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Question Number 29498 by ajfour last updated on 09/Feb/18

Commented by ajfour last updated on 09/Feb/18

$$Relatelinearacceleration,with$$$$position,velocity,angularvelocity,$$$$andangularacceleration.$$

Answered by ajfour last updated on 09/Feb/18

$$\underset{\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}}{\mathit{A}\mathit{c}\mathit{c}\mathit{e}\mathit{l}\mathit{e}\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:}$$$$d\overline{s}=r_{xy}(-\sin \psi \hat{i}+\cos \psi \hat{j})d\psi$$$$+r_{yz}(-\sin \theta \hat{j}+\cos \theta \hat{k})d\theta$$$$+r_{zx}(-\sin \varphi \hat{k}+\cos \varphi \hat{i})d\varphi$$$$+dr\left(\frac{x\hat{i}+y\hat{j}+\hat{k}}{r}\right)$$$$=(zd\varphi -yd\psi )\hat{i}+(xd\psi -zd\theta )\hat{j}$$$$+(yd\theta -xd\varphi )\hat{k}+\frac{dr}{r}(x\hat{i}+y\hat{j}+\hat{k})$$$$\Rightarrow d\overline{s}=\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ d\theta & d\varphi & d\psi \\ x& y& z\end{array}\right]+\frac{dr}{r}(x\hat{i}+y\hat{j}+z\hat{k})$$$$\frac{d\overline{s}}{dt}=\overline{v}=\overline{\omega }\times \overline{r}+\frac{1}{r}\left(\frac{dr}{dt}\right)\overline{r}$$$$\Rightarrow \overline{\mathit{a}}=\overline{\alpha }\times \overline{r}+\overline{\omega }\times \overline{v}-\frac{1}{r^2}{\left(\frac{dr}{dt}\right)}^2\overline{r}$$$$+\frac{1}{r}\left(\frac{d^{2}r}{{dt}^2}\right)\overline{r}+\frac{1}{r}\left(\frac{dr}{dt}\right)\overline{v}$$$$\overline{a}=\overline{\alpha }\times \overline{r}+\overline{\omega }\times \left(\overline{\omega }\times \overline{r}\right)+\frac{1}{r}\left(\frac{dr}{dt}\right)\left(\overline{\omega }\times \overline{r}\right)$$$$-\frac{1}{r^2}{\left(\frac{dr}{dt}\right)}^2\overline{r}+\frac{1}{r}\left(\frac{d^{2}r}{{dt}^2}\right)\overline{r}+$$$$+\frac{1}{r}\left(\frac{dr}{dt}\right)\left(\overline{\omega }\times \overline{r}\right)+\frac{1}{r^2}{\left(\frac{dr}{dt}\right)}^2\overline{r}$$$$\overline{\mathit{a}}=\overline{\mathit{\alpha }}\times \overline{\mathit{r}}+\overline{\mathit{\omega }}\times \left(\overline{\mathit{\omega }}\times \overline{\mathit{r}}\right)$$$$+\frac{1}{\mathit{r}}\left(\frac{{\mathit{d}}^2\mathit{r}}{{\mathit{d}\mathit{t}}^2}\right)\overline{\mathit{r}}+\frac{2}{\mathit{r}}\left(\frac{\mathit{d}\mathit{r}}{\mathit{d}\mathit{t}}\right)\left(\overline{\mathit{\omega }}\times \overline{\mathit{r}}\right)$$$$---------------$$

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