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{\bf Question}

A particle has a path given by $ x = \cos \omega t, \, y =  \sin
\omega t, \, z = t$.  Find its velocity and acceleration in
cylindrical polar coordinates.

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{\bf Answer}

$\ds {\bf v} = \frac{d}{dt}({\bf i} \cos \omega t + {\bf j} \sin
\omega t + t{\bf k}) = -\omega \sin \omega {\bf i} + \omega \cos
\omega t {\bf j} + {\bf k}$

$\ds {\bf r} = {\bf i} \cos \omega t + {\bf j} \sin \omega t + zt$

in cylindrical polar coordinates: $\ds r = \sqrt{x^2 + y^2} = 1;
\hspace{.1in} \phi = \omega t; \hspace{.1in} z = t.$

$ \ds {\bf v} = \dot r {\bf e}_r + r \dot \phi {\bf e}_\phi + \dot
z {\bf e}_z $

$\ds {\bf v}= \omega{\bf e}_\phi + {\bf e}_z$

$\ds {\bf a} = ( \ddot r - r \dot \phi ^2){\bf e}_r +
\frac{1}{r}(\dot r^2 \dot \phi){\bf e}_\phi + \ddot z {\bf e}_z$

$\ds {\bf a} = -\omega^2 {\bf e}_r$



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