9.5: Chapter 5 Answers

Section 5.1

1. \(y=3\cos 4\sqrt{6}t-\frac{1}{2\sqrt{6}}\sin 4\sqrt{6}t\text{ ft}\)

2. \(y=-\frac{1}{4}\cos 8\sqrt{5}t-\frac{1}{4\sqrt{5}}\sin 8\sqrt{5} t\text{ ft}\)

3. \(y=-1.5\cos 14\sqrt{10t}\text{ cm}\)

4. \(y=\frac{1}{4}\cos 8t-\frac{1}{16}\sin 8t\text{ ft};\: R=\frac{\sqrt{17}}{16}\text{ ft};\: \omega _{0}=8\text{ rad/s};\: T= \pi /4\text{ s};\:\phi=\pi-\tan^{-1}4\text{ rad}\)

5. \(y=10\cos 14t+\frac{25}{14}\sin 14t\text{ cm};\: R=\frac{5}{14}\sqrt{809}\text{ cm};\: \omega _{0}=14\text{ rad/s};\: T= \pi /7\text{ s};\:\phi\approx 1.39\text{ rad}\approx 79.88^{\circ}\)

6. \(y=-\frac{1}{4}\cos \sqrt{70}t+\frac{2}{\sqrt{70}}\sin \sqrt{70}t\text{ m};\: R=\frac{1}{4}\sqrt{\frac{67}{35}}\text{ m};\: \omega _{0}=\sqrt{70}\text{ rad/s};\: T= 2\pi /\sqrt{70}\text{ s};\:\phi\approx 5.48\text{ rad}\approx 313.72^{\circ}\)

7. \(y=\frac{2}{3}\cos 16t-\frac{1}{4}\sin 16t\text{ ft}\)

8. \(y=\frac{1}{2}\cos 8t-\frac{3}{8}\sin 8t\text{ ft}\)

9. \(.72\text{ m}\)

10. \(y=\frac{1}{3}\sin t+\frac{1}{2}\cos 2t+\frac{5}{6}\sin 2t\text{ ft}\)

11. \(y=\frac{16}{5}\left(4\sin\frac{t}{4}-\sin t \right)\)

12. \(y=-\frac{1}{16}\sin 8t+\frac{1}{3}\cos 4\sqrt{2}t-\frac{1}{8\sqrt{2}}\sin 4\sqrt{2}t\)

13. \(y=-t\cos 8t-\frac{1}{6}\cos 8t+\frac{1}{8}\sin 8t\text{ ft}\)

14. \(T=4\sqrt{2}\text{ s}\)

15. \(\omega = 8\text{ rad/s}\: y=-\frac{t}{16}(-\cos 8t+2\sin 8t)+\frac{1}{128}\sin 8t\text{ ft}\)

16. \(\omega = 4\sqrt{6}\text{ rad/s};\: y=-\frac{t}{\sqrt{6}}\left[\frac{8}{3}\cos 4\sqrt{6t}+4\sin 4\sqrt{6t} \right]+\frac{1}{9}\sin 4\sqrt{6}t\text{ ft}\)

17. \(y=\frac{t}{2}\cos 2t-\frac{t}{4}\sin 2t+3\cos 2t+2\sin 2t\text{ m}\)

18. \(y=y_{0}\cos\omega_{0}t+\frac{v_{0}}{\omega _{0}}\sin\omega _{0}t;\: R=\frac{1}{\omega _{0}}\sqrt{(\omega _{0}y_{0})^{2}+(v_{0})^{2}};\: \cos\phi=\frac{y_{0}\omega _{0}}{\sqrt{(\omega _{0}y_{0})^{2}+(v_{0})^{2}}};\:\sin\phi=\frac{v_{0}}{\sqrt{(\omega_{0}y_{0})^{2}+(v_{0})^{2}}} \)

19. The object with the longer period weighs four times as much as the other.

20. \(T_{2}=\sqrt{2}T_{1}\), where \(T_{1}\) is the period of the smaller object

21. \(k_{1}=9k_{2}\), where \(k_{1}\) is the spring constant of the system with the shorter period.

22. \(y={1\over 4}\cos 4\sqrt{6}t\)

23. a. \(y=-{1\over 2}\cos 8t\)
     b. \(t={(2n+1)\pi\over 16}\) for \(n=0,1,2,...\)
     c. \(t={7\pi\over 16}\)

24. a. \(y=-{1\over 2}\cos 2t-{3\over 4}\sin 2t\)
     b. \(y={\sqrt{13}\over 4}\sin (2t+\pi+\tan^{-1} {2\over 3})\)
     c. \(t={(n-1)\pi-\tan^{-1}{2\over 3}\over 2}\) for \(n=2,3,4,...\)
     d. \(t={5\pi-\tan^{-1}{2\over 3}\over 2}\)

