Eight particles of mass 'm' each are placed at the verties of a cube of side 'a' . Find gravitational potential energy of this system . .

Options

A. `-2(Gm^(2))/(a)(12+6sqrt(2)+(4)/(sqrt(3)))`
B. `-(Gm^(2))/(a)(12+6sqrt(2)+(4)/(sqrt(3)))`
C. `(Gm^(2))/(a)(12+6sqrt(2)+(4)/(sqrt(3)))`
D. `-(Gm^(2))/(2a)(12+6sqrt(2)+(4)/(sqrt(3)))`

Answer

A, B, C, D

Explanation

Total particles are `n = 8`
`:.` Total numbers of pairs are `n((n-1))/(2)` or `28`.
In `12` pairs , distance between the particles is 'a'. In `12`pairs , distance is `sqrt (2)a` and in remaining `4` pairs distance is `sqrt(3)a`.
`:. U = 12 ((-Gmm)/(a))+12 ((-Gmm)/(sqrt2a))+4((-Gmm)/(sqrt3a))`
`= -(Gm^(2))/(a)(12+6sqrt(2)+(4)/(sqrt(3)))`