Solution manual vector mechanics engineers dynamics 8th beer chapter 15

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P.. Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P.. Vector Mechanics for Engineers: Statics

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

α =

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Chapter 15, Solution 4

Angular coordinate: θ =1800 rev = 3600 radiansπ

Initial angular velocity: ω0 =6000 rpm = 200 rad/sπ

Angular acceleration: d d constant

12

αθ = − ω

2 2

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

π

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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0.2 0

0 30 te− tdt

= + ∫

0.2 0

300.2

t t

0 150 t 1 e− t dt

0.2 0

150150

0.2

t t

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

A O OA

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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A O OA

A O OA

(3.46 in./s2) (27.6 in./s2) (73.1in./s2)

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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B A AB

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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B A ABl

B A AB

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

DA DAd

λλλλ

( )(75 0.8 + 0.48 + 0.36 ) (60 rad/s) (+ 36 rad/s) (+ 27 rad/s)

DAω

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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DA

ADd

λλλλ

( )(75 0.8 + 0.48 + 0.36 ) (60 rad/s) (+ 36 rad/s) (+ 27 rad/s)

DAω

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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2

199.11 10 rad/s31.557 10

π

×Velocity of the earth

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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Chapter 15, Solution 16

3

23 h 56 min = 23.933 h =86.16 10 s,× 1 rev =2 radπ

6 3

2

72.925 10 rad/s86.16 10

a =

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Let v and B a be the belt speed and acceleration These are given as B vB =12 ft/s and aB =96 ft/s 2

These are also the speed and tangential acceleration of periphery of each pulley provided no slipping occurs (a) Angular velocity and angular acceleration of each pulley

Pulley A r =A 8 in 0.6667 ft=

12

18 rad/s0.6667

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Chapter 15, Solution 20

1rev radians, 8 in 0.6667 ft, 5 in 0.41667 ft

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Left pulley

Inner radius r1 = 50 mm Outer radius r2 = 100 mm 0.6 m/s = 600 mm/s

A

v =

1 2

600

6 rad/s100

Avr

Inner radius r3 = 50 mm Outer radius r4 = 100 mm

1 2 4

300

3 rad/s100

vr

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Chapter 15, Solution 22

Left pulley

Inner radius r1 = 50 mm Outer radius r2 = 100 mm 0.6 m/s = 600 mm/sA

v =

( )aA t = −1.8 m/s = 1800 mm/s

1 2

600

6 rad/s100

A

vr

1 2

1800

18 rad/s100

A t

ar

2 4

300

3 rad/s100

vr

2 4

900

9 rad/s100

t

ar

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

( )b On shaft A : aA =rωA

2 1

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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(a) Let point C be the point of contact between the shaft and the ring

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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Let points A, B, and C lie at the axles of gears A, B, and C, respectively

Let D be the contact point between gears A and B

D B

vr

E C

vr

20

2

E B

20

6

E C

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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D t B

D B

ar

E t C

E C

ar

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

( ) For the pulley, a 1 , 1( )0.3 0.15 m

( )

1 0.2 0.1 m2

B

/ /

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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( )a aA B / = aA−aB = −1.9337 m/s2 or 1.9337 m/s2

2 / 1.934 m/s

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

2

28.3 in./s

D =

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

When contact is made, ωA =240 rpm =8 rad/sπ

Let C be the contact point between the two gears

C B B

vr

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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Equating the two expressions for v , C

A A

tt

t

αα

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Motion of disk B ( )ωB 0=500 rpm =52.360 rad/s

Assume that the angular acceleration of disk B is constant

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

A A A

vr

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Let one layer of tape be wound and let v be the tape speed

ωπ

=  −  =

2 0

2

bωπ

=

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

2

bvr

π

=

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

A B AB

vl

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

B A AB

vl

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Geometry

12sin , 36.87

20

12tan , 67.38

5

θ = θ = ° Velocity analysis

4.2 ft/s

A =v

/sin sin 90 sin

θ = ° − β = φ/ sin 4.2sin 67.38 4.5 ft/ssin sin 59.49

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Chapter 15, Solution 40

Geometry

12sin , 36.87

20

12tan , 67.38

5

θ = θ = ° Velocity analysis

4.2 rad/s

AB =ω

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

ω = − − − =

From (2), ( )vB y = − +7 ( )( )0.6 10 = −1 m/s

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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: 0= − +1 10 ,x

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

: 0 175 2 ,= + x x = −87 mmj

2

vr

ω

Circle of 100.0 mm radius centered at x = −87.5 mm,y =50.0 mm

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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7Slope angle of rod tan 0.7, 35

10

θ = = θ = °

10 12.2066 in. 20 7.7934 in.cos

Draw corresponding vector diagram

/ sin 25sin35 14.34 in./s

vAC

vv

φ θ+ = °

1.869 ft/s

B =

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

35.264

φ = ° Velocity analysis

60 24.736

β = ° − =φ °

90 125.264

γ = ° + =φ ° Law of sines sin/ sin30 sin

γ = ° = β/ sin 14.4 sin125.264 28.10 in./ssin sin 24.736

sin sin 24.736

A

B vv

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Label the contact point between gears A and B as 1, the center of gear B

as 2, and the contact point between gears B and C as 3

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

v = +ω d

471.240.5

A

C vd

ω = =942.48 rad/s

14137.2

2 9.4248

lnr

π

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

© 2007 The McGraw-Hill Companies

Chapter 15, Solution 50

Contact point 1 between gears A and B

Contact point 2 between gears B and C

Matching expressions (1) and (3) for v 1,

Vector Mechanics for Engineers: Statics and Dynamics, 8/e, Ferdinand P Beer, E Russell Johnston, Jr.,

Elliot R Eisenberg, William E Clausen, David Mazurek, Phillip J Cornwell

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Let a be the radius of the central gear A, and let b be the radius of the planetary gears B, C, and D The radius of the outer gear E is a+2 bLabel the contact point between gears A and B as 1, the center of gear B

as 2, and the contact point between gears B and E as 3