# Gravitational Potential Energy between atoms within the same object

How can you calculate the gravitational potential energy (GPE) of just one object? I learned that there must be at least two masses in order for gravitational potential energy to exist, but any object is composed of multiple atoms, so isn't there GPE between those atoms within the object? If two spheres have GPE between them, but then I glue the two spheres together, or the two spheres touch at a point, don't they still have GPE?

For atoms and molecules, the electromagnetic force is much stronger than gravity.

So you could calculate the GPE of the molecules making up a baseball, but it would be vanishingly small. Much smaller than the energy required to break the molecular bonds holding the material together (or to break the glue holding the spheres together).

Unless you're looking at something of the size of an asteroid or larger, generally we ignore the internal gravitational energy because other things are more important.

You can calculate the gravitational potential energy of an extended object by using integral calculus to treat it as a collection of infinitesimally small masses $$dm$$, with GPE $$-G(dm_1)(dm_2)/r$$ between each pair. You can think of this as adding up the GPE between each pair of atoms.

When you do this for a uniform-density sphere of mass $$M$$ and radius $$R$$, you find that its gravitational potential energy is

$$U=-\frac{3GM^2}{5R}.$$

A baseball has a mass of about 145 grams and a radius of about 37 millimeters. Its gravitational potential energy is therefore about negative 23 trillionths of a Joule. This is very small; 23 pJ is approximately the energy it takes to lift a flea 0.005 millimeters under Earth gravity.

By contrast, gravitational potential energy is very large for astronomical objects. For the Earth, it is about $$-2\times10^{32}$$ Joules! It would take all of the energy radiated by the Sun for a week to gravitationally disperse the Earth’s atoms, according to Wikipedia.

When a uniform-density sphere of mass $$M_1$$ and radius $$R_1$$ touches another uniform-density sphere of mass $$M_2$$ and radius $$R_2$$, the gravitational potential energy of the whole system is

$$U=-\frac{3GM_1^2}{5R_1}-\frac{3GM_2^2}{5R_2}-\frac{GM_1M_2}{R_1+R_2}.$$

The first two terms are the GPEs of each sphere. The third term is the GPE between the two spheres. This GPE between spheres turns out to be as if the whole mass of each one were concentrated in a point at its center, even though what is really going on is that every atom in the first sphere interacts with every atom in the second sphere.