4.7 Gravitational potential

Description of force having “action at a distance” is best described in terms of force field. The “per unit” measurement is central idea of a force field. The field strength of a gravitational field is the measure of gravitational force experienced by unit mass. On a similar footing, we can associate energy with the force field. We shall define a quantity of energy that is associated with the position of unit mass in the gravitational field. This quantity is called gravitational potential (V) and is different to potential energy as we have studied earlier. Gravitational potential energy (U) is the potential energy associated with any mass - as against unit mass in the gravitational field.

Two quantities (potential and potential energy) are though different, but are closely related. From the perspective of force field, the gravitational potential energy (U) is the energy associated with the position of a given mass in the gravitational field. Clearly, two quantities are related to each other by the equation,

$$U=mV$$

The unit of gravitational potential is Joule/kg.

There is a striking parallel among various techniques that we have so far used to study force and motion. One of the techniques employs vector analysis, whereas the other technique employs scalar analysis. In general, we study motion in terms of force (vector context), using Newton’s laws of motion or in terms of energy employing “work-kinetic energy” theorem or conservation law (scalar context).

In the study of conservative force like gravitation also, we can study gravitational interactions in terms of either force (Newton’s law of gravitation) or energy (gravitational potential energy). It follows, then, that study of conservative force in terms of “force field” should also have two perspectives, namely that of force and energy. Field strength presents the perspective of force (vector character of the field), whereas gravitational potential presents the perspective of energy (scalar character of field).

Gravitational potential

The definition of gravitational potential energy is extended to unit mass to define gravitational potential.

Gravitational potential

The gravitational potential at a point is equal to “negative” of the work by the gravitational force as a particle of unit mass is brought from infinity to its position in the gravitational field.

Or

Gravitational potential

The gravitational potential at a point is equal to the work by the external force as a particle of unit mass is brought from infinity to its position in the gravitational field.

Mathematically,

$$V=-W_G=-\underset{\infty }{\overset{r}{\int }}\frac{F_{G}dr}{m}=-\underset{\infty }{\overset{r}{\int }}Edr$$

Here, we can consider gravitational field strength, “E” in place of gravitational force, “ $F_G$ ” to account for the fact we are calculating work per unit mass.

Change in gravitational potential in a field due to point mass

The change in gravitational potential energy is equal to the negative of work by gravitational force as a particle is brought from one point to another in a gravitational field. Mathematically,