# Acceleration in spherical coordinates

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

#### Solution Preview

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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.