Mechanical Engineering Chapter 17 Problem Active Example Suppose That Given Instant The Hook Moving Downward What

Problem 17.1 In Active Example 17.1, suppose that

at a given instant the hook His moving downward at

2 m/s. What is the angular velocity of gear Aat that

instant?

A

50 mm

B

200 mm

100

mm

Problem 17.2 The angle θ(in radians) is given as a

function of time by θ=0.2πt2.Att=4 s, determine

the magnitudes of (a) the velocity of point Aand (b) the

tangential and normal components of acceleration of

point A.

A

2 ft

θ

Solution: We have

Problem 17.3 The mass Astarts from rest at t=0 and

falls with a constant acceleration of 8 m/s2. When the

mass has fallen one meter, determine the magnitudes of

(a) the angular velocity of the pulley and (b) the tangen-

tial and normal components of acceleration of a point at

the outer edge of the pulley.

100 mm

Solution: We have

Problem 17.4 At the instant shown, the left disk has

an angular velocity of 3 rad/s counterclockwise and an

angular acceleration of 1 rad/s2clockwise.

(a) What are the angular velocity and angular accel-

eration of the right disk? (Assume that there is

no relative motion between the disks at their point

of contact.)

(b) What are the magnitudes of the velocity and accel-

eration of point A?

2.5 m

1 m

A

2 m

3 rad/s

1 rad/s2

Solution:

rLωL=rRωR⇒ωR=rR

rL

ωL=1m

2.5 m (3 rad/s)=1.2 rad/s

Problem 17.5 The angular velocity of the left disk is

given as a function of time by ωA=4+0.2trad/s.

(a) What are the angular velocities ωBand ωCat

t=5s?

(b) Through what angle does the right disk turn from

t=0tot=5s?

100 mm

200 mm

ω

A

ω

B

ω

C

334

Problem 17.6 (a) If the bicycle’s 120-mm sprocket

wheel rotates through one revolution, through how many

revolutions does the 45-mm gear turn? (b) If the angu-

lar velocity of the sprocket wheel is 1 rad/s, what is the

angular velocity of the gear? 45

mm

120

mm

Solution: The key is that the tangential accelerations and tangen-

tial velocities along the chain are of constant magnitude

vB=(0.045)ωBvA=(0.120)(1)

ωB=120

45 rad/s =2.67 rad/s

rB

dθB

dt =rA

dθA

dt

Integrating, we get

rBθB=rAθArA=0.120 m

rB=0.045 m

θB=120

45 (1)rev θA=1 rev.

rA

vA

rB

Problem 17.7 The rear wheel of the bicycle in Prob-

lem 17.6 has a 330-mm radius and is rigidly attached to

the 45-mm gear. It the rider turns the pedals, which are

rigidly attached to the 120-mm sprocket wheel, at one

revolution per second, what is the bicycle’s velocity?

Problem 17.8 The disk is rotating about the origin

with a constant clockwise angular velocity of 100 rpm.

Determine the xand ycomponents of velocity of points

Aand B(in in/s).

A

y

Solution:

Problem 17.9 The disk is rotating about the origin

with a constant clockwise angular velocity of 100 rpm.

Determine the xand ycomponents of acceleration of

points Aand B(in in/s2).

B

A

x

y

8 in

Solution:

336

Problem 17.10 The radius of the Corvette’s tires is

14 in. It is traveling at 80 mi/h when the driver applies

the brakes, subjecting the car to a deceleration of 25 ft/s2.

Assume that the tires continue to roll, not skid, on the

road surface. At that instant, what are the magnitudes

of the tangential and normal components of acceleration

(in ft/s2) of a point at the outer edge of a tire relative

to a nonrotating coordinate system with its origin at the

center of the tire?

y

x

an=rω2=14

12 ft(101 rad/s)2=11,800 ft/s2.

Problem 17.11 If the bar has a counterclockwise

angular velocity of 8 rad/s and a clockwise angular

acceleration of 40 rad/s2, what are the magnitudes of

the accelerations of points Aand B?

0.4 m

0.2 m

A

Solution:

aA=α×rA/O −ω2rA/O

aB=α×rB/O −ω2rB/O

Problem 17.12 Consider the bar shown in Problem

17.11. If |vA|=3 m/s and |aA|=28 m/s2, what are |vB|

and |aB|?

Problem 17.13 A disk of radius R=0.5 m rolls on

a horizontal surface. The relationship between the

horizontal distance xthe center of the disk moves and

the angle βthrough which the disk rotates is x=Rβ.

