Skip to main content

# A.6.1: Section 6.1 Answers

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

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

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.

This page titled A.6.1: Section 6.1 Answers is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

• Was this article helpful?