Mechanical Engineering Chapter 19 Problem The Moment Inertia The Rotor The Medical Centrifuge Kgm The Rotor

subject Type Homework Help
subject Pages 14
subject Words 5256
subject Authors Anthony M. Bedford, Wallace Fowler

Unlock document.

This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
page-pf1
Problem 19.1 The moment of inertia of the rotor of
0.8 N-m on it.
(a) How much work has the motor done on the rotor
when the rotor has rotated through four revolu-
tions?
(b) What is the rotor’s angular velocity (in rpm) when
it has rotated through four revolutions?
Problem 19.2 The 4-lb slender bar is 2 ft in length. It
started from rest in an initial position relative to the iner-
tial reference frame. When it is in the position shown,
the velocity of the end Ais 2i+6j(ft/s) and the bar has
a counterclockwise angular velocity of 12 rad/s. How
much work was done on the bar as it moved from its
initial position to its present position?
y
B
560
page-pf2
Problem 19.3 The 20-kg disk is at rest when the
it has rotated through four revolutions (a) by applying
the equation of angular motion M =1α, and (b) by
applying the principle of work and energy.
0.25 m
Solution:
page-pf3
Problem 19.4 The space station is initially not rotat-
ing. Its reaction control system exerts a constant couple
on it until it has rotated 90, then exerts a constant couple
of the same magnitude in the opposite direction so that
its angular velocity has decreased to zero when it has
undergone a total rotation of 180. The maneuver takes
6 hours. The station’s moment of inertia about the axis
of rotation is I=1.5×1010 kg-m2. How much work is
done in performing this maneuver? In other words, how
much energy had to be expended in the form of reaction
control fuel?
562
page-pf4
Problem 19.5 The helicopter’s rotor starts from rest.
Suppose that its engine exerts a constant 1200 ft-lb cou-
ple on the rotor and aerodynamic drag is negligible. The
rotor’s moment of inertia is I=400 slug-ft2.
(a) Use work and energy to determine the magnitude
of the rotor’s angular velocity when it has rotated
through ve revolutions.
(b) What average power is transferred to the rotor
while it rotates through ve revolutions?
Problem 19.6 The helicopter’s rotor starts from rest.
The moment exerted on it (in N-m) is given as a function
of the angle through which it has turned in radians by
M=6500 20θ. The rotor’s moment of inertia is I=
540 kg-m2. Determine the rotor’s angular velocity (in
rpm) when it has turned through 10 revolutions.
Solution: We will integrate to nd the work.
page-pf5
Problem 19.7 During extravehicular activity, an astro-
naut’s angular velocity is initially zero. She activates two
is 45 kg-m2. Use the principle of work and energy to
determine the angle through which she has rotated when
her angular velocity reaches 15per second.
T
Problem 19.8 The 8-kg slender bar is released from
rest in the horizontal position 1 and falls to position 2.
(a) How much work is done by the bar’s weight as it
falls from position 1 to position 2?
(b) How much work is done by the force exerted on
the bar by the pin support as the bar falls from
position 1 to position 2?
A
2 m
y
x
1
564
page-pf6
Problem 19.9 The 20-lb bar is released from rest in the
2
M
y
Solution: We will use the energy equation in the form
T1+V1+U12 =T2+V2
Problem 19.10 The object consists of an 8-lb slender
2
45
y
page-pf7
Problem 19.11 The object consists of an 8-lb slender
bar welded to a 12-lb circular disk. The object is released
from rest in position 1. Determine the xand ycompo-
nents of force exerted on the object by the pin support
when it is in position 2.
5 in
22 in
1
2
45
x
A
y
Solution: We rst determine the moment of inertia about the xed
point Aand the distance from Ato the center of mass.
22
We now write the equations of motion and the work energy equation
Problem 19.12 The mass of each box is 4 kg. The
radius of the pulley is 120 mm and its moment of inertia
is 0.032 kg-m2. The surfaces are smooth. If the system is
released from rest, how fast are the boxes moving when
the left box has moved 0.5 m to the right?
566
page-pf8
Problem 19.13 The mass of each box is 4 kg. The
radius of the pulley is 120 mm and its moment of iner-
tia is 0.032 kg-m2. The coefcient of kinetic friction
between the boxes and the surfaces is µk=0.12. If the
system is released from rest, how fast are the boxes mov-
