Dynamics in Engineering Practice 6 Edition Chapter 2: SELECTED...
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Dynamics in Engineering Practice 6th Edition Chapter 2: SELECTED ANSWERS:
Problem 2.1
2( ) ( 2 )
r r
r r r r
θ
θ θ θ
= +
= − + +r θ
r θ
r ε ε
r ε ε
Problem 2.2
2 2v vv sρ ρ
= + = +t n t na ε ε ε ε
Problem 2.3 2vsρ
= +t na ε ε
Problem 2.4
2
1 mm/s
2
0.5 mm/s
2
X
Y
X
Y
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
0.2545 mm/s
2.221
0.6733 mm/s
1.948
r
r
vv
aa
θ
θ
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
2.236mm/s
0
2.012mm/s
0.4474
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.5
2
0.383 m/s
4.24
9.24 m/s
84.9
X
Y
X
Y
vv
aa
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
3.86 m/s
1.797
83.01 m/s
20.07
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
4.26 m/s
0
83.72m/s
16.84
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.6
2
551.5 mm/s
1103.1
6927 mm/s
13854
X
Y
X
Y
vv
aa
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
740 mm/s
986.5
15490 mm/s
0
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
1233mm/s
0
9294mm/s
12390
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.7
2
5.4 mm/s
5.25
0.4 mm/s
0.25
X
Y
X
Y
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
3.54 mm/s
6.65
0.305 mm/s
0.360
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
7.53 mm/s
0
0.461mm/s
0.0996
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.8
2
6.928 in/s
6.928
8 in/s
16
X
Y
X
Y
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
9.295 in/s
3.099
0.00158 in/s
17.89
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
9.798in/s
0
5.657in/s
16.97
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.9
a) sec180.2118.3
ftX
Y
vv
⎧ ⎫ ⎡ ⎤=⎨ ⎬ ⎢ ⎥−⎣ ⎦⎩ ⎭
; b) 'sec
'
12.35215.22
ftX
Y
vv
−⎧ ⎫ ⎡ ⎤=⎨ ⎬ ⎢ ⎥−⎣ ⎦⎩ ⎭
Problem 2.10
02
0
02
0 6
r rrv a
rv aθ θ
ωπ ω
⎧ ⎫−⎧ ⎫⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬−⎩ ⎭ ⎩ ⎭⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭ ;
200
20 0
3124
34 12
X X
Y Y
rrv av ar r
πωω
ω πω
⎧ ⎫⎧ ⎫−⎪ ⎪⎧ ⎫ ⎧ ⎫⎪ ⎪
= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬− −⎩ ⎭ ⎩ ⎭⎪ ⎪ ⎪ ⎪
⎩ ⎭ ⎩ ⎭
;0
20
02
0 6
t t
n n
rv arv a
ωπω
⎧ ⎫⎧ ⎫⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭
Dynamics in Engineering Practice Chapter 2 Answers Spring 2007
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Problem 2.11
2
300 mm/s
1600
12800 mm/s
8800
r
r
vv
aa
θ
θ
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
1060 mm/s
1236
6685 mm/s
14021
X
Y
X
Y
vv
aa
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
1628.3mm/s
0
6295mm/s
14200
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.12
2
117.3 m/s
236.4
831 m/s
1174.8
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
263.8 m/s
8.28
722.4 m/s
1245
X
Y
X
Y
vv
aa
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
263.9m/s
0
683mm/s
1267
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.13
2
3265 mm/s
1850
1695090 mm/s
241960
r
r
vv
aa
θ
θ
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
1375 mm/s
3492
1165719 mm/s
1254183
X
Y
X
Y
vv
aa
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
3753mm/s
0
1594042mm/s
625274
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.14
2
500 m/s
220
17 m/s
78
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
509.1 m/s
198
43.13 m/s
67.18
X
Y
X
Y
vv
aa
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
546.2m/s
0
15.78m/s
78.26
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.15
2
1.102 m/s
7.383
53.31 m/s
13.85
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
6.00 m/s
4.441
27.90 m/s
47.49
X
Y
X
Y
vv
aa
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
7.464m/s
0
