Mechanical Engineering Chapter 17 Problem Problem Bar Rotatesyd With Constant Angular Velocity Rads Thecounterclockwise Direction Determine

Problem 17.166 In Problem 17.165, bar AB rotates

with a constant angular velocity of 6 rad/s in the

counterclockwise direction. Determine the acceleration

of point D.

y

D

C

8 in

12 in

angular acceleration of point Cis

Problem 17.167 Point Cis moving to the right at

20 in./s. What is the velocity of the midpoint Gof bar

BC?

A

B

4 in

3 in

y

G

Solution:

Problem 17.168 In Problem 17.167, point Cis mov-

ing to the right with a constant velocity of 20 in/s. What

is the acceleration of the midpoint Gof bar BC?

B

4 in

3 in

y

G

Solution: See the solution of Problem 17.167.

aB=aA+αAB ×rB/A −ω2

AB rB/A

452

Problem 17.169 In Problem 17.167, if the velocity of

point Cis vC=1.0i(in./s), what are the angular veloc-

ity vectors of arms AB and BC?

B

4 in

3 in

y

G

Solution: Use the solution to Problem 17.167: The velocity of the

point Bis determined from the known velocity of point Cand the

The angular velocity of bar AB is determined from the velocity of B.

ij k



1.0−7ωBC =−4ωAB ,−10ωBC =4ωAB .

Problem 17.170 Points Band Care in the x–y

plane. The angular velocity vectors of arms AB and

BC are ωAB =−0.5k(rad/s) and ωBC =−2.0k(rad/s).

Determine the velocity of point C.

760 mm

y

z

B

C

Solution: The vector

rB/A =760(icos 15◦−jsin 15◦)=734.1i−196.7j(mm).

rC/B =900(icos 50◦+jsin 50◦)=578.5i+689.4j(mm).

The velocity of point Bis

Problem 17.171 In Problem 17.170, if the velocity

vector of point Cis vC=1.0i(m/s), what are the

angular velocity vectors of arms AB and BC?

760 mm

y

C

Solution: Use the solution to Problem 17.170.

The vector

rB/A =760(icos 15◦−jsin 15◦)=734.1i−196.7j(mm).

578.5 687.40

1000i=196.7ωAB i+734.1ωAB j−687.4ωBCi+578.5ωBCj(mm/s).

Problem 17.172 The angular velocity vectors of arms

AB and BC are ωAB =−0.5k(rad/s) and ωBC =

2.0k(rad/s), and their angular accelerations are αAB =

1.0k(rad/s2), and αBC =1.0k(rad/s2). What is the

acceleration of point C?

A15⬚

760 mm

900 mm

y

x

z

B

50⬚

C

Solution: Use the solution to Problem 17.170.

454

x

Problem 17.173 The velocity of point Cis vC=

1.0i(m/s) and aC=0. What are the angular velocity

and angular acceleration vectors of arm BC?

A15⬚

760 mm

900 mm

y

x

z

B

50⬚

C

578.5 689.40

1000i=196.7ωAB i+734.1ωAB j−689.4ωBCi

734.1−196.70

−(ω2

AB )(734.1i−196.7j)(mm/s)2.

aB=196.7αAB i+734.1αAB j−639.0i+171.3j(mm/s2)

Solve: αAB =2.24 rad/s2,αBC =−1.465 (rad/s2)

Problem 17.174 The crank AB has a constant

clockwise angular velocity of 200 rpm. What are the

velocity and acceleration of the piston P?

6 in2 in

2 in A

B

P

Solution:

456

Problem 17.175 Bar AB has a counterclockwise

angular velocity of 10 rad/s and a clockwise angular

acceleration of 20 rad/s2. Determine the angular

acceleration of bar BC and the acceleration of point C.

y

6 in

8 in

10 rad/s

AB

Solution: Choose a coordinate system with the origin at the left

The velocity of point Bis

vB=ωBC ×rB/IC =



ij k

00ωBC



=−6ωBCj=80j,

80

6−40

The acceleration of point Cis constrained to be parallel to the xaxis.

