Three mass points each of mass m are placed at the vertices of an equilateral tringale of side l. What is the gravitational field and potential due to three masses at the centroid of the triangle ?

Explanation

`E_1= (GM)/((OA)^2)`
`E_2= (GM)/((OB)^2)`
`E_3= (GM)/((OC)^2)`

From `triangleOBD, " "cos 30^(@)= (BD)/(OB)=(l"/"2)/(OB)`
`OB= (l"/"2)/(cos 30^(@))= (BD)/(OB)= (l"/"2)/(2sqrt(3))= l"/"sqrt(3)`
Gravitational field at O due to m at A, B and C is say `vec(E_1), vec(E_2)" and " vec(E_3)`.
`E= sqrt(E_(2)^(2)+E_(3)^(2)+2E_(2)E_(3) cos 120^(@))`
`=sqrt(((GM3)^2)/(I^2)+((3Gm)/(I))^2+2((3GM)/(I))(-(1)/(2)))`
`=(3GM)/(I)=` along OD
`vecE` is equal and opposite to `vec(E_1)`
net gravitational field = zero
As gravitational potential is scalar
`V " "=V_(1)+V_(2)+V_(3)`
`=(GM)/(OA)-(GM)/(OB)-(GM)/(OC)`
`V" "= -(3GM)/(I"/"sqrt(3))= -3sqrt(3)(Gm)/(I)`.