Mechanical Engineering Chapter 19 Problem The Slender Bar Released From Rest With The Horizontal Surface Smoothwhat

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subject Authors Anthony M. Bedford, Wallace Fowler

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Problem 19.38 The 8-kg slender bar is released from
rest with θ=60. The horizontal surface is smooth.
What is the bar’s angular velocity when θ=30.
2 m
Solution: The bar’s potential energy is
Problem 19.39 The mass and length of the bar are
m=4 kg and l=1.2 m. The spring constant is k=
180 N/m. If the bar is released from rest in the position
θ=10, what is its angular velocity when it has fallen
to θ=20?
k
Solution: If the spring is unstretched when θ=0, the stretch of
the spring is
580
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Problem 19.40 The 4-kg slender bar is pinned to a
2-kg slider at Aand to a 4-kg homogenous cylindrical
disk at B. Neglect the friction force on the slider and
assume that the disk rolls. If the system is released from
rest with θ=60, what is the bar’s angular velocity
when θ=0? (See Example 19.3.) 1 m
A
θ
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Problem 19.41* The sleeve Pslides on the smooth
horizontal bar. The mass of each bar is 4 kg and the
mass of the sleeve Pis 2 kg. If the system is released
from rest with θ=60, what is the magnitude of the
velocity of the sleeve Pwhen θ=40?
1.2 m
O
Q
P
θ
1.2 m
582
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Problem 19.42* The system is in equilibrium in the
position shown. The mass of the slender bar ABC is
6 kg, the mass of the slender bar BD is 3 kg, and the
mass of the slider at Cis 1 kg. The spring constant is
k=200 N/m. If a constant 100-N downward force is
applied at A, what is the angular velocity of the bar
ABC when it has rotated 20from its initial position?
A
1 m
1 m
B
1 m
Solution: Choose a coordinate system with the origin at Dand
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Problem 19.43* The masses of bars AB and BC are
5 kg and 3 kg, respectively. If the system is released
from rest in the position shown, what are the angular
velocities of the bars at the instant before the joint B
hits the smooth oor? 1 m
2 m 1 m
AB
C
Solution: The work done by the weights of the bars as they fall
is
y
A
584
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Problem 19.44* Bar AB weighs 5 lb. Each of the
sleeves Aand Bweighs 2 lb. The system is released
from rest in the position shown. What is the magnitude
of the angular velocity of the bar when sleeve B has
moved 3 in to the right?
9 in
4 in 12 in
y
x
B
A
Solution: Find the geometry in position 2
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Problem 19.45* Each bar has a mass of 8 kg and a
length of l m. The spring constant is k=100 N/m, and
the spring is unstretched when θ=0. If the system is
released from rest with the bars vertical, what is the mag-
nitude of the angular velocity of the bars when
θ=30?
θ
θ
k
586
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Problem 19.46* The system starts from rest with the
crank AB vertical. A constant couple Mexerted on the
crank causes it to rotate in the clockwise direction, com-
pressing the gas in the cylinder. Let sbe the displace-
ment (in meters) of the piston to the right relative to
its initial position. The net force toward the left exerted
on the piston by atmospheric pressure and the gas in
the cylinder is 350/(110s) N. The moment of inertia
of the crank about Ais 0.0003 kg-m2. The mass of the
connecting rod BC is 0.36 kg, and the center of mass of
the rod is at its midpoint. The connecting rod’s moment
of inertia about its center of mass is 0.0004 kg-m2. The
mass of the piston is 4.6 kg. If the clockwise angular
velocity of the crank AB is 200 rad/s when it has rotated
90from its initial position, what is M? (Neglect the
50 mm
A
M
B
C
125 mm
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Problem 19.47* In Problem 19.46, if the system starts
from rest with the crank AB vertical and the couple
M=40 N-m, what is the clockwise angular velocity of
AB when it has rotated 45from its initial position?
