A.6.1: Section 6.1 Answers
- Page ID
- 43772
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.