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
Famous quotes containing the words uniform and/or sphere:
“Thus for each blunt-faced ignorant one
The great grey rigid uniform combined
Safety with virtue of the sun.
Thus concepts linked like chainmail in the mind.”
—Thom Gunn (b. 1929)
“It is in the nature of allegory, as opposed to symbolism, to beg the question of absolute reality. The allegorist avails himself of a formal correspondence between “ideas” and “things,” both of which he assumes as given; he need not inquire whether either sphere is “real” or whether, in the final analysis, reality consists in their interaction.”
—Charles, Jr. Feidelson, U.S. educator, critic. Symbolism and American Literature, ch. 1, University of Chicago Press (1953)