Vector Calculus (13) - Angular Velocity

Vector Calculus (13) - Angular Velocity

33. Angular Velocity

Instantaneous Axis of Rotation

Angular Velocity: $ \omega =\frac{d\Theta }{dt}=\dot{\Theta } $

Velocity: $ v=R\frac{d\Theta }{dt}=R\omega $

$ \dot{\vec{r}}=\vec{v} $

$ R=r\sin \alpha $

$ v=R\omega =r\omega \sin \alpha $

$ \Rightarrow \vec{v}=\vec{\omega }\times \vec{r} $

34. Infinitesimal Rotation

Infinitesimal Rotation can be represented by vector.

Finite Rotation cannot be represented by vector.

$ \delta \vec{r}=\delta \vec{\Theta }\times \vec{r} $

Rotation $ \delta \vec{\Theta_{1}} $: $ \delta \vec{r_{1}}=\delta \vec{\Theta_{1}}\times \vec{r} $

Rotation $ \delta \vec{\Theta_{2}} $: $ \delta \vec{r_{2}}=\delta \vec{\Theta_{2}}\times (\vec{r}+\delta \vec{r_{1}}) $

1) $ \delta \vec{\Theta_{1}} $ followed by $ \delta \vec{\Theta_{2}} $

$ \vec{r}+\delta \vec{r_{12}}=\vec{r}+[\delta \vec{\Theta _{1}}\times \vec{r}+\delta \vec{\Theta_{2}}\times (\vec{r}+\delta \vec{r_{1}})] $

$ \delta \vec{r_{12}}=\delta \vec{\Theta _{1}}\times \vec{r}+\delta \vec{\Theta_{2}}\times \vec{r} $

2) $ \delta \vec{\Theta_{2}} $ followed by $ \delta \vec{\Theta_{1}} $

$ \vec{r}+\delta \vec{r_{21}}=\vec{r}+[\delta \vec{\Theta _{2}}\times \vec{r}+\delta \vec{\Theta_{1}}\times (\vec{r}+\delta \vec{r_{2}})] $

$ \delta \vec{r_{21}}=\delta \vec{\Theta _{2}}\times \vec{r}+\delta \vec{\Theta_{1}}\times \vec{r} $

$ \Rightarrow \delta \vec{r_{21}}=\delta \vec{r_{12}} $

$ \vec{\omega }=\frac{\delta \vec{\Theta }}{\delta t} $

$ \frac{\delta \vec{r}}{\delta t}=\frac{\delta \vec{\Theta }}{\delta t}\times \vec{r} $

($ \delta t\rightarrow 0 $)

$ \vec{v}=\vec{\omega }\times \vec{r} $

from http://coordinate.tistory.com/30 by ccl(A) rewrite - 2021-07-28 18:26:45