Hibeleer

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© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
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R2–1. An automobile transmission consists of theplanetary gear system shown. If the ring gear R is held fixed
so that vR = 0, and the shaft s and sun gear S, rotates at
20 rad> s, determine the angular velocity of each planet gear
P and the angular velocity of the connecting rack D, which
is free to rotate about the center shaft s.

R

P

vR

2 in.
S
20 rad/s

8 in.
4 in.

For planet gear P: The velocity of point A is yA = vs rs =20 a
vB = vA + vB>A

0 = C 6.6 67 D + B vP a
:

;

+
a:b

4
b = 6.667 ft> s.
12

4
bR
12

4
0 = 6.667 - vP a b
12

vP = 20 rad> s

Ans.

For connecting rack D:
vC = vA + vC>A

c y C d = C 6.667 D + B 20 a
:

:

;

+
a:b

2
bR
12

2
yC = 6.667 - 20 a b
12

yC = 3.333 ft> s

The rack is rotating about a fixed axis (shaft s). Hence,
yC = vD rD
3.333= vD a

6
b
12

vD = 6.67 rad> s

Ans.

8 27

s

D
P

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© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing fromthe publisher.

R2–2. An automobile transmission consists of the
planetary gear system shown. If the ring gear R rotates at
vR = 2 rad> s, and the shaft s and sun gear S, rotates at
20 rad> s, determine the angular velocity of each planet gear
P and the angular velocity of the connecting rack D, which
is free to rotate about the center shaft s.

R

P

vR

2 in.
S
20 rad/s

8 in.4 in.

4
For planet gear P: The velocity of points A and B are yA = vS rS = 20 a b
12
8
= 6.667 ft> s and yB = vB rB = 2 a b = 1.333 ft> s.
12
vB = vA + vB>A

c 1.333 d = c 6.667 d + B vP a
;

:

;

+
a:b

- 1.333 = 6.667 - vP a

4
b
12

4
bR
12

vP = 24 rad> s

Ans.

For connecting rack D:
vC = vA + vC>A

c y C d = c 6.667 d + B 24 a
:

:

;

+
a:byC = 6.667 - 24 a

2
b
12

2
bR
12

yC = 2.667 ft> s

The rack is rotating about a fixed axis (shaft s). Hence,
yC = vD rD
2.667 = vD a

6
b
12

vD = 5.33 rad> s

Ans.

8 28

s

D
P

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© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyrightlaws as they currently
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R2–3. The 6-lb slender rod AB is released from rest when
it is in the horizontal position so that it begins to rotate
clockwise. A 1-lb ball is thrown at the rod with a velocity
v = 50 ft> s. The ball strikes the rod at C at the instantthe
rod is in the vertical position as shown. Determine the
angular velocity of the rod just after the impact. Take
e = 0.7 and d = 2 ft.

A

d
3 ft

C

Datum at A:
v

T1 + V1 = T2 + V2
0+0=

11
6
ca
b (3)2 d v2 - 6(1.5)
2 3 32.2

B

v = 5.675 rad> s
a+

(HA)1 = (HA)2

6
1
16
1
1
(50)(2) - c a
b (3)2 d (5.675) = c a
b (3)2 d v2 +
(v )(2)
32.2
3 32.2
3 32.232.2 BL
e = 0.7 =

50 ft/s

vC - vBL
50 - [ - 5.675(2)]

vC = 2v2
Solving,
v2 = 3.81 rad> s

Ans.

vBL = - 35.3 ft> s
vC = 7.61 ft> s

8 29

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© 2010 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently
exist. No portion of this...