Suppose that the center of the disk is moving to the right

with a constant velocity of 2 m/s.

(a) What is the disk’s angular velocity?

(b) Relative to a nonrotating reference frame with

its origin at the center of the disk, what are the

magnitudes of the velocity and acceleration of a

point on the edge of the disk?

x

R

β

R

Problem 17.14 The turbine rotates relative to the coor-

dinate system at 30 rad/s about a ﬁxed axis coincident

with the xaxis. What is its angular velocity vector?

y

338

Problem 17.15 The rectangular plate swings in the x–y

plane from arms of equal length. What is the angular

velocity of (a) the rectangular plate and (b) the bar AB?

y

x

A

B

10 rad/s

Solution: Denote the upper corners of the plate by Band B′, and

denote the distance between these points (the length of the plate) by

since L=L′, the ﬁgure AA′B′Bis a parallelogram. By deﬁnition, the

opposite sides of a parallelogram remain parallel, and since the ﬁxed

side AA′does not rotate, then BB′cannot rotate, so that the plate does

not rotate and

ωBB′=0 .

Similarly, by inspection the angular velocity of the bar AB is

ωAB =10k(rad/s),

where by the right hand rule the direction is along the positive zaxis

(out of the paper).

A

L′

A′

Problem 17.16 Bar OQ is rotating in the clockwise

direction at 4 rad/s. What are the angular velocity vectors

of the bars OQ and PQ?

Strategy: Notice that if you know the angular velocity

of bar OQ, you also know the angular velocity of bar

PQ.

1.2 m

O

Q

1.2 m

y

x

4 rad / s

Problem 17.17 A disk of radius R=0.5 m rolls on

a horizontal surface. The relationship between the hor-

izontal distance xthe center of the disk moves and the

angle βthrough which the disk rotates is x=Rβ. Sup-

pose that the center of the disk is moving to the right

with a constant velocity of 2 m/s.

(a) What is the disk’s angular velocity?

(b) What is the disk’s angular velocity vector?

x

R

β

R

x

y

Problem 17.18 The rigid body rotates with angular

velocity ω=12 rad/s. The distance rA/B =0.4m.

(a) Determine the xand ycomponents of the velocity

of Arelative to Bby representing the velocity as

shown in Fig. 17.10b.

(b) What is the angular velocity vector of the

rigid body?

(c) Use Eq. (17.5) to determine the velocity of Arel-

ative to B.

x

y

BA

ω

rA/B

Solution:

y

340

Problem 17.19 The bar is rotating in the counterclock-

wise direction with angular velocity ω. The magnitude of

the velocity of point Ais 6 m/s. Determine the velocity

of point B.

0.4 m

␻

A

y

Solution:

Problem 17.20 The bar is rotating in the counterclock-

wise direction with angular velocity ω. The magnitude

of the velocity of point Arelative to point Bis 6 m/s.

Determine the velocity of point B.

0.4 m

␻

A

y

Solution:

rA/B =(0.8m)2+(0.6m)2=1m

Problem 17.21 The bracket is rotating about point O

with counterclockwise angular velocity ω. The magni-

tude of the velocity of point Arelative to point Bis

4 m/s. Determine ω.

A

B

120 mm

y

x

O

Solution:

Problem 17.22 Determine the xand ycomponents of

the velocity of point A.y

x

A

5 rad/s

O

2 m

30°

Solution: The velocity of point Ais given by:

Problem 17.23 If the angular velocity of the bar in

Problem 17.22 is constant, what are the xand ycom-

ponents of the velocity of Point A0.1 s after the instant

shown?

342

Problem 17.24 The disk is rotating about the zaxis at

50 rad/s in the clockwise direction. Determine the xand

ycomponents of the velocities of points A,B, and C.

x

y

100 mm

A

Solution: The velocity of Ais given by vA=v0+ω×rA/O ,or

vA=0+(−50k)×(0.1j)=5i(m/s).

=−3.54i−3.54j(m/s),

For C, we have

vc=v0+ω×rC/O =0+

ijk

00−50

−0.1 cos 45◦−0.1 sin 45◦0

=−3.54i+3.54j(m/s).

x

BC

45°45°

100 mm

Problem 17.25 Consider the rotating disk shown in

Problem 17.24. If the magnitude of the velocity of

point Arelative to point Bis 4 m/s, what is the

magnitude of the disk’s angular velocity?

Problem 17.26 The radius of the Corvette’s tires is

14 in. It is traveling at 80 mi/h. Assume that the tires

roll, not skid, on the road surface.

