Cálculo com vectores

A derivada dum vector de magnitude(módulo) constante é sempre perpendicular ao vector.

$\overrightarrow{b}·\overrightarrow{b}=\mathrm{|}\overrightarrow{b}{\mathrm{|}}^2=\mathrm{const}. \Rightarrow \frac{d\mathrm{|}\overrightarrow{b}{\mathrm{|}}^2}{dt}\equiv 2\overrightarrow{b}·\frac{d\overrightarrow{b}}{dt}=0 \therefore \frac{d\overrightarrow{b}}{dt}\perp \overrightarrow{b}$

Existe assim um vector velocidade angular ${\overrightarrow{\omega }}_b$, perpendicular ao plano formado por $\overrightarrow{b}$  e $\frac{d\overrightarrow{b}}{dt}$, de forma a poder-se escrever $\frac{d\overrightarrow{b}}{dt}$=${\overrightarrow{\omega }}_b$×$\overrightarrow{b}$. Como $\mathrm{|}\frac{d\overrightarrow{b}}{dt}\mathrm{|}=\frac{d\theta }{dt}\mathrm{|}\overrightarrow{b}\mathrm{|}$, conclui-se que $\mathrm{|}{\overrightarrow{\omega }}_b\mathrm{|=}\frac{d\theta }{dt}$.

Exemplo-1:

Quando se usam  coordenadas cilíndricas $\mathrm{\{}\mathrm{\rho , \theta , \; z}\mathrm{\}}$ (ditas polares no plano horizontal), o referencial formado em cada ponto $\mathrm{\{}{\rho }_o\mathrm{, }{\theta }_o\mathrm{, }{z}_o\mathrm{\}}$ pelas direcções tangentes às curvas que se obtêem fazendo variar  uma das coordenadas do ponto, enquanto se mantêem as outras constantes, diz-se naturalmente adaptado a estas coordenadas. Para os movimentos dos versores de base deste referencial, pode-se definir um vector único ${\overrightarrow{\omega }}_c$ tal que:

${\overrightarrow{\omega }}_c=\frac{d\theta }{dt}{\overrightarrow{e}}_z$

$\frac{d{\overrightarrow{e}}_{\rho }}{dt}={\overrightarrow{\omega }}_c\times {\overrightarrow{e}}_{\rho }=\frac{d\theta }{dt}{\overrightarrow{e}}_z\times {\overrightarrow{e}}_{\rho }\equiv \frac{d\theta }{dt}{\overrightarrow{e}}_{\theta } ; \frac{d{\overrightarrow{e}}_{\theta }}{dt}={\overrightarrow{\omega }}_c\times {\overrightarrow{e}}_{\theta }=\frac{d\theta }{dt}{\overrightarrow{e}}_z\times {\overrightarrow{e}}_{\theta }\equiv \mathrm{-}\frac{d\theta }{dt}{\overrightarrow{e}}_{\rho } ; \frac{d{\overrightarrow{e}}_z}{dt}={\overrightarrow{\omega }}_c\times {\overrightarrow{e}}_z\equiv 0$

Exemplo-2:

No caso de se usarem coordenadas esféricas $\mathrm{\{}\mathrm{r, \theta , \phi }\mathrm{\}}$ no espaço, o movimento dos versores de base é controlado por um vector ${\overrightarrow{\omega }}_s$. Neste caso:

${\overrightarrow{\omega }}_s=\frac{d\varphi }{dt}{\overrightarrow{e}}_z+\frac{d\theta }{dt}{\overrightarrow{e}}_{\varphi }\equiv \frac{d\varphi }{dt}\left(\cos \left(\theta \right){\overrightarrow{e}}_r\mathrm{-}\sin \left(\theta \right){\overrightarrow{e}}_{\theta }\right)+\frac{d\theta }{dt}{\overrightarrow{e}}_{\varphi }$

$\frac{d{\overrightarrow{e}}_r}{dt}={\overrightarrow{\omega }}_s\times {\overrightarrow{e}}_r=\mathrm{-}\frac{d\varphi }{dt}\sin \left(\theta \right){\overrightarrow{e}}_{\theta }\times {\overrightarrow{e}}_r+\frac{d\theta }{dt}{\overrightarrow{e}}_{\varphi }\times {\overrightarrow{e}}_r\equiv \frac{d\varphi }{dt}\sin \left(\theta \right){\overrightarrow{e}}_{\varphi }+\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }$

$\frac{d{\overrightarrow{e}}_{\theta }}{dt}={\overrightarrow{\omega }}_s\times {\overrightarrow{e}}_{\theta }=\frac{d\varphi }{dt}\cos \left(\theta \right){\overrightarrow{e}}_r\times {\overrightarrow{e}}_{\theta }+\frac{d\theta }{dt}{\overrightarrow{e}}_{\varphi }\times {\overrightarrow{e}}_{\theta }\equiv \frac{d\varphi }{dt}\cos \left(\theta \right){\overrightarrow{e}}_{\varphi }\mathrm{-}\frac{d\theta }{dt}{\overrightarrow{e}}_r$

