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# Acceleration in spherical coordinates

Derive the expression of the acceleration in terms of spherical coordinates, see problem 2 of the attachment.

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begin{document}
section{Acceleration in Spherical Coordinates}
There are different ways to solve this problem.

subsection{A lot of brute force}
This is useful to practice manipulations using inner products, differentiation, chain rule etc. etc. I will give a short outline but I won't work out the problem using this approach. What you do is you write down the fact that the postion vector is given by:
begin{equation}
vec{r}=rhat{r}
end{equation}
Then you differentiate w.r.t. time:
begin{equation}label{velo}
frac{dvec{r}}{dt}=frac{d}{dt}haak{rhat{r}} = frac{d r}{dt}hat{r} + rfrac{dhat{r}}{dt}
end{equation}
We now need to find out how to express the vector \$frac{dhat{r}}{dt}\$ in terms of the unit vectors \$hat{r}\$, \$hat{theta}\$ and \$hat{phi}\$. What you can do is to use the definitions of the unit vectors in terms of the Cartesian unit vectors \$hat{i}\$, \$hat{j}\$ and \$hat{k}\$. You first write:
begin{equation}
frac{dhat{r}}{dt}= frac{partialhat{r}}{partial r}frac{dr}{dt} + frac{partialhat{r}}{partialtheta}frac{dtheta}{dt} + frac{partialhat{r}}{partialphi}frac{dphi}{dt}
end{equation}
Then you carry out the differentations of the unit vector \$hat{r}\$. The result is some linear combination of the unit vectors \$hat{i}\$, \$hat{j}\$ and \$hat{k}\$. You then express that vector back in ...

#### Solution Summary

A detailed derivation is given using both elementary and Largeangian methods.

\$2.19