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Math Help - cylindrical coordinates

  1. #1
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    cylindrical coordinates

    Angular momentum is defined as L = r x mv , where v is contained in the xy plane. show that in cylindrical coordinates angular momentum takes the form  L = mr^{2}\dot{\phi}

    i know v =  \dot{r} \hat{r} + r \hat{\phi} \dot{\phi}
     L =r  \times  m (\dot{r} \hat{r} + r \hat{\phi} \dot{\phi} )

    am not sure what i get when you take the corss product of

     r \times m\dot{r} \hat{r} ?
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  2. #2
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    Re: cylindrical coordinates

    \vec{L} = \vec{r} \times m\vec{v}

    Setting \vec{v} = \frac{dr}{dt} \hat{r} + r \frac{d\phi}{dt} \hat{\phi}, and \vec{r} = r \hat{r} gives:

    \vec{L} = r \hat{r} \times m(\frac{dr}{dt} \hat{r} + r \frac{d\phi}{dt} \hat{\phi})

    \vec{L} = r \hat{r} \times m\frac{dr}{dt} \hat{r} + r \hat{r} \times mr \frac{d\phi}{dt} \hat{\phi}

    and we have \hat{r} \times \hat{r} = 0 and \hat{r} \times \hat{\phi} = \hat{z}, so

    \vec{L} = mr^2 \frac{d\phi}{dt} \hat{z}

    The radial component of velocity doesn't contribute to the angular momentum.

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