ing when the left box has moved 0.5 m to the right? 30
Problem 19.14 The 4-kg bar is released from rest in
the horizontal position 1 and falls to position 2. The
unstretched length of the spring is 0.4 m and the spring
constant is k=20 N/m. What is the magnitude of the
bar’s angular velocity when it is in position 2. 60
1
A
0.6 m 1 m
page-pf9
Problem 19.15 The moments of inertia of gears
that can turn freely on their pin supports are IA=
0.002 kg-m2and IB=0.006 kg-m2. The gears are at
rest when a constant couple M=2 N-m is applied to
gear B. Neglecting friction, use principle of work and
60 mm
A
M
0
The kinetic energy is T2=1
2IAω2
A+1
2IBω2
B,
where ωB=−rA
rBωA,from which
0.09 ωA=399.5 rad/s
Problem 19.16 The moments of inertia of gears A
and Bare IA=0.02 kg-m2and IB=0.09 kg-m2.
Gear Ais connected to a torsional spring with
constant k=12 N-m/rad. If gear Bis given an initial
counterclockwise angular velocity of 10 rad/s with the
torsional spring unstretched, through what maximum
counterclockwise angle does gear Brotate? 200 mm
AB
=0.731 rad =41.9.
568
c
2008 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they
currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.
page-pfa
Problem 19.17 The moments of inertia of three
100 mm
From the principle of work and energy: U=T2T1, where T1=0
since the pulleys start from a stationary position. The work done is
U=θA
0
Mdθ =2(2π)(10)=40πN-m.
0.0065 =139.0 rad/s
Problem 19.18 Model the arm ABC as a single rigid
body. Its mass is 300 kg, and the moment of inertia
from rest with its center of mass 2 m above the ground
1.4 rad/s. How much work do the hydraulic cylinders do
on the arm in moving it from position 1 to position 2?
1.40 m
0.70 m
2.25 m
A
C
Solution: From the principle of work and energy: U=T2T1,
where T1=0 since the system starts from rest. The work done is
U=Ucylinders mg(h2h1), and the kinetic energy is
page-pfb
Problem 19.19 The mass of the circular disk is 5 kg
and its radius is R=0.2 m. The disk is stationary when
a constant clockwise couple M=10 N-m is applied to
it, causing the disk to roll toward the right. Consider the
disk when its center has moved a distance b=0.4m.
(a) How much work has the couple Mdone on the
disk?
(b) How much work has been done by the friction force
exerted on the disk by the surface?
(c) What is the magnitude of the velocity of the center
of the disk?
b
M
Problem 19.20 The mass of the homogeneous cylin-
drical disk is m=5 kg and its radius is R=0.2 m. The
angle β=15. The disk is stationary when a constant
clockwise couple M=10 N-m is applied to it. What is
the velocity of the center of the disk when it has moved
a distance b=0.4 m? (See Active Example 19.1.)
M
R
b
β
570
page-pfc
Problem 19.21 The mass of the stepped disk is 18 kg
and its moment of inertia is 0.28 kg-m2. If the disk is
released from rest, what is its angular velocity when the
center of the disk has fallen 1 m?
0.2
m
0.1
m
176.6=1
2mv2+1
22
Problem 19.22 The 100-kg homogenous cylindrical
disk is at rest when the force F=500 N is applied to a
cord wrapped around it, causing the disk to roll. Use the
principle of work and energy to determine the angular
velocity of the disk when it has turned one revolution.
F
300 mm
Solution: From the principle of work and energy: U=T2T1,
page-pfd
Problem 19.23 The 1-slug homogenous cylindrical
disk is given a clockwise angular velocity of 2 rad/s with
the spring unstretched. The spring constant is k=3 lb/ft.
If the disk rolls, how far will its center move to the right? 1 ft
k
U=−1
2kS2.
The kinetic energy is
T1=1
22+1
2mv2.
By inspection v=ωR, from which
T1=1
2m
2R2+mR2ω2=3
4mR2ω2=0.75(22)=3 ft-lb,
U=−T1,3
2S2=−3,
from which S=2=1.414 ft.
Problem 19.24 The system is released from rest. The
moment of inertia of the pulley is 0.03 slug-ft2. The
slanted surface is smooth. Determine the magnitude of
the velocity of the 10-lb weight when it has fallen 2 ft.
10 lb
6 in
20
5 lb
Solution: Use conservation of energy
572
page-pfe
Problem 19.25 The system is released from rest. The
moment of inertia of the pulley is 0.03 slug-ft2. The
coefcient of kinetic friction between the 5-lb weight
and the slanted surface is µk=0.3. Determine the mag-
nitude of the velocity of the 10-lb weight when it has
fallen 2 ft.
10 lb
6 in
20
5 lb
Solution: Use work energy.
Problem 19.26 Each of the cart’s four wheels weighs
