diff --git a/labs-set-2/template.pdf b/labs-set-2/template.pdf index ff5cd98..099c6ae 100644 Binary files a/labs-set-2/template.pdf and b/labs-set-2/template.pdf differ diff --git a/labs-set-2/template.tex b/labs-set-2/template.tex index cae2a0b..3ede392 100644 --- a/labs-set-2/template.tex +++ b/labs-set-2/template.tex @@ -544,5 +544,59 @@ This corresponds with answer choice B. This corresponds with answer choice D. } +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} &= \langle 1, \frac{\cancel{\sec{t}}\tan{t}}{\cancel{\sec{t}}},0 \rangle \\ + \Big\|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\Big\| &=\sqrt{1+\tan^2{t}}=\sec{t} \\ + \vec{T}(t) &= \langle \cos{t},\sin{t} ,0 \rangle \\ + \frac{\mathrm{d}\vec{T}}{\mathrm{d}t} &= \langle -\sin{t},\cos{t},0\rangle \\ + \Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\| &= 1 \\ + \vec{N}(t) &= \langle -\sin{t},\cos{t},0\rangle \\ + \vec{B}(t) &= \vec{T}(t) \times \vec{N}(t) = \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + \cos{t} & \sin{t} & 0 \\ + -\sin{t} & \cos{t} & 0 + \end{vmatrix} \\ + &= \hat{k}\left[\cos^2{t}+\sin^2{t}\right ] \\ + &= \hat{k} + \end{align*} + This corresponds with answer choice C. +} + +\dfn{Torsion}{ + \begin{align*} + \tau = \frac{(\vec{r}^\prime \times \vec{r}^{\prime\prime})\cdot \vec{r}^{\prime\prime\prime}}{ + \left \| \vec{r}^\prime \times \vec{r}^{\prime\prime} \right \|^2 + } + \end{align*} +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} &= \langle 4\cos{\tfrac{2}{5}t},3,-4\sin{\tfrac{2}{5}t} \rangle \\ + \frac{\mathrm{d}^2\vec{r}}{\mathrm{d}t^2} &= \langle -\tfrac{8}{5}\sin{\tfrac{2}{5}t},0,-\tfrac{8}{5}\cos{\tfrac{2}{5}t} \rangle \\ + \frac{\mathrm{d}^3\vec{r}}{\mathrm{d}t^3} &= \langle -\tfrac{16}{25}\cos{\tfrac{2}{5}t},0,\tfrac{16}{25}\sin{\tfrac{2}{5}t} \rangle \\ + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \times \frac{\mathrm{d}^2\vec{r}}{\mathrm{d}t^2} &= \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + 4\cos{\tfrac{2}{5}t}&3&-4\sin{\tfrac{2}{5}t}\\ + -\tfrac{8}{5}\sin{\tfrac{2}{5}t}& 0 & -\tfrac{8}{5}\cos{\tfrac{2}{5}t} + \end{vmatrix} \\ + &= \hat{i}\left [-\tfrac{24}{5}\cos{\tfrac{2}{5}t} \right]-\hat{j}\left [-\tfrac{32}{5}\cos^2{\tfrac{2}{5}t}-\tfrac{32}{5}\sin^2{\tfrac{2}{5}t}\right]+\hat{k}\left[\tfrac{24}{5}\sin{\tfrac{2}{5}t}] \\ + \left \| \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \times \frac{\mathrm{d}^2\vec{r}}{\mathrm{d}t^2} \right \| &= \sqrt{(\tfrac{24}{5})^2 +(\tfrac{32}{5})^2}=8 \\ + \tau = \frac{(\vec{r}^\prime \times \vec{r}^{\prime\prime})\cdot \vec{r}^{\prime\prime\prime}}{ + \left \| \vec{r}^\prime \times \vec{r}^{\prime\prime} \right \|^2 + } &= \frac{\langle -\tfrac{24}{5}\cos{\tfrac{2}{5}t}, \frac{32}{5}, \tfrac{24}{5}\sin{\tfrac{2}{5}t} \rangle \cdot \langle -\tfrac{-16}{25}\cos{\tfrac{2}{5}t},0,\tfrac{16}{25}\sin{\tfrac{2}{5}t} \rangle }{8^2}\\ + &= \left(-\frac{24}{5}\cos{\tfrac{2}{5}t}\right)\left(-\frac{16}{25}\cos{\tfrac{2}{5}t}\right) + + \left(\frac{32}{5}\right)(0) + + \left(\frac{24}{5}\sin{\tfrac{2}{5}t}\right)\left(\frac{16}{25}\sin{\tfrac{2}{5}t}\right) \\ + &= \frac{384}{125}\cos^2{\tfrac{2}{5}t} + \frac{384}{125}\sin^2{\tfrac{2}{5}t} \\ + &= \frac{384}{125} \\ + \tau &= \frac{(\vec{r}^\prime \times \vec{r}^{\prime\prime})\cdot \vec{r}^{\prime\prime\prime}}{\| \vec{r}^\prime \times \vec{r}^{\prime\prime} \|^2} + = \frac{384/125}{8^2} + = \frac{384}{10000} + = \frac{6}{125}. + \end{align*} + This corresponds with answer choice A. +} \end{document}