# 4.3 Gravitational potential energy  (Page 2/4)

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## Absolute gravitational potential energy

An arrangement of the system is referred to possess zero potential energy with respect to a particular reference. For this we visualize that particles are placed at very large distance. Theoretically, the conservative force like gravitation will not affect bodies which are at infinity. For this reason, zero gravitational reference potential of a system is referred to infinity. The measurement of gravitational potential energy of a system with respect to this theoretical reference is called absolute gravitational potential energy of the system.

$U\left(r\right)=-\underset{\infty }{\overset{r}{\int }}{F}_{G}dr$

As a matter of fact, this integral can be used to define gravitational potential energy of a system:

Gravitational potential energy
The gravitational potential energy of a system of particles is equal to “negative” of the work by the gravitational force as a particle is brought from infinity to its position in the presence of other particles of the system.

For practical consideration, we can choose real specific reference (other than infinity) as zero potential reference. Important point is that selection of zero reference is not a limitation as we almost always deal with change in potential energy – not the absolute potential energy. So long we are consistent with zero potential reference (for example, Earth’s surface is considered zero gravitational potential reference), we will get the same value for the difference in potential energy, irrespective of the reference chosen.

We can also define gravitation potential energy in terms of external force as :

Gravitational potential energy
The gravitational potential energy of a system of particles is equal to the work by the external force as a particle is brought from infinity slowly to its position in the presence of other particles of the system.

## Earth systems

We have already formulated expressions for gravitational potential energy for “Earth – body” system in the module on potential energy.

The potential energy of a body raised to a height “h” has been obtained as :

$U=mgh$

Generally, we refer gravitational potential energy of "Earth- particle system" to a particle – not to a system. This is justified on the basis of the fact that one member of the system is relatively very large in size.

All terrestrial bodies are very small with respect to massive Earth. A change in potential energy of the system is balanced by a corresponding change in kinetic energy in accordance with conservation of mechanical energy. Do we expect a change in the speed of Earth due to a change in the position of ,say, a tennis ball? All the changes due to change in the position of a tennis ball is reflected as the change in the speed of the ball itself – not in the speed of the Earth. So dropping reference to the Earth is not inconsistent to physical reality.

## Gravitational potential energy of two particles system

We can determine potential energy of two particles separated by a distance “r”, using the concept of zero potential energy at infinity. According to definition, the integral of potential energy of the particle is evaluated for initial position at infinity to a final position, which is at a distance “r” from the first particle at the origin of reference.

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