Section 5.4

1. Frequency: 2 rad/s ( $1 / π HZ);$ $1 / π HZ);$ period: $π s$ $π s$
2. Frequency: 8 rad/sec ( $4 / π HZ);$ $4 / π HZ);$ period: $π / 4 sec$ $π / 4 sec$
3. Amplitude: 2m; frequency: 5 rad/s; period: $2 π / 5 s$ $2 π / 5 s$
4. (a) $x (t) = \frac{13}{12} \cos (12 t - α)$ $x (t) = \frac{13}{12} \cos (12 t - α)$ with $α = 2 π - \tan^{- 1} (5 / 12) \approx 5.8884$ $α = 2 π - \tan^{- 1} (5 / 12) \approx 5.8884$ ; (b) Amplitude: $\frac{13}{12}$ $\frac{13}{12}$ m; period: $π / 6 sec$ $π / 6 sec$
6. About 7.33 mi
7. About 10450 ft
8. 29.59 in.
10. Amplitude: 100 cm; period: about 2.01 sec
11. About 3.8 in.
13. (a) $x (t) = 50 (e^{- 2 t / 5} - e^{- t / 2})$ $x (t) = 50 (e^{- 2 t / 5} - e^{- t / 2})$ ; (b) 4.096 exactly
14. (a) $x (t) = 25 e^{- t / 5} \cos (3 t - α)$ $x (t) = 25 e^{- t / 5} \cos (3 t - α)$ with $α = \tan^{- 1} (3 / 4) \approx 0.6435$ $α = \tan^{- 1} (3 / 4) \approx 0.6435$ ; (b) envelope curves $x = \pm 25 e^{- t / 5}$ $x = \pm 25 e^{- t / 5}$ ; pseudoperiod $2 π / 3$ $2 π / 3$
15. $x (t) = 4 e^{- 2 t} - 2 e^{- 4 t}$ $x (t) = 4 e^{- 2 t} - 2 e^{- 4 t}$ , overdamped; $u (t) = 2 \cos (2 \sqrt{2} t)$ $u (t) = 2 \cos (2 \sqrt{2} t)$
16. $x (t) = 4 e^{- 3 t} - 2 e^{- 7 t}$ $x (t) = 4 e^{- 3 t} - 2 e^{- 7 t}$ , overdamped; $u (t) \approx 2 \sqrt{\frac{22}{21}} \cos (\sqrt{21} t - 0.2149)$ $u (t) \approx 2 \sqrt{\frac{22}{21}} \cos (\sqrt{21} t - 0.2149)$
17. $x (t) = 5 e^{- 4 t} (2 t + 1)$ $x (t) = 5 e^{- 4 t} (2 t + 1)$ , critically damped; $u (t) \approx \frac{5}{2} \sqrt{5} \cos (4 t - 5.8195)$ $u (t) \approx \frac{5}{2} \sqrt{5} \cos (4 t - 5.8195)$
18. $x (t) = 2 e^{- 3 t} \cos (4 t - \frac{3 π}{2})$ $x (t) = 2 e^{- 3 t} \cos (4 t - \frac{3 π}{2})$ , underdamped; $u (t) = \frac{8}{5} \cos (5 t - \frac{3 π}{2})$ $u (t) = \frac{8}{5} \cos (5 t - \frac{3 π}{2})$
19. $x (t) \approx \frac{1}{3} \sqrt{313} e^{- 5 t / 2} \cos (6 t - 0.8254)$ $x (t) \approx \frac{1}{3} \sqrt{313} e^{- 5 t / 2} \cos (6 t - 0.8254)$ , underdamped; $u (t) \approx \frac{4}{13} \sqrt{233} \cos (\frac{13}{2} t - 0.5517)$ $u (t) \approx \frac{4}{13} \sqrt{233} \cos (\frac{13}{2} t - 0.5517)$
20. $x (t) \approx 13 e^{- 4 t} \cos (2 t - 1.1760)$ $x (t) \approx 13 e^{- 4 t} \cos (2 t - 1.1760)$ , underdamped; $u (t) \approx \sqrt{\frac{129}{5}} \cos (2 \sqrt{5} t - 0.1770)$ $u (t) \approx \sqrt{\frac{129}{5}} \cos (2 \sqrt{5} t - 0.1770)$
21. $x (t) \approx 10 e^{- 5 t} \cos (10 t - 0.9273)$ $x (t) \approx 10 e^{- 5 t} \cos (10 t - 0.9273)$ , underdamped; $u (t) \approx 2 \sqrt{14} \cos (5 \sqrt{5} t - 0.6405)$ $u (t) \approx 2 \sqrt{14} \cos (5 \sqrt{5} t - 0.6405)$
22. (b) The time-varying amplitude is $\frac{2}{3} \sqrt{3}$ $\frac{2}{3} \sqrt{3}$ , the frequencyis $4 \sqrt{3} rad/s,$ $4 \sqrt{3} rad/s,$ and the phase angle is $π / 6$ $π / 6$ .
23. (a) $k \approx 7018 lb/ft; (b)$ $k \approx 7018 lb/ft; (b)$ After about 2.47 s
34. Damping constant: $c \approx 11.51 lb/ft/s$ $c \approx 11.51 lb/ft/s$ ; spring constant: $k \approx 189.68 lb/ft$ $k \approx 189.68 lb/ft$

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Table of Contents for Section 5.4

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Table of Contents for
Section 5.4