resistance.
Fig. P13.39 and P13.40
A
B
90 ft
r = 240 ft
40°
C
60 ft
D
Fig. P13.41 and P13.42
13.42 A section of track for a roller coaster consists of two circular arcs
AB and CD joined by a straight portion BC. The radius of AB is
90 ft and the radius of CD is 240 ft. The car and its occupants, of
total weight 560 lb, reach point A with practically no velocity and
then drop freely along the track. Determine the maximum and
minimum values of the normal force exerted by the track on the
car as the car travels from A to D. Ignore air resistance and rolling
resistance.
0.8 m
B
O
30°
0.4 m
A
q
13.43 A small sphere B of mass m is released from rest in the position
shown and swings freely in a vertical plane, first about O and then
about the peg A after the cord comes in contact with the peg.
Determine the tension in the cord (a) just before the sphere
comes in contact with the peg, (b) just after it comes in contact
with the peg.
Fig. P13.43
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Methods
v
B
13.44 A small block slides at a speed v 5 8 ft/s on a horizontal surface
at a height h 5 3 ft above the ground. Determine (a) the angle u
at which it will leave the cylindrical surface BCD, (b) the distance
x at which it will hit the ground. Neglect friction and air
resistance.
C
h
q
13.45 A small block slides at a speed v on a horizontal surface. Knowing
D
E
that h 5 2.5 m, determine the required speed of the block if it is
to leave the cylindrical surface BCD when u 5 40°.
x
Fig. P13.44 and P13.45
13.46 (a) A 120-lb woman rides a 15-lb bicycle up a 3-percent slope at
a constant speed of 5 ft/s. How much power must be developed
by the woman? (b) A 180-lb man on an 18-lb bicycle starts down
the same slope and maintains a constant speed of 20 ft/s by braking. How much power is dissipated by the brakes? Ignore air resistance and rolling resistance.
20 ft /s
5 ft /s
3% slope
(a)
(b)
Fig. P13.46
13.47 A power specification formula is to be derived for electric motors
which drive conveyor belts moving solid material at different
rates to different heights and distances. Denoting the efficiency
of a motor by h and neglecting the power needed to drive the
belt itself, derive a formula (a) in the SI system of units for
the power P in kW, in terms of the mass flow rate m in kg/h,
the height b and horizontal distance l in meters and (b) in U.S.
customary units, for the power in hp, in terms of the material
flow rate w in tons/h, and the height b and horizontal distance
l in feet.
b
l
Fig. P13.47
Problems
13.59 An elastic cord is stretched between two points A and B, located
16 in. apart in the same horizontal plane. When stretched directly
between A and B, the tension is 10 lb. The cord is then stretched
as shown until its midpoint C has moved through 6 in. to C9; a force
of 60 lb is required to hold the cord at C9. A 0.2-lb pellet is placed
at C9, and the cord is released. Determine the speed of the pellet
as it passes through C.
A
8 in.
60 lb
C⬘
C
13.60 A 1.5-kg collar is attached to a spring and slides without friction
along a circular rod in a horizontal plane. The spring has an undeformed length of 150 mm and a constant k 5 400 N/m. Knowing
that the collar is in equilibrium at A and is given a slight push to
get it moving, determine the velocity of the collar (a) as it passes
through B, (b) as it passes through C.
8 in.
B
6 in.
Fig. P13.59
150 mm
C
C
125 mm
O
A
100 mm
k
A
175 mm
B
200 mm
Fig. P13.60
13.61 A 500-g collar can slide without friction on the curved rod BC in
a horizontal plane. Knowing that the undeformed length of the
spring is 80 mm and that k 5 400 kN/m, determine (a) the velocity
that the collar should be given at A to reach B with zero velocity,
(b) the velocity of the collar when it eventually reaches C.
13.62 A 3-kg collar can slide without friction on a vertical rod and is
resting in equilibrium on a spring. It is pushed down, compressing
the spring 150 mm, and released. Knowing that the spring constant is k 5 2.6 kN/m, determine (a) the maximum height h
reached by the collar above its equilibrium position, (b) the
maximum speed of the collar.
h
3 kg
k = 2.6 kN/m
Fig. P13.62
B
Fig. P13.61
795
Problems
13.67 The system shown is in equilibrium when f 5 0. Knowing that
initially f 5 90° and that block C is given a slight nudge when
the system is in that position, determine the speed of the block as
it passes through the equilibrium position f 5 0. Neglect the
weight of the rod.
797
D
k = 600 lb/ft
1.1 ft
A
f
B
C
25 lb
2.1 ft
0.3 ft
Fig. P13.67
13.68 A spring is used to stop a 50-kg package which is moving down a
2 m/s
20° incline. The spring has a constant k 5 30 kN/m and is held by
cables so that it is initially compressed 50 mm. Knowing that the
velocity of the package is 2 m/s when it is 8 m from the spring and
neglecting friction, determine the maximum additional deformation of the spring in bringing the package to rest.
50 kg
8m
13.69 Solve Prob. 13.68 assuming the kinetic coefficient of friction
between the package and the incline is 0.2.
Cable
20°
Fig. P13.68
13.70 A 300-g pellet is released from rest at A and slides with friction
along the surface shown. Determine the force exerted on the pellet by the surface (a) just before the pellet reaches B, (b) immediately after it has passed through B.
A
y
30°
r = 0.8 m
1.2
m
D
B
C
r = 6 in.
Fig. P13.70 and P13.71
B
13.71 A 300-g pellet is released from rest at A and slides without friction
along the surface shown. Determine the force exerted on the pellet by the surface (a) just before the pellet reaches C, (b) immediately after it has passed through C.
C
z
13.72 A 1.2-lb collar can slide without friction along the semicircular rod
BCD. The spring is of constant 1.8 lb/in and its undeformed length
is 8 in. Knowing that the collar is released from rest at B, determine (a) the speed of the collar as it passes through C, (b) the
force exerted by the rod on the collar at C.
12 in.
A
Fig. P13.72
3 in.
x
Problems
13.104 A spacecraft is describing an elliptic orbit of minimum altitude
hA 5 2400 km and maximum altitude hB 5 9600 km above the
surface of the earth. Determine the speed of the spacecraft at A.
hA
vB
O
A
hB
B
6370 km
vA
Fig. P13.104
13.105 A spacecraft describing an elliptic orbit about the earth has a maxi-
mum speed vA 5 26.3 3 103 km/h at A and a minimum speed
vB 5 18.5 3 103 km/h at B. Determine the altitude of the spacecraft at B.
13.106 Upon the LEM’s return to the command module, the Apollo
spacecraft of Prob. 13.101 was turned around so that the LEM
faced to the rear. The LEM was then cast adrift with a velocity of
200 m/s relative to the command module. Determine the magnitude and direction (angle f formed with the vertical OC) of the
velocity vC of the LEM just before it crashed at C on the moon’s
surface.
13.107 A satellite is projected into space with a velocity v0 at a distance
r 0 from the center of the earth by the last stage of its launching
rocket. The velocity v0 was designed to send the satellite into a
circular orbit of radius r 0. However, owing to a malfunction of
control, the satellite is not projected horizontally but at an angle
a with the horizontal and, as a result, is propelled into an elliptic
orbit. Determine the maximum and minimum values of the distance from the center of the earth to the satellite.
v0
rmax
a
r0
rmin
Fig. P13.107
f
vC
C
140 km
1740 km
B
vB
Fig. P13.106
O
803

Vector Mechanics for Engineers: Statics and Dynamics

Transcription

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