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Problem 14.106* The 1/4-lb slider Ais pushed along
the circular bar by the slotted bar. The circular bar lies
in the vertical plane. The angular position of the slotted
bar is θ=10t2rad. Determine the polar components of
the total force exerted on the slider by the circular and
−(4 cos θ)(20t)2]t=0.25 =−209 ft/s2.
Problem 14.107* The slotted bar rotates in the hori-
zontal plane with constant angular velocity ω0. The mass
mhas a pin that fits into the slot of the bar. A spring
holds the pin against the surface of the fixed cam. The
surface of the cam is described by r=r0(2−cos θ).
Determine the polar components of the total external
force exerted on the pin as functions of θ.
Cam
k
m
r0
θ
ω
0
Solution: The angular velocity is constant, from which θ=
Problem 14.108* In Problem 14.107, suppose that the
unstretched length of the spring is r0. Determine the
smallest value of the spring constant kfor which the pin
will remain on the surface of the cam.
Solution: The spring force holding the pin on the surface of the
Problem 14.109 A charged particle Pin a magnetic
field moves along the spiral path described by r=1m,
θ=2zrad, where zis in meters. The particle moves
along the path in the direction shown with a constant
speed |v|=1 km/s. The mass of the particle is 1.67 ×
10−27 kg. Determine the sum of the forces on the particle
in terms of cylindrical coordinates.
y
z
x
1 km/s
P
Solution: The force components in cylindrical coordinates are
170
Problem 14.110 At the instant shown, the cylindrical
coordinates of the 4-kg part Aheld by the robotic
manipulator are r=0.6m, θ=25◦, and z=0.8m.
(The coordinate system is fixed with respect to the
earth, and the y axis points upward). A’s radial position
is increasing at dr
dt =0.2 m/s, and d2r
dt2=−0.4 m/s2.
The angle θis increasing at dθ
dt =1.2 rad/s and
d2θ
dt2=2.8 rad/s2. The base of the manipulator arm
is accelerating in the zdirection at d2z
dt2=2.5 m/s2.
Determine the force vector exerted on Aby the
manipulator in cylindrical coordinates.
y
x
A
z
r
z
θ
Solution: The total force acting on part A in cylindrical
y
Problem 14.111 Suppose that the robotic manipulator
in Problem 14.110 is used in a space station to
investigate zero-gmanufacturing techniques. During
an interval of time, the manipulator is programmed
so that the cylindrical coordinates of the 4-kg part
Aare θ=0.15t2rad, r=0.5(1+sin θ) m, and z=
0.8(1+θ) m Determine the force vector exerted on A
by the manipulator at t=2 s in terms of cylindrical
coordinates.
Solution:
dθ
dt2=0.3 rad/s2.
Evaluating these expressions at t=2 s, the acceleration is
dθ
172
Problem 14.112* In Problem 14.111, draw a graph of
the magnitude of the force exerted on part Aby the
manipulator as a function of time from t=0tot=5s
and use it to estimate the maximum force during that
interval of time.
1
0 .5 1 1.5 2 2.5
Time (s)
Fmag (newtons) vs t (s)
3 3.5 4 4.5 5
7
8
9
F
m
a
s
Problem 14.113 The International Space Station is in
a circular orbit 225 miles above the earth’s surface.
(a) What is the magnitude of the velocity of the space
station?
(b) How long does it take to complete one revolution?
=2.21 ×107ft.
(a) From Eq (14.24), the velocity is
v0=gR2
E
r0
(b) Let T be the time required. Then v0T=2πr0,
Problem 14.114 The moon is approximately 383,000
km from the earth. Assume that the moon’s orbit around
the earth is circular with velocity given by Eq. (14.24).
(a) What is the magnitude of the moon’s velocity?
(b) How long does it take to complete one revolution
around the earth?
Problem 14.115 Suppose that you place a satellite into
an elliptic earth orbit with an initial radius r0=6700 km
and an initial velocity v0such that the maximum radius
of the orbit is 13,400 km. (a) Determine v0. (b) What is
the magnitude of the satellite’s velocity when it is at its
maximum radius? (See Active Example 14.10).
r0
v0
Problem 14.116 A satellite is given an initial velocity
v0=6700 m/s at a distance r0=2REfrom the center
of the earth as shown in Fig. 14.18a. Draw a graph of
the resulting orbit.
Solution: The graph is shown.
2.88RE
2.88RE
174
Problem 14.117 The time required for a satellite in a
circular earth orbit to complete one revolution increases
as the radius of the orbit increases. If you choose the
radius properly, the satellite will complete one revolu-
tion in 24 hours. If a satellite is placed in such an orbit
directly above the equator and moving from west to east,
it will remain above the same point on the earth as the
earth rotates beneath it. This type of orbit, conceived
by Arthur C. Clarke, is called geosynchronous, and is
used for communication and television broadcast satel-
lites. Determine the radius of a geosynchronous orbit
in km.