Problem 19.48 The moment of inertia of the disk
about Ois 22 kg-m2.Att=0, the stationary disk is
subjected to a constant 50 N-m torque.
(a) Determine the angular impulse exerted on the disk
from t=0tot=5s.
(b) What is the disk’s angular velocity at t=5s?
50 N-m
O
Problem 19.49 The moment of inertia of the jet
engine’s rotating assembly is 400 kg-m2. The assembly
starts from rest. At t=0, the engine’s turbine exerts
a couple on it that is given as a function of time by
M=6500 125tN-m.
(a) What is the magnitude of the angular impulse
exerted on the assembly from t=0tot=20 s?
(b) What is the magnitude of the angular velocity of
the assembly (in rpm) at t=20 s?
588
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Problem 19.50 An astronaut res a thruster of his
maneuvering unit, exerting a force T=2(1+t)N,
where tis in seconds. The combined mass of the
astronaut and his equipment is 122 kg, and the moment
of inertia about their center of mass is 45 kg-m2.
Modeling the astronaut and his equipment as a
rigid body, use the principle of angular impulse and
momentum to determine how long it takes for his angular
velocity to reach 0.1 rad/s. 300 mm
T
Solution: From the principle of impulse and angular momentum,
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Problem 19.51 The combined mass of the astronaut
and his equipment is 122 kg, and the moment of inertia
about their center of mass is 45 kg-m2. The maneuvering
unit exerts an impulsive force Tof 0.2-s duration, giving
him a counterclockwise angular velocity of 1 rpm.
(a) What is the average magnitude of the impulsive
force?
(b) What is the magnitude of the resulting change in
the velocity of the astronaut’s center of mass?
T
300 mm
Solution:
(a) From the principle of moment impulse and angular momentum,
590
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Problem 19.52 Aywheel attached to an electric
motor is initially at rest. At t=0, the motor exerts
a couple M=200e0.1tN-m on the ywheel. The
moment of inertia of the ywheel is 10 kg-m2.
(a) What is the ywheel’s angular velocity at t=10 s?
(b) What maximum angular velocity will the ywheel
attain?
Problem 19.53 A main landing gear wheel of a Boeing
777 has a radius of 0.62 m and its moment of inertia
is 24 kg-m2. After the plane lands at 75 m/s, the skid
marks of the wheel’s tire is measured and determined to
be 18 m in length. Determine the average friction force
exerted on the wheel by the runway. Assume that the
airplane’s velocity is constant during the time the tire
skids (slips) on the runway.
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Problem 19.54 The force a club exerts on a 0.045-kg
golf ball is shown. The ball is 42 mm in diameter and
can be modeled as a homogeneous sphere. The club is in
contact with the ball for 0.0006 s, and the magnitude of
the velocity of the ball’s center of mass after the ball is
hit is 36 m/s. What is the magnitude of the ball’s angular
velocity after it is hit?
2.5 mm
F
Problem 19.55 Disk Ainitially has a counterclock-
wise angular velocity ω0=50 rad/s. Disks Band Care
initially stationary. At t=0, disk Ais moved into con-
tact with disk B. Determine the angular velocities of
19.4.)
0.2 m
0.4 m 0.3 m
v0
Solution: The FBDs
Given:
mA=4kg,r
A=0.2 m,I
A=1/2mArA2=0.08 kg-m2
F2
F1
592
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Problem 19.56 In Example 19.5, suppose that in a
second test at a higher velocity the angular velocity
of the pole following the impact is ω=0.81 rad/s, the
horizontal velocity of its center of mass is v=7.3
m/s, and the duration of the impact is t =0.009 s.
Determine the magnitude of the average force the car
exerts on the pole in shearing off the supporting bolts.
Do so by applying the principle of angular impulse and
momentum in the form given by Eq. (19.32).
