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298 CHAPTER 16 Planar Kinematics of Rigid Bodies

Problems

16.19 At a certain instant, the velocity of end A of the bar AB is 4 m/s in the

BA60

1 m

4 m/s

Fig. P16.19

direction shown. Knowing that the magnitude of the velocity of end B is 3 m/s,determine the angular velocity of bar AB.

16.20 The wheel rolls without slipping. In the position shown, the vertical com-

CB

0.18 m

0.42 m

Fig. P16.20

ponent of the velocity of point B is 4 m/s directed upward. For this position,calculate the angular velocity of the wheel and the velocity of its center C .

16.21 The disk rolls without slipping with the constant angular velocity .For the position shown, find the angular velocity of link AB and the velocity ofslider A.

B

R

A

C

R

45

Fig. P16.21

16.22 The pinion gear meshes with the two racks. If the racks are moving withthe velocities shown, determine the angular velocity of the gear and the velocityof its center C .

C90 mm

0.6 m/s

0.8 m/s

150 mm O

RB

vO

Fig. P16.22 Fig. P16.23

16.23 The wheel rolls without slipping to the right with constant angular veloc-ity. The velocity of the center of the wheel is vO . Determine the speed of point Bon the rim as a function of its angular position .

16.24 The arm joining the two friction wheels rotates with the constant angular

B

rA

rB0

A

Fig. P16.24, P16.25velocity 0. Assuming that wheel A is stationary and that there is no slippingbetween the wheels, determine the angular velocity of wheel B.

16.1916.39 Problems 299

16.25 Solve Prob. 16.24 if wheel A is rotating clockwise with the angularvelocity A = 20.

16.26 Gear A of the planetary gear train is rotating clockwise at A = 8 rad/s.Calculate the angular velocities of gear B and the arm AB. Note that the outermostgear C is stationary.

A

A B

C

24 2430

Dimensions in mm

B

A

L

Fig. P16.26 Fig. P16.27

16.27 The bar AB is rotating counterclockwise with the constant angularspeed 0. (a) Find the velocities of ends A and B as functions of . (b) Differ-entiate the results of part (a) to determine the accelerations of A and B in termsof .

16.28 End A of bar AD is pushed to the right with the constant velocity

1 m

B

D

A

vA

Fig. P16.28

vA = 0.6 m/s. Determine the angular velocity of AD as a function of .

16.29 The angular speed of link AB in the position shown is 2.8 rad/s clockwise.Compute the angular speeds of links BC and CD in this position.

DC60

B A30

Dimensions in mm

60

2.8 rad/s

0.30 m 0.12 m

0.16

m

A0.12 m

E

6 rad/s

D

B

Fig. P16.29 Fig. P16.30

16.30 The link AB of the mechanism rotates with the constant angular speed of6 rad/s counterclockwise. Calculate the angular velocities of links BD and DE inthe position shown.


16.31 When the mechanism is in the position shown, the velocity of slider D375

125

225

150

vD

B

A

D

Dimensions in mm

Fig. P16.31

is vD = 1.25 m/s. Determine the angular velocities of bars AB and BD at thisinstant.

16.32 When the linkage is in the position shown, bar AB is rotatingcounterclockwise at 16 rad/s. Determine the velocity of the sliding collar C inthis position.

B

C0.40 m

16 rad/s30

0.24 m

0.36 mA

Fig. P16.32

16.33 At the instant shown, end A of the bar ABC has a downward velocity of

C

A

B

0.5 m

0.5 m

30

20

Fig. P16.33

2 m/s. Find the angular velocity of the bar and the speed of end C at this instant.

16.34 Bar AB is rotating counterclockwise with the constant angular velocityB D

A

E

0

30

0.28 m

0.25 m

0.14 m

Fig. P16.34

0 = 30 rad/s. Find the angular velocities of bars BD and DE in the positionshown.

16.35 The wheel is rolling without slipping. Its center has a constant velocityof 0.6 m/s to the left. Compute the angular velocity of bar BD and the velocity ofend D when = 0.

D

0.6 m/s A

B

0.2 m

0.6 m

D A0.36 m

0.18 m

B

E

Fig. P16.35 Fig. P16.36

16.36 Crank AB rotates with a constant counterclockwise angular velocity of16 rad/s. Calculate the angular velocity of bar BE when = 60.

16.6 Instant Center for Velocities 301

16.37 The hydraulic cylinder raises pin B at the constant rate of 30 mm/s.Determine the speed of end D of the bar AD at the instant shown.

Hydrauliccylinder

80 mm

160 mm

80 m

mA

B

D

30

Fig. P16.37

16.38 In the position shown, the speeds of corners A and B of the right trian-gular plate are vA = 3 m/s and vB = 2.4 m/s, directed as shown. Find (a) theangle ; and (b) the speed of corner D.

