Mechanical Engineering Chapter 17 Problem Active Example Suppose That The Merrygoround Has Counterclockwise Angularvelocity And Counterclockwise

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

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

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

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−12 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

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