Share
Explore BrainMass

Acceleration in spherical coordinates

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

Attachments

Solution Preview

documentclass[a4paper]{article}
usepackage{amsmath,amssymb}
newcommand{haak}[1]{!left(#1right)}
newcommand{rhaak}[1]{!left [#1right]}
newcommand{lhaak}[1]{left | #1right |}
newcommand{ahaak}[1]{!left{#1right}}
newcommand{gem}[1]{leftlangle #1rightrangle}
newcommand{gemc}[2]{leftlangleleftlangleleft. #1right | #2
rightranglerightrangle}
newcommand{geml}[1]{leftlangle #1right.}
newcommand{gemr}[1]{left. #1rightrangle}
newcommand{haakl}[1]{left(#1right.}
newcommand{haakr}[1]{left.#1right)}
newcommand{rhaakl}[1]{left[#1right.}
newcommand{rhaakr}[1]{left.#1right]}
newcommand{lhaakl}[1]{left |#1right.}
newcommand{lhaakr}[1]{left.#1right |}
newcommand{ahaakl}[1]{left{#1right.}
newcommand{ahaakr}[1]{left.#1right}}
newcommand{ket}[1]{lhaakl{gemr{#1}}}
newcommand{bra}[1]{lhaakr{geml{#1}}}
newcommand{half}{frac{1}{2}}
newcommand{kwart}{frac{1}{4}}
newcommand{erf}{operatorname{erf}}
newcommand{erfi}{operatorname{erfi}}
renewcommand{Re}{operatorname{Re}}
renewcommand{Im}{operatorname{Im}}
renewcommand{arraystretch}{1.5}
newcommand{brk}[1]{renewcommand{arraystretch}{1}
begin{tabular}{l}#1end{tabular}renewcommand{arraystretch}{1.5}}

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