Chapter 18. Homework problems
18.3 General expressions for velocity and acceleration
You will also need to pick a range
of times. Your plot should include
the instant at which the bug walks
through the origin. Make sure your
and - axes are drawn to the same
scale. A computer plot would be
nice.
Calculate the radius of curvature of
the bug’s path as it goes through the
origin.
Accurately draw (say, on the computer) the osculating circle when
the bug is at the origin on the picture you drew for (a) above.
Force. What is the force on the
bugs feet from the turntable when
she starts her trip? Draw this force
as an arrow on your picture of the
bug’s path.
Force. What is the force on the
bugs feet when she is in the middle of the turntable? Draw this force
as an arrow on your picture of the
bug’s path.
18.3.4 Arm OC rotates with constant rate
. Disc D of radius rotates about point
C at constant rate
measured with respect to the arm OC. What are the absolute
velocity and acceleration of point P on the
disc,
and
? (To do this problem will
require defining a moving frame of reference. More than one choice is possible.)
18.3.3 A small bug is crawling on a
straight line scratched on an old record.
The scratch is a distance
cm from
the center of the turntable. The turntable
is turning clockwise at a constant angular rate
rad s. The bug is walking, relative to the turntable, at a constant
rate
cm s, straight along the
scratch in the -direction. At the instant of
interest, everything is aligned as shown in
figure. The bug has a mass
gram.
a) What is the bug’s velocity?
b) What is the bug’s acceleration?
c) What is the sum of all forces acting
on the bug?
d) Sketch the path of the bug in the
neighborhood of its location at the
time of interest (indicate the direction the bug is moving on this path).
18.3.5 For the configuration in problem 18.3.4 what is the absolute angular velocity of the disk, ?
b)
c)
d)
e)
y
ω
vB/T
bug
x
981
y
z
x
ωD , ω̇D
C
D
a
B
E
a) Pick a suitable moving frame and
do the problem.
b) Pick another suitable moving frame
and redo the problem. Make sure
the answers are the same.
disk D
#
r
ω1
O
C
ω2
P
θ
Filename:Mikes93p1
Problem 18.3.4
18.3.6 For the configuration in problem 18.3.4, taking
to be the angular velocity of disk D relative to the rod, what
is the absolute angular acceleration of the
disk, ? What is the absolute angular acceleration of the disk if
and
are not
constant?
Filename:Mikes93p2
Problem 18.3.7
18.3.8 A small
kg toy train engine is
going clockwise at a constant rate (relative
to the track) of m s on a circular track of
radius m. The track itself is on a turntable
B that is rotating counter- clockwise at a
constant rate of rad s. The dimensions
are as shown. At the instant of interest the
train is pointing due south ( j) and is at
the center of the turntable.
a) What is the velocity of the train relative to the turntable B ?
b) What is the absolute velocity of the
train ?
c) What is the acceleration of the train
relative to the turntable B ?
d) What is the absolute acceleration of
the train ?
e) What is the total force acting on the
train?
f) Sketch the path of the train for
one revolution of the turntable (surprise)?
ˆ North
18.3.7 A turntable oscillates with displacement
sin
. The disc of the
turntable rotates with angular speed and
acceleration
and
. A small bug
walks along line
with velocity relative to the turntable. At the instant shown,
the turntable is at its maximum amplitude
, the line
is currently aligned
with the -axis, and the bug is passing
through point on line
. Point is a
distance from the center of the turntable,
point . Find the absolute acceleration of
the bug, .
turntable, B
track
1m
East
O
ı̂
Filename:pfigure-f93f4
Problem 18.3.8
!
18.3.9 A giant bug walks on a horizontal
disk. An
frame is attached to the disk.
The disk is rotating about the axis (out
of the paper) and simultaneously translating with respect to an inertial frame
Filename:pfigure-blue-55-2
Problem 18.3.3
Mechanics Toolset, Statics, and Dynamics, c Andy Ruina and Rudra Pratap 1994-2018.
