# 7.4.1: Energy in Linear Acceleration Coordinate

The potential is defined as

\[

\label{ene:eq:Fpotetional}

P.E. = - \int_{ref}^2 \pmb{F}\cdot \pmb{dll} \tag{88}

\]

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} \tag{89}

\]

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 \tag{90}

\]

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 \tag{91}

\]

The total work or potential is the integral over the whole mass.

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.