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:
“The Federal Constitution has stood the test of more than a hundred years in supplying the powers that have been needed to make the Central Government as strong as it ought to be, and with this movement toward uniform legislation and agreements between the States I do not see why the Constitution may not serve our people always.”
—William Howard Taft (1857–1930)
“If today there is a proper American “sphere of influence” it is this fragile sphere called earth upon which all men live and share a common fate—a sphere where our influence must be for peace and justice.”
—Hubert H. Humphrey (1911–1978)