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Problem 17.143 In Active Example 17.10, suppose
that the merry-go-round has counterclockwise angular
velocity ωand counterclockwise angular acceleration α.
The person Ais standing still on the ground. Determine
her acceleration relative to your reference frame at the
instant shown.
y
v
A
B
Solution: First the velocity analysis
Problem 17.144 The x–ycoordinate system is body
fixed with respect to the bar. The angle θ(in radians) is
give as a function of time by θ=0.2+0.04t2. The x
Solution: We have
θ=0.2+0.04t2,˙
θ=0.08t
Problem 17.145 The metal plate is attached to a fixed
ball-and-socket support at O. The pin Aslides in a slot
in the plate. At the instant shown, xA=1m,dxA/dt =
2 m/s, and d2xA/dt2=0, and the plate’s angular
velocity and angular acceleration are ω=2k(rad/s) and
α=0. What are the x,y, and zcomponents of the
velocity and acceleration of Arelative to a nonrotating
reference frame with its origin at O?
y
x
A
434
Problem 17.146 Suppose that at the instant shown in
Problem 17.145, xA=1m, dxA/dt =−3 m/s,
d2xA/dt2=4 m/s2, and the plate’s angular velocity and
angular acceleration are ω=−4j+2k(rad/s), and α=
3i−6j(rad/s2). What are the x, y,z components of the
velocity and acceleration of Arelative to a nonrotating
reference frame that is stationary with respect to O?
y
x
A
Problem 17.147 The coordinate system is fixed
relative to the ship B. At the instant shown, the ship
is sailing north at 5 m/s relative to the earth, and its
angular velocity is 0.26 rad/s counterclockwise. Using
radar, it is determined that the position of the airplane is
1080i+1220j+6300k(m) and its velocity relative to
the ship’s coordinate system is 870i−45j−21k(m/s).
What is the airplane’s velocity relative to the earth? (See
Example 17.11.)
x
y
A
B
N
Solution:
Problem 17.148 The space shuttle is attempting to
recover a satellite for repair. At the current time, the
satellite’s position relative to a coordinate system fixed
to the shuttle is 50i(m). The rate gyros on the shuttle
indicate that its current angular velocity is 0.05j+
0.03k(rad/s). The Shuttle pilot measures the velocity of
the satellite relative to the body-fixed coordinate system
and determines it to be −2i−1.5j+2.5k(rad/s). What
are the x,y, and zcomponents of the satellite’s velocity
relative to a nonrotating coordinate system with its origin
fixed to the shuttle’s center of mass?
50 m
x
y
436
Problem 17.149 The train on the circular track is
traveling at a constant speed of 50 ft/s in the direction
shown. The train on the straight track is traveling at
20 ft/s in the direction shown and is increasing its speed
at 2 ft/s2. Determine the velocity of passenger Athat
y
50 ft/s
Problem 17.150 In Problem 17.149, determine the
acceleration of passenger Athat passenger Bobserves
relative to the coordinate system fixed to the car in which
Bis riding.
Solution:
Rearrange:
Problem 17.151 The satellite Ais in a circular polar
orbit (a circular orbit that intersects the earth’s axis of
rotation). The radius of the orbit is R, and the magnitude
of the satellite’s velocity relative to a non-rotating refer-
ence frame with its origin at the center of the earth is vA.
At the instant shown, the satellite is above the equator.
An observer Bon the earth directly below the satellite
measures its motion using the earth-fixed coordinate sys-
tem shown. What are the velocity and acceleration of the
satellite relative to B’s earth-fixed coordinate system?
N
A
BA x
y
R
RE
438
Problem 17.152 A car Aat north latitude Ldrives
north on a north–south highway with constant speed v.
The earth’s radius is RE, and the earth’s angular velocity
is ωE. (The earth’s angular velocity vector points north.)
The coordinate system is earth fixed, and the xaxis
N
y
x
Problem 17.153 The airplane Bconducts flight tests
of a missile. At the instant shown, the airplane is trav-
eling at 200 m/s relative to the earth in a circular path
of 2000-m radius in the horizontal plane. The coordinate
system is fixed relative to the airplane. The xaxis is tan-
gent to the plane’s path and points forward. The yaxis
points out the plane’s right side, and the zaxis points
out the bottom of the plane. The plane’s bank angle (the
inclination of the zaxis from the vertical) is constant
and equal to 20◦.Relative to the airplane’s coordinate
system, the pilot measures the missile’s position and
velocity and determines them to be rA/B =1000i(m)
and vA/B =100.0i+94.0j+34.2k(m/s).
(a) What are the x,y, and zcomponents of the air-
plane’s angular velocity vector?
(b) What are the x,y, and zcomponents of the mis-
sile’s velocity relative to the earth?
x
z
y
20°
2000 m
A
B
440
Problem 17.154 To conduct experiments related to
long-term spaceflight, engineers construct a laboratory
on earth that rotates about the vertical axis at Bwith
a constant angular velocity ωof one revolution every
6 s. They establish a laboratory-fixed coordinate system
with its origin at Band the zaxis pointing upward.
An engineer holds an object stationary relative to the
laboratory at point A, 3 m from the axis of rotation, and
releases it. At the instant he drops the object, determine
its acceleration relative to the laboratory-fixed coordinate
system,
(a) assuming that the laboratory-fixed coordinate sys-
tem is inertial and
(b) not assuming that the laboratory-fixed coordinate
system is inertial, but assuming that an earth-fixed
coordinate system with its origin at Bis inertial.
(See Example 17.13.)
