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.

Transclude of Potential-Energy#potential-energy-of-many-particle-systems

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: