# 7.4.1: Energy in Linear Acceleration Coordinate

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

The potential is defined as
$\label{ene:eq:Fpotetional} P.E. = - \int_{ref}^2 \pmb{F}\cdot \pmb{dll}$

In Chapter 3 a discussion about gravitational energy potential was presented. For example, for the gravity force is

$\label{ene:eq:gravity} F = - \dfrac{G\,M\,m}{r^2}$ Where $$G$$ is the gravity coefficient and $$M$$ is the mass of the Earth. $$r$$ and $$m$$ are the distance and mass respectively. The gravity potential is then

$\label{ene:eq:gavityPotential} PE_{gravity} = - \int_{\infty}^r - \dfrac{G\,M\,m}{r^2} dr$ The reference was set to infinity. The gravity force for fluid element in small distance then is $$g\, dz\, dm$$. The work this element moving from point 1 to point 2 is

$\label{ene:eq:eleGravity} \int_1^2 g\, dz\, dm = g\,\left(z_2 - z_1 \right) dm$ The total work or potential is the integral over the whole mass.