Finding the curl of velocity in spherical coordinates

  • #1
37
0

Homework Statement


The angular velocity vector of a rigid object rotating about the z-axis is given by
ω = ω z-hat. At any point in the rotating object, the linear velocity vector is given by v = ω X r, where r is the position vector to that point.

a.) Assuming that ω is constant, evaluate v and X v in cylindrical coordinates.

b.) Evaluate v in spherical coordinates.

c.) Evaluate the curl of v in spherical coordinates and show that the resulting expression is equivalent to that given for X v in part a.

Homework Equations


The expressions for the curl in cylindrical and spherical coordinates. Since I don't know how to put the determinant here ill just leave them out.

For spherical

x = r sinθ cosΦ

y = r sinθ sinΦ

z = r cos θ

The Attempt at a Solution


So I worked out part a correctly (I think) which is in the attachment, but I'm stuck on part b.

b.) So for this part I have to convert ω to spherical coordinates. Since ω only lies along the z-axis, that means that Φ and θ are equal to zero, so

ω = ω r-hat

and the position vector in spherical polars is

R =
r r-hat

so that means that when I cross ω and R I get zero, I don't know what I'm missing.
 

Attachments

Last edited:

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
12,799
3,156
part (a) looks good.
b.) So for this part I have to convert ω to spherical coordinates. Since ω only lies along the z-axis, that means that Φ and θ are equal to zero, so

ω = ω r-hat
Go to an arbitrary point in space and try to write ω in terms of the unit vectors ##\hat{r}##, ##\hat{\theta}##, and ##\hat{\phi}## at that point.

and the position vector in spherical polars is

R =
r r-hat
OK
 
  • #3
37
0
part (a) looks good.


Go to an arbitrary point in space and try to write ω in terms of the unit vectors ##\hat{r}##, ##\hat{\theta}##, and ##\hat{\phi}## at that point.

OK
Is there some resource you can point me to so I can learn how to type out symbols and equations neatly like you just did?

I can't really picture it in the way you're asking me too. What if I substitute z-hat = r-hat cosθ - θ-hat sinθ?
 
  • #4
TSny
Homework Helper
Gold Member
12,799
3,156
Is there some resource you can point me to so I can learn how to type out symbols and equations neatly like you just did?
https://www.physicsforums.com/help/latexhelp/
You can learn a lot by just examining how others have used Latex in their posts. That's how I picked it up. I still have a lot to learn.

I can't really picture it in the way you're asking me too. What if I substitute z-hat = r-hat cosθ - θ-hat sinθ?
Yes, that's it.
 

Related Threads on Finding the curl of velocity in spherical coordinates

Replies
2
Views
1K
Replies
0
Views
3K
Replies
9
Views
1K
Replies
7
Views
14K
Replies
11
Views
16K
Replies
3
Views
7K
  • Last Post
Replies
9
Views
3K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
2
Views
1K
Top