420 mm

640 mm

D

A

B

vA

vB20

B D

E

A

0

35

0.60 m

0.50

m 0.30 m

Fig. P16.38 Fig. P16.39

16.39 Bar DE is rotating counterclockwise with the constant angular velocity0 = 5 rad/s. Find the angular velocities of bars AB and BD in the position shown.

16.6 Instant Center for Velocities

The instant center for velocities of a body undergoing plane motion is defined tobe the point that has zero velocity at the instant under consideration.* This pointmay be either in a body or outside the body (in the body extended). It is oftenconvenient to use the instant center of the body in computing the velocities ofpoints in the body.

*Three centers are sometimes used in the kinematic analysis of plane motion: the instant center ofrotation for virtual motion (see Art. 10.6), the instant center for velocities, and the instant center foraccelerations. Each of these points is called simply the instant center when it is clear from the contextwhich center is being used. The discussion of instant center for velocities presented here parallels thediscussion of instant center of rotation for virtual motion in Art. 10.6.


Problems

Note: The following problems are to be solved using instant centers for velocities.16.40 The end of the cord that is wrapped around the hub of the wheel is pulledto the right with the velocity v0 = 0.7 m/s. Find the angular velocity of the wheel,assuming no slipping.

v0

0.45 m

A0.2

4 m

Fig. P16.40

16.41 The wheel rolls without slipping with the angular velocity = 8 rad/s.

0.25 m

y

xA

Fig. P16.41

Determine the coordinates of a point B on the wheel for which the velocity vectoris vB = 2.4i + 0.7j m/s.16.42 The unbalanced wheel rolls and slips along the horizontal plane. At the

Dimensions in mm

400OG

vO = 600 mm/s

= 2 rad/s

200

Fig. P16.42

instant shown, the angular velocity of the wheel and velocity vO of its center areas indicated. Find the magnitude and direction of the velocity of G at this instant.

16.43 A 500-mm diameter wheel rolls and slips on a horizontal plane. Theangular velocity of the wheel is = 12 rad/s (counterclockwise), and the velocityof the center of the wheel is 1.8 m/s to the left. (a) Find the instant center forvelocities of the wheel. (b) Calculate the velocity of the point on the wheel that isin contact with the plane.

16.44 Determine the coordinates of the instant center for velocities of the barAB in (a) and (b).

90 mm

120 mm

A

Bx

y

(a)

O

B

A

y

x

450 mm

240 mm

(b)

Fig. P16.44

16.4016.60 Problems 309

16.45 Find the coordinates of the instant center for velocities of bar AB in(a) and (b).

84 mm

BA

y

x45 60

40 mm

(a)

A

3R

B

R R

No slipping

x

y

(b)

Fig. P16.45

16.46 The arm connected between the centers of gears A and B is rotatingcounterclockwise with the angular velocity of 4.8 rad/s. At the same time, A isrotating at 24 rad/s, also counterclockwise. Determine the angular velocity of B.

BA120

mm80 mm

Fig. P16.46

16.47 The pinion gear meshes with the two racks. If the racks are moving with

C90 mm

0.6 m/s

0.8 m/s

150 mm

Fig. P16.47

the velocities shown, determine the angular velocity of the gear and the velocityof its center C . (Note: This problem was solved as Prob. 16.22 by the method ofrelative velocity.)

16.48 Bar AB is rotating counterclockwise at the constant angular velocity of

A

B C

D

0.24 m

0.12

m

0.18 m

60

Fig. P16.48

6 rad/s. Determine the angular velocity of bar CD when the mechanism is in theposition shown.

16.49 Sketch the locus of the instant center of velocities of bar AB inFig. P16.44(a) as varies from 0 to 90. (This curve is called a space centrode.)

16.50 The 3-m wooden plank is tumbling as it falls in the vertical plane.When the plank is in a horizontal position, the velocities of ends A and B areas shown in the figure. For this position, determine the location of the instant


center for velocities, the angular velocity of the plank, and the velocity of themidpoint G.

3 m

y

xA G B

5 m/s2 m/s

Fig. P16.50

16.51 For the triangular plate undergoing plane motion, vA and the direction of

30

30

60 m

m

60 mm

60 mm

vA = 3 m/s

vB

BA

C

Fig. P16.51

vB are known. Calculate the angular speed of the plate and the speeds of cornersB and C.

16.52 At the instant shown, the angular velocity of the cylinder, which is rollingwithout slipping, is 2 rad/s, counterclockwise. Find the velocity of end B of therod that is pinned to the cylinder at A.

A

45

y

xB

30

0.3 mL

A

B

50 30

Fig. P16.52 Fig. P16.53

16.53 When bar AB is in the position shown, end B is sliding to the right witha velocity of 0.8 m/s. Determine the velocity of end A in this position.

