Next: Hydrostatic pressure Up: Notes on gravitational potential Previous: Notes on gravitational potential   Contents   Index

Potential of a homogeneous gravitational field

A homogeneous gravitational field is characterized by a constant vector field $ \color{white} \vec{g}$, the well known gravitational acceleration. Let us consider some volume or vessel filled with an incompressible fluid of density $ \color{white} \varrho$. The gravitational field exerts a force on the fluid. Its force density $ \color{white} \vec{f}$ is given by the following well known formula
6 3

It is a simple task to  potential to ([*]). Let us remind, a potential (if it exists) of some given force field $ \color{white} \vec{F}$ is defined as a certain function $ \color{white} \phi$ satisfying the following equation

7 2
We can see that to the $ \color{white} \vec{f}$ given by ([*]) there exists a potential, let us denote it by $ \color{white} \varphi$, stating
8 3
where $ \color{white} C \in \mathbb{R}$ is a constant of integration3.3 and $ \color{white} \vec{r}$ is a position vector3.4.