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Work done by an external agent in bringi...

Work done by an external agent in bringing three particles each of mass `m` at the vertices of an equilateral triangle of length `l` is

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We know that gravitational field is conservative.
`W=-(DeltaPE)_("system")`………..`(1)`
`DeltaPE=(PE_("final")-PE_("initial"))`
Since initially the particles `A`, `B` and `C` were at infinity, their initial gravitational potential energy was zero
`implies DeltaPE=PE_("final")`
The gravitational potential energy of the configuration of the particles `=PE` between `A` and `B+PE` between `B` and `C+PE` between `C` and `A`.
`impliesDeltaPE=|(-(Gm_(A)+m_(B))/(l))+(-(Gm_(B)m_(C ))/(l))+(-(Gm_(C )m_(A))/(l))|`
Putting `m_(A)=m_(B)=m_(C )=m`, we obtain
`DeltaPE=(3Gm^(2))/(l)`............`(2)`
Using `(1)` and `(2)` , we obtain , `W=3Gm^(2)//l`.
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