Problems
END-OF-SECTION PROBLEMS
15.220 A flight simulator is used to train pilots on how to recognize spatial
disorientation. It has four degrees of freedom and can rotate around
a planetary axis as well as in yaw, pitch, and roll. The pilot is seated
so that her head B is located at r 5 2i 1 1j ft with respect to the
center of the cab A. Knowing that the cab is rotating about the planetary axis with a constant angular velocity of 20 rpm counterclockwise as seen from above, and pitches with a constant angular velocity
of 13k rad/s, determine (a) the velocity of the pilot’s head, (b) the
angular acceleration of the cab, (c) the acceleration of the pilot’s head.
y
Planetary
axis
Yaw
B
A
Roll
Pitch
x
z
O
8 ft
Fig. P15.220 and P15.221
15.221 A flight simulator is used to train pilots on how to recognize spatial
disorientation. It has four degrees of freedom and can rotate around
a planetary axis as well as in yaw, pitch, and roll. The pilot is seated
so that her head B is located at r 5 2i 1 1j ft with respect to the
center of the cab A. The cab is rotating about the planetary axis with
an angular velocity of 20 rpm counterclockwise as seen from above
and this is increasing by 1 rad/s2. Knowing that the cab rolls with a
constant angular velocity of 24i rad/s, determine (a) the velocity of
the pilot’s head, (b) the angular acceleration of the cab, (c) the acceleration of the pilot’s head.
Y
90 mm
C
E
135 mm
D
15.222 and 15.223 The rectangular plate shown rotates at the constant
rate v2 5 12 rad/s with respect to arm AE, which itself rotates at
the constant rate v1 5 9 rad/s about the Z axis. For the position
shown, determine the velocity and acceleration of the point of the
plate indicated.
15.222 Corner B
15.223 Corner C
B
Z
A
ω1
ω2
X
135 mm
Fig. P15.222 and P15.223
1091
586
CHAPTER 20
T H R E E -D I M E N S I O N A L K I N E M AT I C S
RIGID BODY
OF A
PROBLEMS
20–37. Solve Example 20.5 such that the x, y, z axes move
with curvilinear translation, ! = 0 in which case
the collar appears to have both an angular velocity
! xyz = V1 + V2 and radial motion.
20–38. Solve Example 20.5 by fixing x, y, z axes to rod BD
so that ! = V1 + V2. In this case the collar appears only to
move radially outward along BD; hence ! xyz = 0.
20–39. At the instant u = 60!, the telescopic boom AB of
the construction lift is rotating with a constant angular
velocity about the z axis of v1 = 0.5 rad>s and about the pin
at A with a constant angular speed of v2 = 0.25 rad>s.
Simultaneously, the boom is extending with a velocity of
1.5 ft>s, and it has an acceleration of 0.5 ft>s2, both measured
relative to the construction lift. Determine the velocity and
acceleration of point B located at the end of the boom at
this instant.
*20–40. At the instant u = 60!, the construction lift is
rotating about the z axis with an angular velocity of
v1 = 0.5 rad>s and an angular acceleration of
#
v1 = 0.25 rad>s2 while the telescopic boom AB rotates
about the pin at A with an angular velocity of v2 = 0.25 rad>s
#
and angular acceleration of v2 = 0.1 rad>s2. Simultaneously,
the boom is extending with a velocity of 1.5 ft>s, and it has
an acceleration of 0.5 ft>s2, both measured relative to the
frame. Determine the velocity and acceleration of point B
located at the end of the boom at this instant.
z
v1 ! 4 rad/s
v" 1 ! 3 rad/s2
A
1.5 m
3 m/s
2 m/s2
D
B
0.6 m
C
x
v2 ! 5 rad/s
v" 2 ! 7 rad/s2
y
Prob. 20–41
20–42. At the instant u = 30!, the frame of the crane and
the boom AB rotate with a constant angular velocity of
v1 = 1.5 rad>s and v2 = 0.5 rad>s, respectively. Determine
the velocity and acceleration of point B at this instant.