16.54 Slider C of the mechanism has a constant downward velocity of 0.8 m/s.

A

C

B

0.18 m

0.36

m

45

Fig. P16.54

Determine the angular velocity of crank AB when it is in the position shown.

16.55 Bar BC of the linkage slides in the collar D. If bar AB is rotating clock-wise with the constant angular velocity of 12 rad/s, determine the angular velocityof BC when it is in the horizontal position shown.

A

B C

D

60

0.81 m12 rad/s

0.45 m

Fig. P16.55

16.4016.60 Problems 311

16.56 Bar BC of the linkage slides in the collar D. If bar AB is rotating clock-wise with the constant angular velocity of 12 rad/s, determine the angular velocityof bar BC in the position shown.

A B

C

D

0.30 m0.40 m

0.35 m

12 rad/s

Fig. P16.56

16.57 When the mechanism is in the position shown, the angular velocity of barAB is 72 rad/s, clockwise. For this position, compute the angular velocity of theplate BCD and the velocity of corner D.

72 rad/s35

C

D

B

900

mm

600 mm

600 m

m

500 m

m

A

Fig. P16.57

16.58 The crank AB of the mechanism rotates counterclockwise at 8 rad/s.Calculate the velocities of sliders C and D at the instant shown.

D

B

600 mm

45 60 30

8 rad/sA C

Fig. P16.58


16.59 Bar AB of the mechanism rotates clockwise with the angular velocity 0.Compute the angular velocities of bars BD and DE for the position shown.

70

D

A

B

E

b

45

0

b

Fig. P16.59

16.60 When the mechanism is in the position shown, the velocity of the

60 mm

80 mm

OA

B

160 m

m

Fig. P16.60center O of the disk is 0.4 m/s to the right. Assuming that the disk rolls withoutslipping, calculate the velocity of the collar B in this position.

16.7 Method of Relative Acceleration

In Arts. 16.5 and 16.6, we analyzed the velocities of points in a rigid body under-going plane motion. Two methods were presented: the method of relative velocityand instant centers for velocities. In this article, we introduce the method of rela-tive acceleration, which employs the equation aB = aA + aB/A for two points inthe same rigid body.

Figure 16.12(a) shows a rigid body that is undergoing general plane motion.The angular velocity and angular acceleration vectors of the body are and ,respectively. Letting A and B be two points in the body, the acceleration of B withrespect to A is, according to Eqs. (16.8),

aB/A = (aB/A)n + (aB/A)t (16.14a)

where the normal and tangential components of the relative acceleration are

(aB/A)n = ( rB/A) [(aB/A)n = rB/A2](aB/A)t = rB/A [(aB/A)t = rB/A]

(16.14b)

(16.14c)

Substituting Eqs. (16.14) into aB = aA + aB/A gives

aB = aA + ( rB/A) + rB/A (16.15)


Problems

16.61 At a given instant, the endpoints of the bar AB have the accelerations

30

8 m/s2 6 m/s230

200 mm

A

B

Fig. P16.61

shown. Determine the angular velocity and angular acceleration of the bar at thisinstant.

16.62 The wheel rolls on its 0.36 m-radius hub without slipping. The angular

AD

0.60 m 0.36 m

Fig. P16.62

velocity of the wheel is 3 rad/s. Determine the acceleration of point D on the rimof the wheel if the angular acceleration of the wheel is (a) 6.75 rad/s2 clockwise;and (b) 6.75 rad/s2 counterclockwise.

16.63 A string is wrapped around the hub of the spool. A pull at the end ofthe string causes the spool to roll on the horizontal plane without slipping. At acertain instant, the angular velocity and angular acceleration of the spool are asshown in the figure. For this instant, find (a) the acceleration of point D on thespool; (b) the acceleration of point B; and (c) the acceleration a0 of the end of thestring.

16.64 A string is wrapped around the hub of the spool. A pull at the end of thestring causes the spool to roll and slip on the horizontal plane. At a certain instant,the angular velocity and angular acceleration of the spool are as shown in thefigure, while the velocity and acceleration of the end of the string are v0 = 1 m/sand a0 = 2 m/s2, respectively. For this instant, find the acceleration of (a) point Don the spool; (b) point A; and (c) point B.

y

x

0.15 m= 6 rad/s

v0, a0

0.06 m

B

A

D

= 15 rad/s2

B

A

1.2 m

Fig. P16.63, P16.64 Fig. P16.65

16.65 When = 30, the angular velocity of the bar is 2 rad/s counterclock-AvA, aA

B2 m

5050

Fig. P16.66

wise, and the acceleration of slider B is 8 m/s2, directed to the right. Calculate theacceleration of slider A at this instant.

16.66 When the rod AB is in the horizontal position shown, the velocity andacceleration of collar A are vA = 2 m/s and aA = 6 m/s2, directed as shown. Cal-culate the acceleration of collar B and the angular acceleration of the rod in thisposition.