Solution: We have
Problem 14.118* You can send a spacecraft from the
earth to the moon in the following way. First, launch
the spacecraft into a circular “parking” orbit of radius
r0around the earth (Fig. a). Then increase its velocity
in the direction tangent to the circular orbit to a value
v0such that it will follow an elliptic orbit whose maxi-
mum radius is equal to the radius rMof the moon’s orbit
around the earth (Fig. b). The radius rM=238,000 mi.
Let r0=4160 mi. What velocity v0is necessary to send
a spacecraft to the moon? (This description is simplified
in that it disregards the effect of the moon’s gravity.)
Parking
orbit
r0r0
v0
rM
Elliptic
orbit
Moon’s
orbit
(a) (b)
Problem 14.119* At t=0, an earth satellite is a dis-
tance r0from the center of the earth and has an initial
velocity v0in the direction shown. Show that the polar
equation for the resulting orbit is
r
r0=(ε +1)cos2β
[(ε +1)cos2β−1] cos θ−(ε +1)sin βcos βsin θ+1,
where ε=r0v2
0
gR2
E−1.
r0
0
RE
β
u=Asin θ+Bcos θ+gR2
r2
0v2
0cos2β.(2)
176
Problem 14.120 The Acura NSX can brake from
60 mi/h to a stop in a distance of 112 ft. (a) If you
assume that the vehicle’s deceleration is constant, what
are its deceleration and the magnitude of the horizontal
force its tires exert on the road? (b) If the car’s tires are
at the limit of adhesion (i.e., slip is impending), and the
normal force exerted on the car by the road equals the
car’s weight, what is the coefficient of friction µs? (This
analysis neglects the effects of horizontal and vertical
aerodynamic forces).
Solution:
(a) 60 mi/h =88 ft/s.
Problem 14.121 Using the coefficient of friction
obtained in Problem 14.120, determine the highest speed
at which the NSX could drive on a flat, circular track of
600-ft radius without skidding.
Problem 14.122 A cog engine hauls three cars of
sightseers to a mountain top in Bavaria. The mass of
each car, including its passengers, is 10,000 kg and the
friction forces exerted by the wheels of the cars are
negligible. Determine the forces in couplings 1, 2, and 3
if: (a) the engine is moving at constant velocity; (b) the
engine is accelerating up the mountain at 1.2 m/s2.
40°
1
2
3
Solution: (a) The force in coupling 1 is
F3
178
Problem 14.123 In a future mission, a spacecraft
approaches the surface of an asteroid passing near the
earth. Just before it touches down, the spacecraft is
moving downward at a constant velocity relative to
the surface of the asteroid and its downward thrust is
0.01 N. The computer decreases the downward thrust to
0.005 N, and an onboard laser interferometer determines
that the acceleration of the spacecraft relative to the
surface becomes 5 ×10−6m/s2downward. What is the
gravitational acceleration of the asteroid near its surface?
(0.01 −0.005)=1×10−5N/kg2
Problem 14.124 A car with a mass of 1470 kg, includ-
ing its driver, is driven at 130 km/h over a slight rise
in the road. At the top of the rise, the driver applies the
brakes. The coefficient of static friction between the tires
and the road is µs=0.9 and the radius of curvature of
the rise is 160 m. Determine the car’s deceleration at the
instant the brakes are applied, and compare it with the
deceleration on a level road.
Problem 14.125 The car drives at constant velocity up
the straight segment of road on the left. If the car’s tires
continue to exert the same tangential force on the road
after the car has gone over the crest of the hill and is on
the straight segment of road on the right, what will be
the car’s acceleration?
8°
5°
Problem 14.126 The aircraft carrier Nimitz weighs
91,000 tons. (A ton is 2000 lb.) Suppose that it is
traveling at its top speed of approximately 30 knots (a
knot is 6076 ft/h) when its engines are shut down. If
the water exerts a drag force of magnitude 20,000vlb,
where vis the carrier’s velocity in feet per second, what
distance does the carrier move before coming to rest?
v(0)=50.63, and v(x) =v(0)−gK
180
Problem 14.127 If mA=10 kg, mB=40 kg, and the
coefficient of kinetic friction between all surfaces is
µk=0.11, what is the acceleration of Bdown the
inclined surface?
A
B
(2) Fy=NA−mAgcos θ=0. From Newton’s second law for
(4) Fy=NB−NA−mBgcos θ=0. Since the pulley is one-to-
one, the sum of the displacements is xB+xA=0. Differentiate
twice:
(5) aB+aA=0. Solving these five equations in five unknowns, T=
49.63 N, NA=92.2N,NB=460.9N,aA=−0.593 m/s2,
aB=0.593 m/s2
NA
WAWB
NB
Problem 14.128 In Problem 14.127, if Aweighs
20 lb, Bweighs 100 lb, and the coefficient of kinetic
friction between all surfaces is µk=0.15, what is the
tension in the cord as Bslides down the inclined surface?