Separate components:

Problem 17.176 The angular velocity of arm AC is

1 rad/s counterclockwise. What is the angular velocity

of the scoop?

D

C

1 m

Problem 17.177 The angular velocity of arm AC in

Problem 17.176 is 2 rad/s counterclockwise, and its

angular acceleration is 4 rad/s2clockwise. What is the

angular acceleration of the scoop?

D

C

1 m

Problem 17.178 If you want to program the robot so

that, at the instant shown, the velocity of point Dis

vD=0.2i+0.8j(m/s) and the angular velocity of arm

CD is 0.3 rad/s counterclockwise, what are the neces-

sary angular velocities of arms AB and BC?

x

D

C

y

B

30°

20°

A

250 mm

300 mm

250 mm

Solution: The position vectors are:

460

Problem 17.179 The ring gear is stationary, and the

sun gear rotates at 120 rpm (revolutions per minute) in

the counterclockwise direction. Determine the angular

velocity of the planet gears and the magnitude of the

velocity of their centerpoints.

20 in

Sun gear

34 in

Planet gears (3)

Ring gear

7 in

from which ωS=4πk(rad/s). At the point of contact between the sun

The velocity of the centerpoint of the top most planet gear is

S

Problem 17.180 Arm AB is rotating at 10 rad/s in the

clockwise direction. Determine the angular velocity of

the arm BC and the velocity at which the arm slides

relative to the sleeve at C.

30⬚

A

B

C

2 m

1.8 m

1.56 0.90

The unit vector from Bto Cis

462

Problem 17.181 In Problem 17.180, arm AB is rotat-

ing with an angular velocity of 10 rad/s and an angular

acceleration of 20 rad/s2, both in the clockwise direction.

Determine the angular acceleration of arm BC.

30⬚

A

B

C

2 m

1.8 m

Solution: Use the solution to 17.180. The vector

Collecting terms,

Problem 17.182 Arm AB is rotating with a constant

counterclockwise angular velocity of 10 rad/s. Deter-

mine the vertical velocity and acceleration of the rack R

of the rack-and-pinion gear.

B

C

12 in

D

6 in

10 in

Solution: The vectors:

464

Problem 17.183 The rack Rof the rack-and-pinion

gear is moving upward with a constant velocity of

10 ft/s. What are the angular velocity and angular

acceleration of bar BC ?

B

C

12 in

6 in

10 in

Problem 17.184 Bar AB has a constant counterclock-

wise angular velocity of 2 rad/s. The 1-kg collar C slides

on the smooth horizontal bar. At the instant shown, what

is the tension in the cable BC?

A

B

C

1 m 2 m

2 m

Solution:

ijk



466

Problem 17.185 An athlete exercises his arm by rais-

ing the 8-kg mass m. The shoulder joint A is stationary.

The distance AB is 300 mm, the distance BC is 400 mm,

and the distance from Cto the pulley is 340 mm. The

angular velocities ωAB =1.5 rad/s and ωBC =2 rad/s

are constant. What is the tension in the cable?

A

B

m

C

ω

AB

ω

BC 30°

60°

Solution:

y

e

Problem 17.186 The hydraulic actuator BC of the

crane is extending (increasing in length) at a constant

rate of 0.2 m/s. When the angle β=35o, what is the

angular velocity of the crane’s boom AD ?

3m 2m

2 m

AB

C

D

b

Solution: Using AD we have

Problem 17.187 The coordinate system shown is ﬁxed

relative to the ship B. The ship uses its radar to measure

the position of a stationary buoy A and determines it to

be 400i+200j(m). The ship also measures the velocity

of the buoy relative to its body-ﬁxed coordinate sys-

tem and determines it to be 2i−8j(m/s). What are the

ship’s velocity and angular velocity relative to the earth?

(Assume that the ship’s velocity is in the direction of the

yaxis).

y

x

A

B

Solution:

468