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Relation between r ,ω and θ for rotation around fixed axis.

  1. Nov 10, 2013 #1
    relation between r ,ω and θ for rotation around fixed axis.

    [tex] \frac{d\textbf {r}}{dt} = \textbf {ω} [/tex]


    [tex] \frac{dθ}{dt} = ω [/tex]


    [tex] \lvert\frac{d\textbf {r}}{dt}\rvert = \frac{dθ}{dt} [/tex]

    bold means vector. Is this right?
     
    Last edited: Nov 10, 2013
  2. jcsd
  3. Nov 11, 2013 #2

    tiny-tim

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    Hi AakashPandita! :smile:

    (type \left and and \right before two ordinary ||s, and they resize to fit! :wink:)

    [tex] \frac{dθ}{dt} = ω = |\textbf{ω}|[/tex]
    ω (the angular velocity vector) is along the axis of rotation (ie, out of the page)
    [tex] \left|\frac{d\hat{\textbf {r}}}{dt}\right| = r\frac{dθ}{dt} = rω [/tex]
    and
    [tex] \frac{d\hat{\textbf {r}}}{dt} = rω\hat{\textbf{θ}} [/tex]
    where ##\hat{\textbf {r}}## and ##\hat{\textbf {θ}}## are the unit vectors in the radial and tangential directions
     
  4. Nov 11, 2013 #3

    vanhees71

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    Let the rotation be around the [itex]z[/itex] axis of a Cartesian coordinate system. Then
    [tex]\vec{r}(t)=r(t) \begin{pmatrix}
    \cos[\alpha(t)] \\
    \sin[\alpha(t)] \\
    0
    \end{pmatrix}.
    [/tex]
    This gives
    [tex]\vec{v}(t)=\dot{r}(t) = \dot{r}(t) \begin{pmatrix}
    \cos[\alpha(t)] \\
    \sin[\alpha(t)] \\
    0
    \end{pmatrix} + r(t) \dot{\alpha}(t) \begin{pmatrix}
    -\sin[\alpha(t)] \\
    \cos[\alpha(t)] \\
    0
    \end{pmatrix} = \dot{r} \hat{r} + r \omega \hat{\theta}.
    [/tex]
    If the mass is fixed to a circle, then you have [itex]\dot{r}=0[/itex] and you can write
    [tex]\vec{v}=\vec{\omega} \times \vec{r} \quad \text{with} \quad \vec{\omega}=\omega \vec{e}_z=\dot{\theta} \vec{e}_z.[/tex]
     
  5. Nov 11, 2013 #4

    ZapperZ

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    You should already know that something is not quite right here, because the dimensions are all wrong! Always check that first, because that is your first "line-of-defense"!

    Zz.
     
  6. Nov 11, 2013 #5
    how did you get this?
     
  7. Nov 11, 2013 #6
    i dont understand matrices
     
  8. Nov 11, 2013 #7

    D H

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    vanhees71 wasn't using matrices, AakashPandita. He was using vectors.
     
  9. Nov 11, 2013 #8
    i dont understand those brackets
     
  10. Nov 11, 2013 #9

    vanhees71

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    How are you supposed to solve this question without vectors? I guess you just use a different notation than I use. In your original posting you used abstract vectors, but the equations are unfortunately not correct. So I thought, it's best to use components wrt. a Cartesian basis to explicitly calculate the derivatives.
     
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