Gravitational Self Energy of a Sphere
Whenever a stable system is formed, it loses some of its energy. Similarly, to break it and separate its entities to infinity we must give it at least the same amount of energy. This energy is called the self-potential energy of the system. Technically, self-potential energy is defined as the amount of work done by the internal forces in assembling a system from infinity or work done by an external agent to slowly break the system and separate its entities to infinity.
The gravitational self-potential energy of a solid sphere of mass density and radius is . What is the gravitational self-potential energy of a sphere of mass density and radius ?
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Relevant wiki: Evaluating Gravitational Potential Energy
The formula of the gravitational potential energy is E = 5 R 3 G M 2 . where M is the mass, R is the radius and G is the gravitational constant. Looking at the following equation M = 3 4 π R 3 ρ . since both R and ρ are doubled, then M doubled = 3 4 π ( 2 R ) 3 ( 2 ρ ) = 1 6 ( 3 4 R 3 ρ ) = 1 6 M . Thus, E doubled = 5 ( 2 R ) 3 G ( 1 6 M ) 2 . = 2 1 6 2 ⋅ 5 R 3 G M 2 . = 1 2 8 E .
Relationship Between E , M and R
If we were to generalize this for arbitrary constants k 1 and k 2 for ρ n = k 1 ρ and R n = k 2 R (where k 1 , k 2 > 0 ), and E is the original potential energy, then the new potential energy is E general = 5 3 G ⋅ k 2 R ( 3 4 π ( k 2 ) 3 R 3 k 1 ρ ) 2 . = 5 3 G ⋅ k 2 R ( k 2 ) 6 ( k 1 ) 2 ( 3 4 π R 3 ρ ) 2 . = ( k 1 ) 2 ( k 2 ) 5 ⋅ 5 R 3 G M 2 . = ( k 1 ) 2 ( k 2 ) 5 E . So E depends on the constants k 1 and k 2 .