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### 2.6 Gravitational potential.

In the previous section we discussed about the gravitational field. In this section we'll discuss another important concept in physics called the concept of potential, in particular the gravitational potential.

Firstly, the definition of the gravitational potential. I can't emphaisze enough to remind how important it is to understand the definition before we go ahead. "Gravitational potential at any point is the work done to bring a unit mass from infinity to the point". Its SI unit is joules per kilogram (Jkg$^{-1}$)

By this definition the gravitational potential is zero at infinity and negative near the point source. The reson for it being negative can be understood on the basis of the fact that the gravitaional force is attractive force which means that the mass does the work in moving from infinity to the point. The potential $\phi$ in the field of a point mass $M$ is given by the equation.
$$\phi=-\frac{GM}{r}$$
where,  $\phi$=gravitational potential, in Jkg$^{-1}$
$M$=mass, in kg
$r$= distance from the point mass
The above definition can be understood as coming from the following derivation,
$$\phi=\frac{Work done}{mass}=\frac{\int_{-\infty}^{r}F_{G}dx}{m}=-\frac{GM}{r}$$
The expression for gravitational force has been used in the above derivation. The plot of the gravitational potential can be seen below.

The change in the gravitational potential energy, $E_{p}$ of a mass $m$ that is moved through a change in gravitational potential of $\Delta \phi$, is the work done on the mass to produce the move, and is given by:
$$E_{p}=m\Delta \phi$$
Ex. Calculate the gain in the potential energy of a iron ball of 30kg when on the surface vs when at a height of 10$^{5}$m above the ground. Given earth radius=$6.38\times 10^{6}$ and mass of the earth =$5.98\times 10^{24}$

Soln: gravitational potential= $9.18\times 10^{5} Jkg^{-1}$
gravitational potential energy=m$\Delta \phi$=$2.75\times10^7J$