Solution: Using the data from Example 19.5 we write the linear
and angular impulse momentum equations for the pole. Fis the force
of the car and Sis the shear force in the bolts
(F S)t =mv
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Problem 19.57 The force exerted on the cue ball by
the cue is horizontal. Determine the value of hfor which
the ball rolls without slipping. (Assume that the frictional
force exerted on the ball by the table is negligible.)
h
R
Solution: From the principle of moment impulse and angular
594
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Problem 19.58 In Example 19.6, we neglected the
moments of inertia of the two masses mabout the axes
through their centers of mass in calculating the total
angular momentum of the person, platform, and masses.
Suppose that the moment of inertia of each mass about
the vertical axis through its center of mass is IM=
0.001 kg-m2. If the person’s angular velocity with her
arms extended to r1=0.6misω1=1 revolution per
second, what is her angular velocity ω2when she pulls
the masses inward to r2=0.2 m? Compare your result
r1
r1
r2
Solution: Using the numbers from Example 19.6, we conserve
angular momentum
HO1=(IP+2mr2
1+2IM1
=(0.4 kg-m2+2[4 kg][0.6m]
2+2[0.001 kg-m2])1rev
s
HO2=(IP+2mr2
2+2IM2
s.
Problem 19.59 Two gravity research satellites (mA=
250 kg, IA=350 kg-m2;mB=50 kg, IB=16 kg-m2)
are tethered by a cable. The satellites and cable
rotate with angular velocity ω0=0.25 rpm. Ground
controllers order satellite Ato slowly unreel 6 m of
additional cable. What is the angular velocity afterward?
A
B
0
ω
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Problem 19.60 The 2-kg bar rotates in the horizontal
plane about the smooth pin. The 6-kg collar Aslides on
the smooth bar. Assume that the moment of inertia of
the collar Aabout its center of mass is negligible; that
is, treat the collar as a particle. At the instant shown, the
angular velocity of the bar is ω0=60 rpm and the dis-
tance from the pin to the collar is r=1.8 m. Determine
the bar’s angular velocity when r=2.4m.
r
A
3 m
k
v0
Problem 19.61 The 2-kg bar rotates in the horizontal
plane about the smooth pin. The 6-kg collar Aslides
Solution: Angular momentum is conserved
1
Problem 19.62* The 2-kg bar rotates in the horizontal
plane about the smooth pin. The 6-kg collar Aslides
on the smooth bar. The moment of inertia of the collar
r=2.4m.
Solution: Angular momentum is conserved
1
T1=1
21
3(2kg)(3m)2+(6kg)(1.8 m)2+(0.2 kg-m2)(2 rad/s)2
596
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Problem 19.63 The circular bar is welded to the verti-
cal shafts, which can rotate freely in bearings at Aand B.
Let Ibe the moment of inertia of the circular bar and
shafts about the vertical axis. The circular bar has an
A
Problem 19.64 The 10-lb bar is released from rest
in the 45position shown. It falls and the end of the
bar strikes the horizontal surface at P. The coefcient
of restitution of the impact is e=0.6. When the bar
rebounds, through what angle relative to the horizontal
will it rotate?
3 ft
45
P
Solution: We solve the problem in three phases.
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Problem 19.65 The 10-lb bar is released from rest in
the 45position shown. It falls and the end of the bar
strikes the horizontal surface at P. The bar rebounds to
a position 10relative to the horizontal. If the duration
of the impact is 0.01 s, what is the magnitude of the
average vertical force the horizontal surface exerted on
the bar at P?
3 ft
45
P
598
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Problem 19.66 The 4-kg bar is released from rest in
the horizontal position above the xed projection at A.
The distance b=0.35 m. The impact of the bar with the
projection is plastic; that is, the coefcient of restitution
of the impact is e=0. What is the bar’s angular velocity
immediately after the impact?
1 m
0.2 m
b
A
Problem 19.67 The 4-kg bar is released from rest in
the bar’s center of mass to be zero immediately after the
impact? What is the bar’s angular velocity immediately
after the impact?
1 m
0.2 m
b
A

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