Gravitational Potential Energy

Gravitational Potential Energy is the Potential Energy of a System concerning the Force of Gravitation.

Close to the ground, it can be simplified to:

However, we cannot simplify this in space. We must use Newton’s Law of Universal Gravitation:

to determine the gravitational potential energy.

For this task, we can use Work, where where we get the work done as the system changes from configuration to .

Remember that Potential Energy is a Conservative Force, so we only care about the start and end result.

For that, we use the work in integral form:

Because and point in opposite directions, we can change our Dot Product to a negative sign:

Continuing:

Our difference in potential energy is negative work, so:

We can consider the value of potential energy at a single point by taking an infinite separation of the particles (where the potential energy is zero).

For Systems of Particles with more than two particles considered, we use a similar logic.

Take a look at the example from the Potential Energy page.

Potential Energy of Many Particle Systems

Interpreted differently, imagine we have two particles who start separated by an infinitely large distance at rest.

Let’s consider Gravitation as an example force in this scenario.

When we bring the two particles together at a constant velocity until the separation is a distance apart, we can model them as:

Where is the work done by gravitation. The work that we apply, , is:

Which is our equation for gravitational potential energy, which we derive in Gravitational Potential Energy. We can say that:

The potential energy of a system of particles is equal to the work done by an external agent to assemble the system, starting from the standard reference configuration.

(Physics 5th Edition, Halliday, Resnick, Krane)

In other words, potential energy can be defined as the work needed to bring the system in its current configuration from a “default” configuration.

In our case, the standard reference configuration is the initial infinite separation.

This holds for a system containing more than two particles. We can calculative potential energy

Link to original

Lets consider a system for three masses, , , , which start infinitely separated from each other. To find the work needed to bring it to a set of distances from each other (detailed below) we move each particle, one by one, at a constant velocity, our external force counteracting the gravitational force.

Our potential energy is:

We just combine the external work done for each particle and then use that to find gravitational potential energy, similar to before. This makes sense: potential energy is based off the configuration of the system.

If we wanted to separate our masses once again, we would apply an energy:

Also known as the binding energy.

Info

The derivative of gravitational potential energy with respect to is , shown: