A hypothetical planet of mass `M` has three moons each of equal mass 'm' each revolving in the same circular orbit of radius `R`. The masses are equally spaced and thus form an equilateral triangle. Find :

(i) the total `P.E.` of the system
(ii) the orbital speed of each moon such that they maintain this configuration.

Answer

`(i) -(3Gm)/R(m/(sqrt(3))+M)), (ii) sqrt(G/R(m/(sqrt(3)+M))))`

Explanation

(i) `U`=potential energy of `(1)` in the fileld `M+` potential energy of `(2)` in filed of `[(1)+M]`+potential energy of `(3)` in field of `[(1)+(2)+M]`
`U=-(GMm)/R+m[-(GM)/R-(Gm)/(sqrt(3)R)]+m[-(Gm)/R-(Gm)/(sqrt(3)R)-(Gm)/(sqrt(3)R)]`
`=-(2GMm)/R-(3Gm^(2))/(sqrt(3)R)`
`=-(3Gm)/R[M+m/(sqrt(3))]`

`(ii) F_(R)=F_(mM)+2F_(mm)cos 30^(@)`
`F_(R)=(GMm)/(R^(2))+sqrt(3)(Gm^(2))/((sqrt(3)R)^(2))`
`F_(R)=(Gm)/(R^(2))[M+m/(sqrt(3))]`
This force will provided required centrepetal force
`(mv^(2))/R=(Gm)/(R^(2))[M+m/(sqrt(3))]`
`v=sqrt(G/R(M+m/(sqrt(3))))`