Problem 14.129 A gas gun is used to accelerate
projectiles to high velocities for research on material
properties. The projectile is held in place while gas is
pumped into the tube to a high pressure p0on the left
and the tube is evacuated on the right. The projectile is
then released and is accelerated by the expanding gas.
Assume that the pressure pof the gas is related to the
volume Vit occupies by pV γ=constant, where γis
a constant. If friction can be neglected, show that the
velocity of the projectile at the position xis
v=
2p0Axγ
0
m(γ −1)1
xγ−1
0−1
xγ−1,
where mis the mass of the projectile and Ais the cross-
sectional area of the tube.
Projectile
p0
x0
p
x
182
Problem 14.130 The weights of the blocks are WA=
120 lb, and WB=20 lb and the surfaces are smooth.
Determine the acceleration of block Aand the tension
in the cord.
A
B
Fx=TA=WA
g+WB
gaA.
For block B: Fy=TA−WB=WB
gaB. Since the pulley is one-
to-one, as the displacement of B increases downward (negatively) the
displacement of A increases to the right (positively), from which xA=
−xB. Differentiate twice to obtain aA=−aB. Equate the expressions
to obtain:
aWA
g+WB
g=WB+WB
ga, from which
a=gWB
WA+2WB=g20
160 =32.17
g=4.02 ft/s2
mAmB
wB
Problem 14.131 The 100-Mg space shuttle is in orbit
when its engines are turned on, exerting a thrust force
T=10i−20j+10k(kN) for 2 s. Neglect the resulting
change in mass of the shuttle. At the end of the 2-s burn,
fuel is still sloshing back and forth in the shuttle’s tanks.
What is the change in the velocity of the center of mass
of the shuttle (including the fuel it contains) due to the
2-s burn?
Problem 14.132 The water skier contacts the ramp
with a velocity of 25 mi/h parallel to the surface of
the ramp. Neglecting friction and assuming that the tow
rope exerts no force on him once he touches the ramp,
estimate the horizontal length of the skier’s jump from
the end of the ramp. 20 ft
8 ft
Problem 14.133 Suppose you are designing a roller-
coaster track that will take the cars through a vertical
loop of 40-ft radius. If you decide that, for safety, the
downward force exerted on a passenger by his seat at the
top of the loop should be at least one-half the passenger’s
weight, what is the minimum safe velocity of the cars at
the top of the loop?
40 ft
Solution: Denote the normal force exerted on the passenger by the
184
Problem 14.134 As the smooth bar rotates in the hor-
izontal plane, the string winds up on the fixed cylinder
and draws the 1-kg collar Ainward. The bar starts from
rest at t=0 in the position shown and rotates with con-
stant acceleration. What is the tension in the string at
t=1s?
100 mm
400 mm
6 rad/s2
A
Solution: The angular velocity of the spool relative to the bar is
d2r
dt2=−Rα =−0.05(6)=−0.3 m/s2. The take up velocity of the
spool is
vs=Rα dt =−0.05(6)t =−0.3tm/s.
The velocity of the collar relative to the bar is
dr
dt =−0.3tm/s.
The velocity of the collar relative to the bar is dr/dt =−0.3tm/s.
The position of the collar relative to the bar is r=−0.15t2+0.4m.
The angular acceleration of the collar is d2θ
dt2=6 rad/s2. The angu-
lar velocity of the collar is dθ
dt =6trad/s. The radial acceleration is
ar=d2r
dt2−rdθ
dt 2
=−0.3−(−0.15t2+0.4)(6t)2.Att=1 s the
radial acceleration is ar=−9.3 m/s2, and the tension in the string is
|T|=|mar|=9.3N
A
T
Problem 14.135 In Problem 14.134, suppose that the
coefficient of kinetic friction between the collar and the
bar is µk=0.2. What is the tension in the string at
t=1s?
Problem 14.136 If you want to design the cars of a
train to tilt as the train goes around curves in order to
achieve maximum passenger comfort, what is the rela-
tionship between the desired tilt and θ, the velocity vof
the train, and the instantaneous radius of curvature, ρ,
of the track?
θ
Problem 14.137 To determine the coefficient of static
friction between two materials, an engineer at the U.S.
National Institute of Standards and Technology places a
small sample of one material on a horizontal disk whose
surface is made of the other material and then rotates
the disk from rest with a constant angular acceleration
of 0.4 rad/s2. If she determines that the small sample
slips on the disk after 9.903 s, what is the coefficient of
friction?
200 mm
186
Problem 14.138* The 1-kg slider Ais pushed along
the curved bar by the slotted bar. The curved bar lies
in the horizontal plane, and its profile is described by
r=2θ
2π+1m, where θis in radians. The angular
position of the slotted bar is θ=2trad. Determine the
radial and transverse components of the total external
force exerted on the slider when θ=1200.
A
θ
Problem 14.139* In Problem 14.138, suppose that the
curved bar lies in the vertical plane. Determine the radial
and transverse components of the total force exerted on
Aby the curved and slotted bars at t=0.5s.