Ax
y
x
y
B
A
B
3 m
ω
Solution: (a) If the laboratory system is inertial, Newton’s second
law is F=ma. The only force is the force of gravity; so that as the
Problem 17.155 The disk rotates in the horizontal
plane about a fixed shaft at the origin with constant
angular velocity w=10 rad/s. The 2-kg slider A
moves in a smooth slot in the disk. The spring
is unstretched when x=0 and its constant is k=
400 N/m. Determine the acceleration of Arelative to
the body-fixed coordinate system when x=0.4m.
Strategy: Use Eq. (17.30) to express Newton’s second
law for the slider in terms of the body-fixed coordinate
system.
x
x
kA
ω
y
Solution:
y
442
Problem 17.156* Engineers conduct flight tests of a
rocket at 30◦north latitude. They measure the rocket’s
motion using an earth-fixed coordinate system with
the xaxis pointing upward and the yaxis directed
northward. At a particular instant, the mass of the rocket
is 4000 kg, the velocity of the rocket relative to the
engineers’ coordinate system is 2000i+2000j(m/s),
and the sum of the forces exerted on the rocket by its
thrust, weight, and aerodynamic forces is 400i+400j
(N). Determine the rocket’s acceleration relative to the
engineers’ coordinate system,
(a) assuming that their earth-fixed coordinate system
is inertial and
(b) not assuming that their earth-fixed coordinate
system is inertial.
N
30°
y
x
Solution: Use Eq. (17.22):
(b) If the earth fixed system is not assumed to be inertial, aB=
−REω2
Ecos2λi+REω2
Ecos λsin λj, the angular velocity of the
rotating coordinate system is ω=ωEsin λi+ωEcos λj(rad/s).
Collect terms,
=maArel
Problem 17.157* Consider a point Aon the surface of
the earth at north latitude L. The radius of the earth is RE
and its angular velocity is ωE. A plumb bob suspended
just above the ground at point Awill hang at a small
angle βrelative to the vertical because of the earth’s
rotation. Show that βis related to the latitude by
tan β=ω2
EREsin Lcos L
g−ω2
EREcos2L.
Strategy: Using the earth-fixed coordinate system
shown, express Newton’s second law in the form given
by Eq. (17.22).
N
x
y
B
RE
A
L
x
A
β
apparent acceleration due to gravity is ghorizontal =REω2
Ecos Lsin L.
From equation of angular motion, the moments about the
bob suspension are Mvertical =(λ sin β)mgvertical and Mhorizontal =
444
Problem 17.158* Suppose that a space station is in
orbit around the earth and two astronauts on the station
toss a ball back and forth. They observe that the ball
appears to travel between them in a straight line at
constant velocity.
(a) Write Newton’s second law for the ball as it travels
between the astronauts in terms of a nonrotating
coordinate system with its origin fixed to the
station. What is the term "F? Use the equation
you wrote to explain the behavior of the ball
observed by the astronauts.
(b) Write Newton’s second law for the ball as it travels
between the astronauts in terms of a nonrotating
coordinate system with its origin fixed to the center
of the earth. What is the term "F? Explain the
difference between this equation and the one you
obtained in part (a).
Solution: An earth-centered, non-rotating coordinate system can
Problem 17.159 If θ=60◦and bar OQ rotates in the
counterclockwise direction at 5 rad/s, what is the angular
velocity of bar PQ?
O
P
Q
200 mm 400 mm
u
Solution: By applying the law of sines, β=25.7◦. The velocity
The velocity of Pis
vpi=vQ+ωPQ ×rP/Q
=−sin 60◦i+cos 60◦j+
ijk
00ωPQ
0.4 cos β−0.4 sin β0
.
Equating iand jcomponents vP=−sin 60◦+0.4ωPQ sin β, and
0=cos 60◦+0.4ωPQ cos β. Solving, we obtain vP=−1.11 m/s and
ωPQ =−1.39 rad/s.
y
Problem 17.160 Consider the system shown in Prob-
lem 17.159. If θ=55◦and the sleeve Pis moving to
the left at 2 m/s, what are the angular velocities of bars
OQ and PQ?
O
P
Q
200 mm 400 mm
u
446
Problem 17.161 Determine the vertical velocity vHof
the hook and the angular velocity of the small pulley.
H
120 mm/s
40 mm
2.
Problem 17.162 If the crankshaft AB is turning in the
counterclockwise direction at 2000 rpm, what is the ve-
locity of the piston?
C
C
y
x
B
A
5 in
45°
2 in
448
Problem 17.163 In Problem 17.162, if the piston is
moving with velocity vC=20j(ft/s), what are the angu-
lar velocities of the crankshaft AB and the connecting
rod BC?
C
C
y
B
5 in
45⬚
Solution: Use the solution to Problem 17.162. The vector location
of point B(the main rod bearing) rB=1.414(−i+j)in. From the law
if sines the interior angle between the connecting rod and the vertical
at the piston is
θ=sin−12 sin 45◦
5=16.43◦.
−110
=−1.414ωAB (i+j)(in/s).
Problem 17.164 In Problem 17.162, if the piston
is moving with velocity vC=20j(ft/s), and its
acceleration is zero, what are the angular accelerations
of crankshaft AB and the connecting rod BC?
C
C
y
B
5 in
45⬚
Solution: Use the solution to Problem 17.163. rB/A =1.414(−i+
jin., ωAB =−131.1 rad/s, rB/C =rB−rC=−1.414i−4.796j(in.),
ωBC =38.65 rad/s. For point B,
aB=aA+αAB ×rB/A −ω2
AB rB/A
−110
aB=−1.414αAB (i+j)+24291(i−j)(in/s2).
Equate expressions and separate components:
Problem 17.165 Bar AB rotates at 6 rad/s in the
counterclockwise direction. Use instantaneous centers
to determine the angular velocity of bar BCD and the
velocity of point D.
y
D
C
8 in
12 in
450
