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{\bf Question}

Derive the components of
\begin{description}
\item[(a)] velocity and acceleration in cylindrical polar
coordinates,
\item[(b)] velocity in spherical polar coordinates.
\end{description}

The expression for acceleration in spherical polar coordinates is
\begin{eqnarray*} {\bf a} & = & (\ddot r - r \dot \theta^2 - r \sin^2 \theta \dot
\phi^2){\bf r}_r + \left[ \frac{1}{r} \frac{d}{dt}(r^2 \dot
\theta) - r \sin \theta \cos \theta \dot \phi^2 \right] {\bf
e}_\theta \\ & & + \frac{1}{r \sin \theta}\frac{d}{dt}(r^2 \sin^2
\theta \dot \phi){\bf e}_\phi \end{eqnarray*}


\vspace{.25in}

{\bf Answer}

\begin{description}
\item[(a)]
\begin{eqnarray*} {\bf r} & = & r{\bf e}_r + z{\bf e}_z \\ {\bf v} =
\dot {\bf r} & = & \dot r {\bf e}_r + r \dot {\bf e}_r + \dot z
{\bf e}_z + z \dot{\bf e}_z \\ {\rm Now\ } \dot {\bf e}_z & = & 0
{\rm \ as\ in\ a\ fixed\ direction } \\ {\bf e}_r & = & {\bf i}
\cos \theta + {\bf j} \sin \theta \\ \dot {\bf e}_r & = & \dot
\theta(-\sin \theta {\bf i} + \cos \theta {\bf j})  =  \dot \theta
{\bf e}_\theta \\ {\rm So\ \ } {\bf  v} & = & \dot r {\bf e}_r + r
\dot \theta {\bf e}_\theta + \dot z {\bf e}_z \\ {\bf a} = \dot
{\bf v} & = & \ddot r {\bf e}_r + \dot r \dot {\bf e}_r + \dot{(r
\dot \theta)}{\bf e}_\theta + r \dot \theta \dot {\bf e}_\theta +
\ddot z {\bf e}_z \\ {\rm Now\ } \dot {\bf e}_\theta & = &
\frac{d}{dt}(-\sin \theta {\bf i} + \cos \theta {\bf j}) = -\dot
\theta{\bf e}_r \\{\rm So\ \ } {\bf a} & = & \ddot r {\bf e}_r +
\dot r \dot \theta {\bf e}_\theta + \dot{(r \dot \theta)}{\bf
e}_\theta - r \dot \theta^2 {\bf e}_r + \ddot z {\bf e}_z \\
\Rightarrow {\bf a} & = & (\ddot r - r \dot \theta^2){\bf e}_r +
(r \ddot \theta + 2 \dot r \dot \theta){\bf e}_\theta + \ddot z
{\bf e}_z
\end{eqnarray*}


\item[(b)]
\begin{eqnarray*} & & {\bf r} = r {\bf e}_r \hspace{.2in} {\bf v } = \dot
r {\bf e}_r + r \dot {\bf e}_r \\ {\rm Now\ \ } {\bf e}_r & = &
{\bf i} \sin \theta \cos \phi + {\bf j} \sin \theta \sin \phi +
{\bf k} \cos \theta \\ \dot {\bf e}_r & = & \dot \theta ({\bf i}
\cos \theta \cos \phi + {\bf j} \cos \theta \sin \phi - {\bf k}
\sin \theta) \\ & & + \dot \phi(-{\bf i} \sin \theta \sin \phi +
{\bf j}\sin \theta \cos \phi) \\ & = & \dot \theta {\bf e}_\theta
+ \dot \phi \sin \theta {\bf e}_\phi \\ {\rm So\ \ }{\bf v} & = &
\dot r {\bf e}_r + r \dot \theta {\bf e}_\theta + r \dot \phi \sin
\theta {\bf e}_\phi
\end{eqnarray*}
\end{description}



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