3 lb, has a radius of 5 in, and has moment of inertia
I=0.01 slug-ft2. The cart (not including its wheels)
weighs 20 lb. The cart is stationary when the constant
horizontal force F=10 lb is applied. How fast is the
cart going when it has moved 2 ft to the right?
F
page-pff
Problem 19.27 The total moment of inertia of car’s
two rear wheels and axle is IR, and the total moment of
inertia of the two front wheels is IF. The radius of the
tires is R, and the total mass of the car, including the
wheels, is m. The car is moving at velocity v0when
the driver applies the brakes. If the car’s brakes exert a
constant retarding couple Mon each wheel and the tires
do not slip, determine the car’s velocity as a function
of the distance sfrom the point where the brakes are
applied.
s
Solution: When the car rolls a distance s, the wheels roll through
Problem 19.28 The total moment of inertia of the
car’s two rear wheels and axle is 0.24 kg-m2. The total
moment of inertia of the two front wheels is 0.2 kg-m2.
The radius of the tires is 0.3 m. The mass of the car,
including the wheels, is 1480 kg. The car is moving at
100 km/h. If the car’s brakes exert a constant retarding
couple of 650 N-m on each wheel and the tires do not
slip, what distance is required for the car to come to a
stop? (See Example 19.2.)
574
page-pf10
Problem 19.29 The radius of the pulley is R=
100 mm and its moment of inertia is I=0.1 kg-m2. The
mass m=5 kg. The spring constant is k=135 N/m.
The system is released from rest with the spring
unstretched. Determine how fast the mass is moving
when it has fallen 0.5 m.
R
km
Problem 19.30 The masses of the bar and disk are
14 kg and 9 kg, respectively. The system is released
1.2 m 0.3 m
Solution: The work done by the weights of the bar and disk as
they fall is
=6.72 kg-m2,
so the disk’s nal kinetic energy is
we obtain
page-pf11
Problem 19.31 The masses of the bar and disk are
14 kg and 9 kg, respectively. The system is released
from rest with the bar horizontal. Determine the angular
Problem 19.32 The 45-kg crate is pulled up the
inclined surface by the winch. The coefcient of kinetic
friction between the crate and the surface is µk=0.4.
The moment of inertia of the drum on which the cable is
being wound is IA=4 kg-m2. The crate starts from rest,
and the motor exerts a constant couple M=50 N-m
on the drum. Use the principle of work and energy to
determine the magnitude of the velocity of the crate
when it has moved 1 m.
0.15 m
AM
20°
Solution: The normal force is
M
576
page-pf12
Problem 19.33 The 2-ft slender bars each weigh 4 lb,
and the rectangular plate weighs 20 lb. If the system is
released from rest in the position shown, what is the
velocity of the plate when the bars are vertical? 45°
is the change in height, from which U=14.06 ft-lb. The kinetic
energy is
T2=1
2Wplate
gv2+21
2WbarL2
3gω2=0.3523v2,
where ω=v
Lhas been used.
Substitute into U=T2and solve: v=6.32 ft/s.
v
Problem 19.34 The mass of the 2-m slender bar is
8 kg. A torsional spring exerts a counterclockwise cou-
ple on the bar, where k=40 N-m/rad and θis in
radians. The bar is released from rest with θ=5. What
is the magnitude of the bar’s angular velocity when
θ=60?
k
u
page-pf13
Problem 19.35 The mass of the suspended object Ais
8 kg. The mass of the pulley is 5 kg, and its moment of
inertia is 0.036 kg-m2. If the force T=70N is applied
to the stationary system, what is the magnitude of the
velocity of Awhen it has risen 0.2 m?
T
0.4 m.
T1=0,V
1=0,T
2=1
Problem 19.36 The mass of the left pulley is 7 kg,
and its moment of inertia is 0.330 kg-m2. The mass of
the right pulley is 3 kg, and its moment of inertia is
0.054 kg-m2. If the system is released from rest, how
fast is the 18-kg mass moving when it has fallen 0.1 m?
0.3 m
0.2 m
9 kg
18 kg
A
B
Solution: When mass Cfalls a distance x, the center of pulley A
578
page-pf14
Problem 19.37 The 18-kg ladder is released from rest
with θ=10. The wall and oor are smooth. Modeling
the ladder as a slender bar, use conservation of energy to
determine the angular velocity of the bar when θ=40.
θ
2. The angular velocity
about this center is ω=2
Lv, where vis the velocity of the center
of mass of the ladder. The kinetic energy of the ladder is

Trusted by Thousands of
Students

Here are what students say about us.

Copyright ©2022 All rights reserved. | CoursePaper is not sponsored or endorsed by any college or university.