$\frac{d{\overrightarrow{e}}_{\varphi }}{dt}={\overrightarrow{\omega }}_s\times {\overrightarrow{e}}_{\varphi }=\frac{d\varphi }{dt}\left(\cos \left(\theta \right){\overrightarrow{e}}_r\times {\overrightarrow{e}}_{\varphi }\mathrm{-}\sin \left(\theta \right){\overrightarrow{e}}_{\theta }\times {\overrightarrow{e}}_{\varphi }\right)\equiv \mathrm{-}\frac{d\varphi }{dt}\cos \left(\theta \right){\overrightarrow{e}}_{\theta }\mathrm{-}\frac{d\varphi }{dt}\sin \left(\theta \right){\overrightarrow{e}}_r$

Resumo das derivadas de versores móveis

$\begin{array}{c}\begin{array}{l}\frac{d{\overrightarrow{e}}_{\rho }}{dt}\equiv \frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }\\ \frac{d{\overrightarrow{e}}_{\theta }}{dt}\equiv \mathrm{-}\frac{d\theta }{dt}{\overrightarrow{e}}_{\rho }\\ \frac{d{\overrightarrow{e}}_z}{dt}\equiv 0\end{array}\\ \underset{\mathrm{Coord}.\mathrm{Cilíndricas}}{⏟}\end{array}\begin{array}{c}\begin{array}{l}\frac{d{\overrightarrow{e}}_r}{dt}\equiv \frac{d\varphi }{dt}\sin \left(\theta \right){\overrightarrow{e}}_{\varphi }+\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }\\ \frac{d{\overrightarrow{e}}_{\theta }}{dt}\equiv \frac{d\varphi }{dt}\cos \left(\theta \right){\overrightarrow{e}}_{\varphi }\mathrm{-}\frac{d\theta }{dt}{\overrightarrow{e}}_r\\ \frac{d{\overrightarrow{e}}_{\varphi }}{dt}\equiv \mathrm{-}\frac{d\varphi }{dt}\cos \left(\theta \right){\overrightarrow{e}}_{\theta }\mathrm{-}\frac{d\varphi }{dt}\sin \left(\theta \right){\overrightarrow{e}}_r\end{array}\\ \underset{\mathrm{Coord}.\mathrm{Esféricas}}{⏟}\end{array}$

Velocidade e aceleração

$\mathrm{Decomposição}\mathrm{radial}\mathrm{-}\mathrm{transversal}$

$\overrightarrow{r}=\mathrm{|}\overrightarrow{r}\mathrm{|}{\overrightarrow{e}}_r\equiv r{\overrightarrow{e}}_r;\overrightarrow{v}\equiv \frac{d\overrightarrow{r}}{dt}\equiv \frac{dr}{dt}{\overrightarrow{e}}_r+\mathrm{|}\overrightarrow{r}\mathrm{|}\frac{d{\overrightarrow{e}}_r}{dt}\equiv \frac{dr}{dt}{\overrightarrow{e}}_r+\overrightarrow{\omega }\times \overrightarrow{r}$

$v\equiv \mathrm{|}\overrightarrow{v}\mathrm{|}=\frac{ds}{dt}\equiv \underset{\mathrm{\Delta t}\to 0}{\lim }\frac{\mathrm{|}\Delta \overrightarrow{r}\mathrm{|}}{\mathrm{\Delta t}}$

$v=\mathrm{\rho \omega }=\sqrt{{\left(\frac{dr}{dt}\right)}^2+r^2\left(\sin {\left(\theta \right)}^2{\left(\frac{d\varphi }{dt}\right)}^2+{\left(\frac{d\theta }{dt}\right)}^2\right)}$

$\mathrm{Decomposição}\mathrm{tangente}\mathrm{-}\mathrm{normal}$

$\overrightarrow{v}\equiv \mathrm{|}\overrightarrow{v}\mathrm{|}{\overrightarrow{e}}_t\equiv v{\overrightarrow{e}}_t$

$\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}=\frac{dv}{dt}{\overrightarrow{e}}_t+\overrightarrow{\Omega }\times \overrightarrow{v}=a_t{\overrightarrow{e}}_t+a_n{\overrightarrow{e}}_{\perp }\text{Curvatura κ e Raio de Curvatura ρ}:{\displaystyle \kappa =\frac{1}{\rho }}={\displaystyle \frac{\mathrm{|}\overrightarrow{v}\times \overrightarrow{a}\mathrm{|}}{v^3}\equiv \frac{a_n}{v^2}}$

$\overrightarrow{\Omega }\mathrm{(}\overrightarrow{r}\mathrm{)}$ é um vector perpendicular ao plano osculador da trajectória no ponto $\overrightarrow{r}$. A sua magnitude é $\mathrm{|}\overrightarrow{\Omega }\mathrm{|}\equiv \mathrm{|}\frac{d\varphi }{dt}\mathrm{|}$.

$\mathrm{Curvatura\; e\; velocidade\; para\; curva\; 3D.}$

© Amaro Rica da Silva, Prof. Dep. Física-IST with Mathematica  (September 20, 2005)