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.