20–43. At the instant u = 30!, the frame of the crane is
rotating with an angular velocity of v1 = 1.5 rad>s and
#
angular acceleration of v1 = 0.5 rad>s2, while the boom AB
rotates with an angular velocity of v2 = 0.5 rad>s and
#
angular acceleration of v2 = 0.25 rad>s2. Determine the
velocity and acceleration of point B at this instant.
z
B
20
20–41. At the instant shown, the arm AB is rotating about the
fixed pin A with an angular velocity v1 = 4 rad>s and angular
#
acceleration v 1 = 3 rad>s2. At this same instant, rod BD is
rotating relative to rod AB with an angular velocity v2 = 5 rad>s,
#
which is increasing at v 2 = 7 rad>s2. Also, the collar C is moving
along rod BD with a velocity of 3 m>s and an acceleration of
2 m>s2, both measured relative to the rod. Determine the
velocity and acceleration of the collar at this instant.
v1, v1
z
15 ft
V1, V1
1.5 m
12 m
B
u
v2, v2
x
2 ft
A
O
C
Probs. 20–39/40
y
O
A
u
V2, V2
Probs. 20–42/43
y
20.4
*20–44. At the instant shown, the rod AB is rotating about
the z axis with an angular velocity v1 = 4 rad>s and an
angular acceleration v# 1 = 3 rad>s2. At this same instant, the
circular rod has an angular motion relative to the rod as
shown. If the collar C is moving down around the circular
rod with a speed of 3 in.>s, which is increasing at 8 in.>s2,
both measured relative to the rod, determine the collar’s
velocity and acceleration at this instant.
20–46. At the instant shown, the industrial manipulator is
rotating about the z axis at v1 = 5 rad>s, and about joint B at
v2 = 2 rad>s. Determine the velocity and acceleration of the
grip A at this instant, when f = 30°, u = 45°, and r = 1.6 m.
20–47. At the instant shown, the industrial manipulator is
#
rotating about the z axis at v1 = 5 rad>s, and v1 = 2 rad>s2;
#
and about joint B at v2 = 2 rad>s and v2 = 3 rad>s2.
Determine the velocity and acceleration of the grip A at
this instant, when f = 30°, u = 45°, and r = 1.6 m.
z
z
v1 ! 4 rad/s
v" 1 ! 3 rad/s2
v2 B
A
f
B
1.2 m
5 in.
v1
u
x
y
v2 ! 2 rad/s
v" 2 ! 8 rad/s2
x
587
RELATIVE-MOTION ANALYSIS USING TRANSLATING AND ROTATING AXES
A
r
4 in.
C
y
Prob. 20–44
Probs. 20–46/47
20–45. The particle P slides around
the circular hoop with
#
a constant angular velocity of u = 6 rad>s, while the hoop
rotates about the x axis at a constant rate of v = 4 rad>s. If
at the instant shown the hoop is in the x–y plane and the
angle u = 45°, determine the velocity and acceleration of the
particle at this instant.
*20–48. At the given instant, the rod is turning about the z
axis with a constant angular velocity v1 = 3 rad>s. At this
same instant, the disk is spinning at v2 = 6 rad>s when
v# 2 = 4 rad>s2, both measured relative to the rod. Determine
the velocity and acceleration of point P on the disk at
this instant.
z
20
z
v1 ! 3 rad/s
x
200 mm O
u
O
2m
P
y
0.5 m
1.5 m
P
x
V ! 4 rad/s
Prob. 20–45
v2 ! 6 rad/s
" ! 4 rad/s2
v
2
0.5 m
y
Prob. 20–48
15.224 Rod AB is welded to the 0.3-m-radius plate that rotates at the
constant rate v1 5 6 rad/s. Knowing that collar D moves toward
end B of the rod at a constant speed u 5 1.3 m/s, determine, for
the position shown, (a) the velocity of D, (b) the acceleration of D.
Y
0.2 m
u
B
D
A
0.3 m
Z
C
0.25 m
ω1
X
15.225 The bent rod shown rotates at the constant rate of v1 5 5 rad/s and
collar C moves toward point B at a constant relative speed of
u 5 39 in./s. Knowing that collar C is halfway between points B and
D at the instant shown, determine its velocity and acceleration.
