Mechanical Engineering Chapter 14 Problem The Slider Pushed Along The Circular Bar The Slotted Bar The

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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

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.

1 km/s

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.

Solution: The total force acting on part A in cylindrical

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.

0 .5 1 1.5 2 2.5

Time (s)

Fmag (newtons) vs t (s)

3 3.5 4 4.5 5

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

(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).

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

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

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

r0=(ε +1)cos2β

[(ε +1)cos2β−1] cos θ−(ε +1)sin βcos βsin θ+1,

where ε=r0v2

gR2

E−1.

u=Asin θ+Bcos θ+gR2

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°

Solution: (a) The force in coupling 1 is

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?

(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

WAWB

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

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

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.

Fx=TA=WA

g+WB

gaA.

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:

aWA

g+WB

g=WB+WB

ga, from which

a=gWB

WA+2WB=g20

160 =32.17

g=4.02 ft/s2

mAmB

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

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

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−rdθ

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

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?

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=2θ

2π+1m, 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.

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.