A.6.1: Section 6.1 Answers

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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. \(y=\frac{1}{4}\cos 8t-\frac{1}{16}\sin 8t\text{ ft};\: R=\frac{\sqrt{17}}{16}\text{ ft};\: \omega _{0}=8\text{ rad/s};\: T= \pi /4\text{ s};\:\phi\approx -.245\text{ rad}\approx -14.04^{\circ}\)

5. \(y=10\cos 14t+\frac{25}{14}\sin 14t\text{ cm};\: R=\frac{\sqrt{5}}{14}\sqrt{809}\text{ cm};\: \omega _{0}=14\text{ rad/s};\: T= \pi /7\text{ s};\:\phi\approx .177\text{ rad}\approx 10.12^{\circ}\)

6. \(y=-\frac{1}{4}\cos \sqrt{70}t+\frac{2}{\sqrt{70}}\sin \sqrt{70}t\text{ m};\: R=\frac{1}{4}\sqrt{\frac{67}{35}}\text{ m};\: \omega _{0}=\sqrt{70}\text{ rad/s};\: T= 2\pi /\sqrt{70}\text{ s};\:\phi\approx 2.38\text{ rad}\approx 136.28^{\circ}\)

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.