Four particles each of mass `m` are kept at the four vertices of a square of side 'a' . Find gravitational potential energy of this system.

Options

A. `-(sqrt(2)Gm^(2))/l (2-1/(sqrt(2)))`
B. `-(2Gm^(2))/l (2+1/(sqrt(2)))`
C. `-(2Gm^(2))/l (sqrt(2)+1/(sqrt(2)))`
D. `-(2Gm^(2))/l (sqrt(2)-1/(sqrt(2)))`

Answer

B

Explanation

Form figure
`AB=BC=CD =AD =I`
`:. AC=BD=sqrt(l^(2)+l^(2))=lsqrt(2)`
Total potential energy of the system of four particles each of mass `m` placed at the vertices `A,B,C` and `D` of a square is
`U=(-(Gxxmxxm)/(AB))+(-(Gxxmxxm)/(AC))`
`+(-(Gxxmxxm)/(AD))+(-(Gxxmxxm)/(BC))`
`=(-(Gm^(2))/l)+(-(Gm^(2))/(lsqrt(2)))+(-(Gm^(2))/l)`
`+(-(Gm^(2))/l)+(-(Gm^(2))/(lsqrt(2)))+(-(Gm^(2))/l)`
`=-(4Gm^(2))/l-(2Gm^(2))/(lsqrt(2))=-(2Gm^(2))/l(2+1/(sqrt(2)))`