16.6116.82 Problems 323

16.67 The crank AB is rotating clockwise with the constant angular velocity of20 rad/s. Determine the acceleration of piston C when = 90.

A

B

C

0.15 m

0.30

m

A

B

C

1.5 m

1.5 m

1 m

Fig. P16.67 Fig. P16.68

16.68 In the position shown, the angular velocity and angular accelerationof the bar AB are AB = 3 rad/s CW and AB = 12 rad/s2 CCW. Calculate theacceleration of roller C in this position.

16.69 When the mechanism is in the position shown, bar AB is rotating with theangular velocity and angular acceleration , both counterclockwise. Determinethe angular acceleration of bar BC and the acceleration of roller C in this position.

BC

b 2b

, A

Fig. P16.69

16.70 Rod AB of the mechanism is sliding to the right with a constant velocity B

C

A

30

160 mm

Fig. P16.70

of 4 m/s. Determine the acceleration of roller C in the position shown.

16.71 When the mechanism is in the position shown, the velocity of the slidingcollar is vA = 2 m/s, and it is increasing at the rate of 1.2 m/s2. For this position,calculate the angular accelerations of bars AB and BC.

A

B C0.6 m

0.5 m0.4 m

vA

Fig. P16.71


16.72 As the hydraulic cylinder elongates, it raises pin B of the mechanism.

30160

mm

80 mm

80 m

m

A

E

B

D

Hydrauliccylinder

Fig. P16.72

When the system is in the position shown, the velocity of pin B is 40 mm/supward, and it is increasing at the rate of 80 mm/s2. For this instant, determinethe angular accelerations of bars AD and AE.

16.73 Bar AB is rotating clockwise with the constant angular velocity of20 rad/s. For the position shown, determine the angular accelerations of bars BDand DE.

0.3 m

0.8 m

0.4 m

E D

B

A20 rad/s

Fig. P16.73

16.74 The wheel rolls without slipping with the constant clockwise angularvelocity of 0.8 rad/s, as end B of bar AB slides on the ground. Calculate theacceleration of B in the position shown.

0.6 m

OA

0.8 rad/s

B1.5 m

Fig. P16.74

16.75 Bar BC of the mechanism rotates clockwise with the constant angularvelocity of 24 rad/s. Determine the angular accelerations of bars AB and CD inthe position shown.

B C

A D

4545 0.75 m

0.3 m

0.3 m

Fig. P16.75

16.76 In the position shown, the angular velocity and angular acceleration of

A

B

C

D

45

0.5 m

0.3 m

0.3 m

45

Fig. P16.76

bar CD are 6 rad/s and 20 rad/s2, respectively, both counterclockwise. Computethe angular accelerations of bars AB and BC in this position.

16.77 Bar AB of the mechanism rotates with the constant angular velocity of

0.3 m 0.12 m

0.16

m

A0.12 m

E

3 rad/s

D

B

Fig. P16.77

3 rad/s counterclockwise. For the position shown, calculate the angular accelera-tions of bars BD and DE.

16.6116.82 Problems 325

16.78 The wheel rolls without slipping on the horizontal surface. In the posi- 0.6 m

0.8 m0.3 m

A

O

B

G

Fig. P16.78

tion shown, the angular velocity of the wheel is 4 rad/s counterclockwise, and itsangular acceleration is 5 rad/s2 clockwise. Find the angular acceleration of rod ABand the acceleration of slider B in this position.

16.79 The disk is rotating counterclockwise with the constant angular speedof 2 rad/s. For the position shown, find the angular accelerations of bars ABand BD.

B0.3 m

2 rad/s

D

A0.9 m 0.3 m

O

Fig. P16.79

16.80 The arm joining the friction wheels A and B is rotating with the angular B

A rA = 60 mm

rB = 25 mm

Fig. P16.80

velocity = 5 rad/s and the angular acceleration = 12.5 rad/s2, both coun-terclockwise. Assuming that wheel A is stationary and that there is no slipping,determine the magnitude of the acceleration of the point on the rim of B that is incontact with A.

16.81 When the mechanism is in the position shown, the angular velocity of thegear is 2 rad/s clockwise, and its angular acceleration is 4 rad/s2 counterclockwise.Determine the angular accelerations of bars AB and BD in this position.

30O60

A

150

B

D

60

150

Dimensions in mm

Fig. P16.81

16.82 Bar AB of the mechanism rotates with the constant angular velocity

0.6

m

DB

y

x45

A E

0.8 m

0.5 m

1.2 rad/s

Fig. P16.82

1.2 rad/s clockwise. For the position shown, (a) verify that the angular velocitiesof the other two bars are BD = 1.358 rad/s counterclockwise and DE = 1.131rad/s clockwise; and (b) determine the acceleration vector of point D.

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