25. a. \(y={2\over 3}\cos 10t-{1\over 2}\sin 10t\)
     b. \(y={5\over 6}\sin (10t+\pi-\tan^{-1} {4\over 3})\)
     c. The amplitude is \({5\over 6}\) and the period is \({\pi\over 5}\) s
     d. 15
     e. \(t={(n-1)\pi+\tan^{-1}{4\over 3}\over 10}\) s for \(n=1,2,3,...\)
     f. \(t={2\pi+\tan^{-1}{4\over 3}\over 10}\) s
     g. \({25\over 3}\) ft/s
     h. \(t={1\over 10}({\pi\over 6}+2n\pi+\tan^{-1} {4\over 3})\) and \(t={1\over 10}({5\pi\over 6}+2n\pi+\tan^{-1} {4\over 3})\) for \(n=0,1,2,...\)

26. \(y=-{1\over 2}\cos 4t+{9\over 4}\sin 4t+{1\over 2}e^{-2t}\cos 4t-2e^{-2t}\sin 4t\)

27. \(y=\frac{e^{-2t}}{2}(3\cos 2t-\sin 2t)\text{ ft};\:\sqrt{\frac{5}{2}}e^{-2t}\text{ ft}\)

28. \(y=-e^{-t}\left(3\cos 3t+\frac{1}{3}\sin 3t \right)\text{ ft}\frac{\sqrt{82}}{3}e^{-t}\text{ ft}\)

29. \(y=e^{-16t}\left(\frac{1}{4}+10t \right)\text{ ft}\)

30. \(y=-\frac{e^{-3t}}{4}(5\cos t+63\sin t)\text{ ft}\)

31. \(0\leq c<8\text{ lb-sec/ft}\)

32. \(y=\frac{1}{2}e^{-3t}\left(\cos\sqrt{91}t+\frac{11}{\sqrt{91}}\sin\sqrt{91}t \right)\text{ ft}\)

33. \(y=-\frac{e^{-4t}}{3}(2+8t)\text{ ft}\)

34. \(y=e^{-10t}\left(9\cos 4\sqrt{6}t+\frac{45}{2\sqrt{6}}\sin 4\sqrt{6}t \right)\text{ cm}\)

35. \(y=e^{-3t/2}\left(\frac{3}{2}\cos\frac{\sqrt{41}}{2}t+\frac{9}{2\sqrt{41}}\sin\frac{\sqrt{41}}{2}t \right)\text{ ft}\)

36. \(y=e^{-\frac{3}{2}t}\left(\frac{1}{2}\cos\frac{\sqrt{119}}{2}t-\frac{9}{2\sqrt{119}}\sin\frac{\sqrt{119}}{2}t \right)\text{ ft}\)

37. \(y=e^{-8t}\left(\frac{1}{4}\cos 8\sqrt{2}t-\frac{1}{4\sqrt{2}}\sin 8\sqrt{2}t \right)\text{ ft}\)

38. \(y=e^{-t}\left(-\frac{1}{3}\cos 3\sqrt{11}t+\frac{14}{9\sqrt{11}}\sin 3\sqrt{11}t \right)\text{ ft}\)

39. \(y_{p}=\frac{22}{61}\cos 2t+\frac{2}{61}\sin 2t\text{ ft}\)

40. \(y=-\frac{2}{3}(e^{-8t}-2e^{-4t})\)

41. \(y=e^{-2t}\left(\frac{1}{10}\cos 4t-\frac{1}{5}\sin 4t \right)\text{ m}\)

42. \(y=e^{-3t}(10\cos t-70\sin t)\text{ cm}\)

43. \(y_{p}=-\frac{2}{15}\cos 3t+\frac{1}{15}\sin 3t\text{ ft}\)

44. \(y_{p}=\frac{11}{100}\cos 4t+\frac{27}{100}\sin 4t\text{ cm}\)

45. \(y_{p}=\frac{42}{73}\cos t+\frac{39}{73}\sin t\text{ ft}\)

46. \(y=-\frac{1}{2}\cos 2t+\frac{1}{4}\sin 2t\text{ m}\)

47. \(y_{p}=\frac{1}{c\omega _{0}}(-\beta\cos\omega _{0}t+\alpha\sin\omega _{0}t)\)

48. a. \(t=1/4\) s
     b. \(t=1/2\) s and \(y=-e^{-2}\) ft

49. \(y=-{4\over 3}e^{-2t}+{1\over 3}e^{-8t}\)

50. \(y={2\over 3}e^{-2t}-{5\over 3}e^{-8t}\)

51. a. \(y=e^{-2t}\cos t+2e^{-2t}\sin t\)
     b. \(y=\sqrt{5}e^{-2t}\sin (t+\tan^{-1} {1\over 2})\)
     c. \(t=n\pi-\tan^{-1}{1\over 2}\)
     d. \(t=2\pi-\tan^{-1} {1\over 2}\) s

52. \(y=-{16\over 3}e^{-t/2}\cos {\sqrt{47}\over 2}t-{76\over 3\sqrt{47}}e^{-t/2}\sin {\sqrt{47}\over 2}t+{10\over 3}\cos 3t+{10\over 3}\sin 3t\)

53. \(y={1\over 4}e^{-4t}+te^{-4t}-{1\over 4}\cos 4t\)

54. \(y={1\over 2}\cos 4t-{9\over 4}\sin 4t-{1\over 2}e^{-2t}\cos 4t+2e^{-2t}\sin 4t\)