(a) What is the angular velocity of its wheels?

(b) In terms of the earth-ﬁxed coordinate system shown,

determine the velocity (in ft/s) of the point of the

tire with coordinates (−14 in, 0,0).

y

x

Problem 17.27 Point Aof the rolling disk is moving

toward the right. The magnitude of the velocity of point

Cis 5 m/s. determine the velocities of points Band D.

y

45⬚

C

D

0.6 m

Solution: Point Bis the center of rotation (zero velocity).

rC/B =√2(0.6m)=0.849 m,

Problem 17.28 The helicopter is in planar motion in

the x–yplane. At the instant shown, the position

of its center of mass, G,isx=2m, y=2.5m,

and its velocity is vG=12i+4j(m/s). The position

of point T, where the tail rotor is mounted, is x=

−3.5m,y=4.5 m. The helicopter’s angular velocity is

0.2 (rad/s) clockwise. What is the velocity of point T?

y

x

T

G

344

Problem 17.29 The bar is moving in the x–yplane

and is rotating in the counterclockwise direction. The

velocity of point Arelative to the reference frame is

vA=12i−2j(m/s). The magnitude of the velocity of

point Arelative to point Bis 8 m/s. what is the velocity

of point Brelative to the reference frame?

y

B

2 m

30⬚

Solution:

ω=v

Problem 17.30 Points Aand Bof the 2-m bar slide

on the plane surfaces. Point Bis moving to the right at

3 m/s. What is the velocity of the midpoint Gof the bar?

Strategy: First apply Eq. (17.6) to points Aand B

to determine the bar’s angular velocity. Then apply

Eq. (17.6) to points Band G.

y

A

Solution: Take advantage of the constraints (B stays on the ﬂoor,

Problem 17.31 Bar AB rotates at 6 rad/s in the clock-

wise direction. Determine the velocity (in in/s) of the

slider C.

Solution:

vB=vA+ωAB ×rB/A

Problem 17.32 If θ=45◦and the sleeve Pis moving

to the right at 2 m/s, what are the angular velocities of

the bars OQ and PQ?

1.2 m

O

Q

P

θ

1.2 m

Solution: From the ﬁgure, v0=0, vP=vPi=2i(m/s)

Problem 17.33 In Active Example 17.2, consider the

instant when bar AB is vertical and rotating in the clock-

wise direction at 10 rad/s. Draw a sketch showing the

positions of the two bars at that instant. Determine the

angular velocity of bar BC and the velocity of point C.

0.4 m

10 rad/s

A

B

C

0.4 m 0.8 m

346

Problem 17.34 Bar AB rotates in the counterclockwise

direction at 6 rad/s. Determine the angular velocity of

bar BD and the velocity of point D.

y

0.32 m

D

Problem 17.35 At the instant shown, the piston’s velo-

city is vC=−14i(m/s). What is the angular velocity of

the crank AB?

y

Solution:

vB=vA+ωAB ×rB/A

Problem 17.36 In Example 17.3, determine the angu-

lar velocity fo the bar AB that would be necessary so

that the downward velocity of the rack VR=10 ft/s at

the instant shown.

Dv

R

6 in

B

C

A

12 in

10 rad/s

16 in6 in 6 in

10 in

Solution: We have

348

Problem 17.37 Bar AB rotates at 12 rad/s in the

clockwise direction. Determine the angular velocities of

bars BC and CD.

A

C

B

D

350

mm

200

mm

300 mm 350 mm

12 rad/s

Solution: The strategy is analogous to that used in Problem 17.36.

Problem 17.38 Bar AB is rotating at 10 rad/s in the

counterclockwise direction. The disk rolls on the circular

surface. Determine the angular velocities of bar BC and

the disk at the instant shown.

2 m

A

B

C

3 m

1 m

Solution: The point “D” at the bottom of the wheel has zero veloc-

ity.

Problem 17.39 Bar AB rotates at 2 rad/s in the coun-

terclockwise direction. Determine the velocity of the

midpoint Gof bar BC.

C

B

10 in

12 in

G

y

350

Problem 17.40 Bar AB rotates at 10 rad/s in the

counterclockwise direction. Determine the velocity of

point E.

x

y

A

B

C

10 rad/s E

D

400 mm

700 mm 700 mm400

mm

Problem 17.41 Bar AB rotates at 4 rad/s in the

counterclockwise direction. Determine the velocity of

point C.

A

B

C

D

Ex

y

600 mm

400 mm

500 mm

300

mm

300

mm

200

mm

352