Y
B
14.4 in.
Fig. P15.224
u
C
A
6 in.
E
X
20.8 in.
ω1
D
Z
Fig. P15.225
15.226 The bent pipe shown rotates at the constant rate v1 5 10 rad/s.
Knowing that a ball bearing D moves in portion BC of the pipe
toward end C at a constant relative speed u 5 2 ft/s, determine at
the instant shown (a) the velocity of D, (b) the acceleration of D.
Y
8 in.
u
B
C
D
6 in.
ω1
12 in.
A
Z
Fig. P15.226
1092
X
588
CHAPTER 20
T H R E E -D I M E N S I O N A L K I N E M AT I C S
20–49. At the instant shown, the backhoe is traveling
forward at a constant speed vO = 2 ft>s, and the boom ABC
is rotating about the z axis with an angular velocity
#
v1 = 0.8 rad>s and an angular acceleration v1 = 1.30 rad>s2.
At this same instant the boom is rotating with v2 = 3 rad>s
#
when v2 = 2 rad>s2, both measured relative to the frame.
Determine the velocity and acceleration of point P on the
bucket at this instant.
OF A
RIGID BODY
*20–52. The crane is rotating about the z axis with a
constant rate v1 = 0.25 rad>s, while the boom OA is rotating
downward with a constant rate v2 = 0.4 rad>s. Compute the
velocity and acceleration of point A located at the top of the
boom at the instant shown.
20–53. Solve Prob. 20–52 if the angular motions are
#
#
increasing at v1 = 0.4 rad>s2 and v2 = 0.8 rad>s2 at the
instant shown.
z
B
u
A
v1 " 0.25 rad/s
vO ! 2 ft/s
O
A
2 ft
y
C 4 ft
P
z
v1 ! 0.8 rad/s
v1 ! 1.30 rad/s2
v2 ! 3 rad/s
v2 ! 2 rad/s2
40 ft
15 ft
5 ft
v2 " 0.4 rad/s
3 ft
x
30!
O
x
Prob. 20–49
y
20–50. At the instant shown, the arm OA of the conveyor
belt is rotating about the z axis with a constant angular
velocity v1 = 6 rad>s, while at the same instant the arm is
rotating upward at a constant rate v2 = 4 rad>s. If the
#
conveyor is running at a constant rate r = 5 ft>s, determine
the velocity and acceleration of the package P at the instant
shown. Neglect the size of the package.
20
20–51. At the instant shown, the arm OA of the conveyor
belt is rotating about the z axis with a constant angular velocity
v1 = 6 rad>s, while at the same instant the arm is rotating
upward at a constant rate v2 = 4 rad>s. If the conveyor is
#
$
running at a rate r = 5 ft>s, which is increasing at r = 8 ft>s2,
determine the velocity and acceleration of the package P at
the instant shown. Neglect the size of the package.
Probs. 20–52/53
20–54. At the instant shown, the arm AB is rotating about
the fixed bearing with an angular velocity v1 = 2 rad>s and
#
angular acceleration v1 = 6 rad>s2. At the same instant, rod
BD is rotating relative to rod AB at v2 = 7 rad>s, which is
#
increasing at v2 = 1 rad>s2. Also, the collar C is moving
#
along rod BD with a velocity r = 2 ft>s and a deceleration
$
2
r = - 0.5 ft>s , both measured relative to the rod.
Determine the velocity and acceleration of the collar at
this instant.
z
v2 ! 7 rad/s
v2 ! 1 rad/s2
A
z
P
V1 ! 6 rad/s
O
x
r ! 1 ft
B
C
u ! 30"
r ! 6 ft
u ! 45"
V2 ! 4 rad/s
Probs. 20–50/51
1.5 ft
v1 ! 2 rad/s
v1 ! 6 rad/s2
x
y
2 ft
A
y
Prob. 20–54
D