5.88m/s
54.85
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.16
26.9498 14.398 ft/s ft/s
0 17.28t t
n n
v av a⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫
= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭
; 26.214 20.61ft/s ft/s
3.112 9.00X X
Y Y
v av a
⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬−⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭
; 26.838 11.08 ft/s ft/s
1.242 19.57r rv a
v aθ θ
⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭
Problem 2.17 Let θ=3.810 in the CCW direction
2;27.78 2
0 0.7617t t
n n
ms
v am s
v a= =
⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬
⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭; 27.78
m/s0
X
Y
vv
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
; 22m/s
0.7617X
Y
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
27.72m/s
1.846rv
vθ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
; 21.945m/s
0.893ra
aθ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.18
2200 1 m/s m/s
0 0.4565t t
n n
v av a
−⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭
; 248.51 0.685m/s m/s
194.03 0.859X X
Y Y
v av a
−⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬−⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭
; 2195.24 1.075 m/s m/s
43.39 0.229r rv a
v aθ θ
−⎧ ⎫ ⎧ ⎫⎧ ⎫ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭⎩ ⎭ ⎩ ⎭
Problem 2.19
2
20 m/s
0
2 m/s
1.432
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
17.89 m/s
8.95
1.1483 m/s
2.175
X
Y
X
Y
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
19.53m/s
4.34
1.164m/s
1.831
r
r
vv
aa
θ
θ
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.20
2
40 ft/s
0
0.32 ft/s
27.31
t
n
t
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
22.19 ft/s
33.3
22.89 ft/s
14.89
X
Y
X
Y
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
5.78ft/s
39.6
26.98ft/s
4.252
r
r
vv
aa
θ
θ
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Dynamics in Engineering Practice Chapter 2 Answers Spring 2007
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Problem 2.21
2
sec
sec
50
1.55.674
t m
n
t m
n
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭−⎧ ⎫ ⎧ ⎫
=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
2
sec
sec
1.7334.69
5.8420.559
X m
Y
X m
Y
vv
aa
⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
2
sec
sec
3.3483.713
3.2094.913
r m
r m
vv
aa
θ
θ
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬−⎩ ⎭⎩ ⎭
−⎧ ⎫ ⎧ ⎫=⎨ ⎬ ⎨ ⎬⎩ ⎭⎩ ⎭
Problem 2.6
a
v
θ
γ
β
r
Y
X
β’
εt
εn
εθ
εr
a
v
θ
γ
β
r
Y
X
β’
Problem 2.7a
εθ
εr
γ
v
β’ r
Y
X
a
θ β
Problem 2.9
Y X’
Y’ vX
v vX
vY’
X
vX’
θ
n
γ
v β’ r
Y
X a θ
t
θ
r
Problem 2.4
γ
v
β’r
Y
X
a
θβ
εn’
εt
εn
Problem 2.7b Problem 2.8
v
β
r
Y
X
a
γ
θ
t
n
θ
r
Problem 2.5
εθ
εr
a
θ
v
γ
r
Y
X
β β’
εn
εt
a
θ
v
γ
r
Y
X
β β’
Dynamics in Engineering Practice Chapter 2 Answers Spring 2007
620
Problem 2.10
J
Y θ εθ εr
θ
θ
a r
I
X
v
β
et
en
Problem 2.15
εθ
εr
v
a
β
θ
Y
X
r εn
εt v
a
β
θ
Y
X
r
εr
εn
εθ
εt
v
a
β θ
Xr
Y
Problem 2.11
εr
εθ
v
a
β
θ
Y
X
r
β’
εn
εt v
a
β
θ
Y
X
r
β’
Problem 2.12
εθ εr
β
Y
r
θ
v a
X
εn
εt
β
Y
r
θ
va
X
β’
Problem 2.13
a
θ
r
θy
A
vr
β
X
Y y
x
r
t
n
vθ
θ v
Problem 2.14
Dynamics in Engineering Practice Chapter 2 Answers Spring 2007
621
X
εθ
εr
εt
Y εn
β’
θ’
r
v
a
Problem 2.16
εt
εn
v
X
Y
θr
a
Problem 2.17
Problem 2.18 εt
εn
v
a
X
Y
θ
β
r
εr
εθ
v
a
X
Y
θ
β
r
εr
εθ
θ
a v
X
Y
β
r
εt εn
θ
a v
X
Y
β
r
Problem 2.19
β
εt
εn
β’ r
Y
X
θ
a
v
Problem 2.20
εθ εr
β’r
Y
X
θ
a
v
β
0 1 2 3 4 5 60
1
2
3
4
5
6
pathobject
X (m)
Y (m
)
6
0
Y j dx.( )
Y X( )
60 j dx. X,
Y
X
v
θr
r
θ
aXaY
vX
vY
aθ
ar
vθvr
Problem 2.21
v
X
Y
θ r
a
εr
εθ
Dynamics in Engineering Practice Chapter 2 Answers Spring 2007
