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Finding the curl of velocity in spherical coordinates

  1. Feb 4, 2017 #1
    1. The problem statement, all variables and given/known data
    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.
    2. Relevant 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

    3. 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.
     

    Attached Files:

    Last edited: Feb 4, 2017
  2. jcsd
  3. Feb 4, 2017 #2

    TSny

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    Homework Helper
    Gold Member

    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
     
  4. Feb 4, 2017 #3
    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?
     
  5. Feb 4, 2017 #4

    TSny

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    Homework Helper
    Gold Member

    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.

    Yes, that's it.
     
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