Simple cylindrical coords problem

[ognik]

Rigid body rotating about a fixed axis with constant $\omega$ along the z axis. Express position vector $\vec{r}$ in cyl. circ. cords and using cyl. circ. cords find (a) $\vec{v}=\omega \times \vec{r}$ (b) $\nabla \times \vec{v}$

So $ \vec{r} = \vec{\rho}\rho + \vec{z}z $

(a) = $\begin{vmatrix}
\hat{\rho}&\hat{\phi}&\hat{z}\\0&0&\omega\\\ \rho&\ 0&z
\end{vmatrix} = \vec{\phi}\omega \rho $

But the solution shown is $ \vec{\phi} \omega \rho $? Shouldn't it be the unit vector $\hat{\phi}$?

(b) $ \nabla = \left( \partial_{\rho}, \frac{1}{\rho} \partial_{\phi}, \partial_z \right) $

$ \therefore \nabla \times \vec{v} =
\begin{vmatrix}
\hat{\rho}&\hat{\phi}&\hat{z}\\\partial_{\rho}, & \frac{1}{\rho} \partial_{\phi}, &\partial_z \\\ 0&\ \omega \rho & 0
\end{vmatrix}$
which is obviously not going to give me the answer in the book ($2\omega$), so what am I missing please? Also isn't the book's answer missing a direction?

 

[Sudharaka]

Rigid body rotating about a fixed axis with constant $\omega$ along the z axis. Express position vector $\vec{r}$ in cyl. circ. cords and using cyl. circ. cords find (a) $\vec{v}=\omega \times \vec{r}$ (b) $\nabla \times \vec{v}$

So $ \vec{r} = \vec{\rho}\rho + \vec{z}z $

(a) = $\begin{vmatrix}
\hat{\rho}&\hat{\phi}&\hat{z}\\0&0&\omega\\\ \rho&\ 0&z
\end{vmatrix} = \vec{\phi}\omega \rho $

But the solution shown is $ \vec{\phi} \omega \rho $? Shouldn't it be the unit vector $\hat{\phi}$?

(b) $ \nabla = \left( \partial_{\rho}, \frac{1}{\rho} \partial_{\phi}, \partial_z \right) $

$ \therefore \nabla \times \vec{v} =
\begin{vmatrix}
\hat{\rho}&\hat{\phi}&\hat{z}\\\partial_{\rho}, & \frac{1}{\rho} \partial_{\phi}, &\partial_z \\\ 0&\ \omega \rho & 0
\end{vmatrix}$
which is obviously not going to give me the answer in the book ($2\omega$), so what am I missing please? Also isn't the book's answer missing a direction?

Hi ognik,

I am not sure what your problem is with the first part. The solution you have obtained; $\vec{\phi} \omega \rho $ seems to be correct.

For the second part the notation $\nabla \times \vec{v}$ seems to indicate you have to find the Curl of $\vec{v}$ not the gradient.

 

[ognik]

Hi, thanks - must be a typo in the book for part a, left out the \hat, good to have confirmation.

Part b - oops, yes curl, which should be: $\frac{1}{\rho}\begin{vmatrix}
\hat{\rho}&\rho\hat{\phi}&\hat{z}\\ \partial_{\rho}&\partial_{\phi}&\partial_z \ \\0&\rho \omega \rho &0
\end{vmatrix}$ for $ \vec{v}=\hat{\phi}\omega \rho $

I get $ \nabla \times \vec{v} = \hat{z}2\omega $, the book again appears to have a typo - no $\hat{z}$, assume mine is OK? Thanks for the help.

 

[Sudharaka]

Hi, thanks - must be a typo in the book for part a, left out the \hat, good to have confirmation.

Part b - oops, yes curl, which should be: $\frac{1}{\rho}\begin{vmatrix}
\hat{\rho}&\rho\hat{\phi}&\hat{z}\\ \partial_{\rho}&\partial_{\phi}&\partial_z \ \\0&\rho \omega \rho &0
\end{vmatrix}$ for $ \vec{v}=\hat{\phi}\omega \rho $

I get $ \nabla \times \vec{v} = \hat{z}2\omega $, the book again appears to have a typo - no $\hat{z}$, assume mine is OK? Thanks for the help.

Well, the Curl should always be a vector. If the book only asks for the magnitude of Curl the answer is $2\omega$ otherwise the vector component should be included. :)