978-0077687342 Chapter 15 Part 2

subject Type Homework Help
subject Pages 14
subject Words 2512
subject Authors Brian Self, E. Johnston, Ferdinand Beer, Phillip Cornwell

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page-pf1
PROBLEM 15.17
The earth makes one complete revolution on its axis in 23 h 56 min. Knowing that the mean radius of the
earth is 3960 mi, determine the linear velocity and acceleration of a point on the surface of the earth
(a) at the equator, (b) at Philadelphia, latitude 40° north, (c) at the North Pole.
SOLUTION
( )
3600 s
h
6
6
23 h 56 m 23.933 h
2 rad
(23.933 h)
72.925 10 rad/s
5280 ft
(3960 mi) mi
20.91 10 ft
radius of path
cos
R
r
R
π
f
=
=
= ×

=

= ×
=
=
ω
page-pf2
PROBLEM 15.18
A series of small machine components being moved by a
conveyor belt pass over a 120 mm radius idler pulley. At the
instant shown, the velocity of Point A is 300 mm/s to the left
and its acceleration is 180 mm/s2 to the right. Determine (a) the
angular velocity and angular acceleration of the idler pulley,
(b) the total acceleration of the machine component at B.
SOLUTION
300 mm/s
BA
vv= =
120 mm
B
r=
( ) 180 mm/s
Bt A
aa= =
(a)
,
BB
vr
ω
=
300 2.5 rad/s
120
B
B
v
r
ω
= = =
2.50 rad/s=ω
() ,
Bt B
ar
α
=
() 180 1.5 rad/s
120
Bt
B
a
r
α
= = =
2
1.500 rad/s=α
(b)
222
( ) (120)(2.5) 750 mm/s
Bn B
ar
ω
= = =
22 2 2 2
( ) ( ) (180) (750) 771 mm/s
750
tan , 76.5
180
B Bt Bn
aaa
ββ
= += + =
= = °
2
771 mm/s
B
=a
76.5°
page-pf3
PROBLEM 15.19
A series of small machine components being moved by a
conveyor belt pass over a 120-mm-radius idler pulley. At the
instant shown, the angular velocity of the idler pulley is 4 rad/s
clockwise. Determine the angular acceleration of the pulley for
which the magnitude of the total acceleration of the machine
component at B is
2
2400 mm/s .
page-pf4
PROBLEM 15.20
The belt sander shown is initially at rest. If the driving drum B
has a constant angular acceleration of 120 rad/s2 counter-
clockwise, determine the magnitude of the acceleration of the
belt at Point C when (a) t = 0.5 s, (b) t = 2 s.
SOLUTION
2
Bt n
B
page-pf5
PROBLEM 15.21
The rated speed of drum B of the belt sander shown is 2400 rpm.
When the power is turned off, it is observed that the sander
coasts from its rated speed to rest in 10 s. Assuming uniformly
decelerated motion, determine the velocity and acceleration of
Point C of the belt, (a) immediately before the power is turned
off, (b) 9 s later.
SOLUTION
02400 rpm
ω
=
C
page-pf6
PROBLEM 15.22
The two pulleys shown may be operated with the V belt in any of three
positions. If the angular acceleration of shaft A is 6 rad/s2 and if the
system is initially at rest, determine the time required for shaft B to
reach a speed of 400 rpm with the belt in each of the three positions.
6
A AA
Belt at left:
2 in.
4 in.
B
A
d
d=
3.49 st=
Belt in middle:
3 in.
3 in.
B
A
d
d=
6.98 st=
4 in.
2 in.
B
A
d
d=
page-pf7
PROBLEM 15.23
Three belts move over two pulleys without slipping in the speed
reduction system shown. At the instant shown the velocity of
Point A on the input belt is 2 ft/s to the right, decreasing at the
rate of 6 ft/s2. Determine, at this instant, (a) the velocity and
acceleration of Point C on the output belt, (b) the acceleration
of Point B on the output pulley.
( )
24
412
r
PROBLEM 15.23 (Continued)
page-pf8
(a) Velocity and acceleration of Point C.
2(3) 0.5 ft/s

