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Engineering LibreTexts

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.