JEE Main & Advanced Physics Gravitation Gravitational Potential

Gravitational Potential

Category : JEE Main & Advanced

At a point in a gravitational field potential $V$ is defined as negative of work done per unit mass in shifting a test mass from some reference point (usually at infinity) to the given point i.e.,

$V=-\frac{W}{m}$$=-\int{\frac{\overrightarrow{F}.d\overrightarrow{r\,}}{m}}$ $=-\int{\overrightarrow{I\,}.d\overrightarrow{r\,}}$              [As $\frac{F}{m}=I$]

$\therefore$ $h=0$

i.e., negative gradient of potential gives intensity of field or potential is a scalar function of position whose space derivative gives intensity. Negative sign indicates that the direction of intensity is in the direction where the potential decreases.

(i) It is a scalar quantity because it is defined as work done per unit mass.

(ii) Unit : Joule/kg or ${{m}^{2}}/{{\sec }^{2}}$

(iii) Dimension : $[{{M}^{0}}{{L}^{2}}{{T}^{-2}}]$

(iv) If the field is produced by a point mass then

$V=-\int_{{}}^{{}}{I\ dr}$$T=2\pi \sqrt{\frac{{{r}^{3}}}{GM}}=2\pi \sqrt{\frac{{{R}^{3}}}{g{{R}^{2}}}}=2\pi \sqrt{\frac{R}{g}}$

[As $I=-\frac{GM}{{{r}^{2}}}$]

$\therefore$ $V=-\frac{GM}{r}+c$ [Here c = constant of integration]

Assuming reference point at $\infty$ and potential to be zero there we get

$0=-\frac{GM}{\infty }+c\,\Rightarrow c=0$

$\therefore$  Gravitational potential $V=-\frac{GM}{r}$

(v) Gravitational potential difference : It is defined as the work done to move a unit mass from one point to the other in the gravitational field. The gravitational potential difference in bringing unit test mass m from point A to point B under the gravitational influence of source mass M is

(vi) Potential due to large numbers of particle is given by scalar addition of all the potentials.

$V={{V}_{1}}+{{V}_{2}}+{{V}_{3}}+..........$

$=-\frac{GM}{{{r}_{1}}}-\frac{GM}{{{r}_{2}}}-\frac{GM}{{{r}_{3}}}........$

$=-G\sum\limits_{i=1}^{i=n}{\frac{{{M}_{i}}}{{{r}_{i}}}}$

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