B=a
page-pf9
PROBLEM 15.24
A gear reduction system consists of three gears A, B, and C. Knowing
that gear A rotates clockwise with a constant angular velocity
A
ω
= 600 rpm, determine (a) the angular velocities of gears B and C,
(b) the accelerations of the points on gears B and C which are in contact.
SOLUTION
(600)(2 )
600 rpm 20 rad/s.
π
ωπ
= = =
/
6
C
EC
r
2
658 in./s
C
=a
page-pfa
PROBLEM 15.25
A belt is pulled to the right between cylinders A and B. Knowing that the
speed of the belt is a constant 5 ft/s and no slippage occurs, determine (a) the
angular velocities of A and B, (b) the accelerations of the points which are in
contact with the belt.
SOLUTION
page-pfb
PROBLEM 15.26
Ring C has an inside radius of 55 mm and an outside radius of 60 mm and is
positioned between two wheels A and B, each of 24-mm outside radius.
Knowing that wheel A rotates with a constant angular velocity of 300 rpm
and that no slipping occurs, determine (a) the angular velocity of the ring C
and of wheel B, (b) the acceleration of the Points on A and B that are in
contact with C.
2
300 rpm 60
31.416 rad/s
24 mm
A
A
r
π
ω

=

=
=
1
2
24 mm
60 mm
55 mm
B
r
r
r
=
=
=
page-pfc
PROBLEM 15.26 (Continued)
B=a
page-pfd
PROBLEM 15.27
Ring B has an inside radius
2
r
and hangs from the horizontal
shaft A as shown. Shaft A rotates with a constant angular velocity
of 25 rad/s and no slipping occurs. Knowing that r1 = 12 mm,
r2 = 30 mm, and r3 = 40 mm, determine (a) the angular velocity
of ring B, (b) the accelerations of the points of shaft A and ring B
which are in contact, (c) the magnitude of the acceleration of
point D.
30 mm
page-pfe
PROBLEM 15.27 (Continued)
2
D
D
page-pff
PROBLEM 15.28
A plastic film moves over two drums. During a 4-s interval the
speed of the tape is increased uniformly from v0 = 2 ft/s to
v1 = 4 ft/s. Knowing that the tape does not slip on the drums,
determine (a) the angular acceleration of drum B, (b) the
number of revolutions executed by drum B during the 4-s
interval.
2
6 in./s
15 in.
t
B
B
a
r
α
= =
2
0.400 rad/s
B=α
(b) Angular displacement of drum B.
0
24 in./s 1.6 rad/s
v
page-pf10
PROBLEM 15.29
Cylinder A is moving downward with a velocity of 3 m/s when the brake is
suddenly applied to the drum. Knowing that the cylinder moves 6 m
downward before coming to rest and assuming uniformly accelerated motion,
determine (a) the angular acceleration of the drum, (b) the time required for
the cylinder to come to rest.
page-pf11
PROBLEM 15.30
The system shown is held at rest by the brake-and-drum system shown. After
the brake is partially released at
0,t=
it is observed that the cylinder moves
5 m in 4.5 s. Assuming uniformly accelerated motion, determine (a) the
angular acceleration of the drum, (b) the angular velocity of the drum at
3.5t=
s.
page-pf12
PROBLEM 15.31
A load is to be raised 20 ft by the hoisting system shown. Assuming gear A
is initially at rest, accelerates uniformly to a speed of 120 rpm in 5 s, and
then maintains a constant speed of 120 rpm, determine (a) the number of
revolutions executed by gear A in raising the load, (b) the time required to
raise the load.
4
f
ωπ
Total time elapsed:
5s
f
tt= +∆
10.14 s
f
t=
page-pf13
PROBLEM 15.32
A simple friction drive consists of two disks A and B. Initially, disk B has a
clockwise angular velocity of 500 rpm, and disk A is at rest. It is known that
disk B will coast to rest in 60 s. However, rather than waiting until both
disks are at rest to bring them together, disk A is given a constant angular
acceleration of 3 rad/s2 counterclockwise. Determine (a) at what time the
disks can be brought together if they are not to slip, (b) the angular velocity
of each disk as contact is made.
( )
2
0
0 52.360 0.87266 rad/s
60
BB
B
t
ωω
α
= = = −
( )( )
3 52.360 0.87266 157.08 2.618 in./s
vr t t
ω
== −=
page-pf14
PROBLEM 15.32 (Continued)

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