Mechanical Engineering Chapter 16 Homework Peg The Gear Slides Freely Along The

781

16–143.

SOLUTION

Gear Motion:The IC of the gear is located at the point where the gear and

Since the gear rolls on the gear rack, .By referring to Fig. b,

Reference Frame: The rotating reference frame is attached to link AB and

coincides with the XYZ fixed reference frame, Figs. cand d.Thus, vBand aBwith

respect to the XYZ frame is

For motion of the frame with reference to the XYZ reference frame,

For the motion of point Bwith respect to the frame is

Equating the iand jcomponents yields

Acceleration: Applying the relative acceleration equation.

x¿y¿z¿

a=aO

r=1.5

0.15 =10 rad>s

Peg Bon the gear slides freely along the slot in link AB.If

the gear’s center Omoves with the velocity and

acceleration shown, determine the angular velocity and

angular acceleration of the link at this instant.

Equating the icomponents,

vO  3 m/s

aO  1.5 m/s2

A

O

B

600 mm

150 mm

Ans:

782

Ans:

*16–144.

The cars on the amusement-park ride rotate around the

axle at Awith a constant angular velocity

measured relative to the frame AB. At the same time the

frame rotates around the main axle support at Bwith a

constant angular velocity Determine the

velocity and acceleration of the passenger at Cat the

instant shown.

vf=1 rad>s.

vA>f=2 rad>s,

D

8ft

v

f

1 rad/s

A

B

x

y

Motion of

moving refernce

Motion of C with respect

to moving reference

Motion of A:

Substitute the data into Eqs.(1) and (2) yields:

vA=v*rA>B

SOLUTION

783

16–145.

Aride in an amusement park consists of a rotating arm AB

having a constant angular velocity about point

Aand a car mounted at the end of the arm which has a

constant angular velocity measured

relative to the arm. At the instant shown, determine the

velocity and acceleration of the passenger at C.

V¿=5–0.5k6rad>s,

vAB =2 rad>s

SOLUTION

rB>A=(10 cos 30° i+10 sin 30° j)={8.66i+5j}ft

60

30

B

C

A

x

y2ft

10 ft

vAB 2 rad/s

v¿0.5 rad/s

Ans:

784

16–146.

SOLUTION

rB>A=(10 cos 30°i+10 sin 30°j)={8.66i+5j}ft

Aride in an amusement park consists of a rotating arm AB

that has an angular acceleration of when

at the instant shown. Also at this instant the

car mounted at the end of the arm has an angular

acceleration of and angular velocity of

measured relative to the arm.

Determine the velocity and acceleration of the passenger C

at this instant.

V¿=5–0.5k6rad>s,

A

= 5–0.6k6rad>s2

vAB =2 rad>s

aAB =1 rad>s2

60

30

B

C

A

x

y2ft

10 ft

vAB 2 rad/s

v¿0.5 rad/s

Ans:

785

16–147.

If the slider block C is fixed to the disk that has a constant

counterclockwise angular velocity of 4 rad

>

s, determine the

angular velocity and angular acceleration of the slotted arm

AB at the instant shown.

Ans:

180 mm

B

40 mm

C60 mm

v  4 rad/s

30



SOLUTION

vC=–(4)(60) sin 30°i–4(60) cos 30°j=–120i–207.85j

786

*16–148.

SOLUTION

Reference Frame: The xyz rotating reference frame is attached to car Cand

coincides with the XYZ fixed reference frame at the instant considered, Fig. a. Since

At the instant shown, car Atravels with a speed of ,

which is decreasing at a constant rate of , while car C

travels with a speed of , which is increasing at a

constant rate of . Determine the velocity and

acceleration of car Awith respect to car C.

3m>s2

15 m>s

2m>s2

25 m

>

s

250 m

15 m/s

2m/s2

200 m

A

B

15 m/s

3m/s2

25 m/s

2m/s2

C

45

Also, the angular velocity and angular acceleration of the xyz reference frame is

The velocity and accdeleration of car Awith respect to the XYZ frame is

Velocity: Applying the relative velocity equation,

Acceleration: Applying the relative acceleration equation,

787

16–149.

At the instant shown, car Btravels with a speed of 15 ,

which is increasing at a constant rate of , while car C

travels with a speed of which is increasing at a

constant rate of . Determine the velocity and

acceleration of car Bwith respect to car C.

3 m>s2

15 m>s,

2m>s2

m

>

s

SOLUTION

Reference Frame: The xyz rotating reference frame is attached to Cand coincides

250 m

15 m/s

2m/s2

200 m

A

B

15 m/s

3m/s2

25 m/s

C

45

Also, the angular velocity and angular acceleration of the xyz reference frame with

respect to the XYZ reference frame are

From the geometry shown in Fig. a,

Velocity: Applying the relative velocity equation,

Acceleration: Applying the relative acceleration equation,

aB=aC+v

#*rB>C+v*(v*rB>C)+2v*(vrel)xyz +(arel)xyz

788

16–150.

SOLUTION

AB 2.5 rad/s

45

30

150 mm

C

A

v

B

D

rBA

sin 120° =0.15 m

sin 45°

The two-link mechanism serves to amplify angular motion.

Link AB has a pin at B which is confined to move within the

slot of link CD. If at the instant shown, AB (input) has an

angular velocity of vAB = 2.5 rad>s, determine the angular

velocity of CD (output) at this instant.

789

16–151.

The disk rotates with the angular motion shown. Determine

the angular velocity and angular acceleration of the slotted

link AC at this instant. The peg at B is fixed to the disk.

A

C

v  6 rad/s

a  10 rad/s2

30



30



0.3 m

0.75 m

B

SOLUTION

vB=–6(0.3)i=–1.8i

Ans:

790

*16–152.

The Geneva mechanism is used in a packaging system to

convert constant angular motion into intermittent angular

motion. The star wheel Amakes one sixth of a revolution

for each full revolution of the driving wheel Band the

attached guide C.To do this, pin P, which is attached to B,

slides into one of the radial slots of A, thereby turning

wheel A, and then exits the slot. If Bhas a constant angular

velocity of , determine and of wheel A

at the instant shown.

AA

VA

vB=4 rad>s

SOLUTION

The circular path of motion of Phas a radius of

Thus,

Solving,

A

vB

BC

P

4 in.

u 30

4 rad/s

Ans: