Derivation For A Uniform Sphere
The gravitational binding energy of a sphere is found by imagining that it is pulled apart by successively moving spherical shells to infinity, the outermost first, and finding the total energy needed for that.
If we assume a constant density then the masses of a shell and the sphere inside it are:
- and
The required energy for a shell is the negative of the gravitational potential energy:
Integrating over all shells we get:
Remembering that is simply equal to the mass of the whole divided by its volume for objects with uniform density we get:
And finally, plugging this in to our result we get:
Read more about this topic: Gravitational Binding Energy
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—James Madison (1751–1836)