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Three particles, each of mass m, are sit...

Three particles, each of mass m, are situated at the vertices of equilateral triangle of side length a. The only forces. It is desired that each particle moves in a circle while maintaining the original mutual speration a. Find the intial velocity that should be given to each particle and also the time period of the circular motion.

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Refer to figure, force of attraction on body at `C` due to body at `A` is
`F_(1)=(Gmm)/(r^(2))=(Gm^(2))/(r^(2))` along `CA`
Force of attraction on body at `C` due to body at `B` is
`F_(2)=(Gmm)/(r^(2))=(Gm^(2))/(r^(2))` along `CB`.

These force `vecF_(1)` and `vecF_(2)` are inclined at an angle `60^@`. the resultant force on the body at `C` is
`F=sqrt(F_(1)^(2)+F_(2)^(2)+2F_(1)F_(2)cos60^@)`
`=sqrt(F_(1)^(2)+F_(2)^(2)+2F_(1)F_(2)(1/2))`
`sqrt(3)F_(1)=sqrt(3)(Gm^(2))/(r^(2))` acting along CD
`:'F_(1)=F_(2)=[(Gm^(2))/(r^(2))]`
Here `OC=2/3 C=2/3ACsin60^@`
`=2/3rxx(sqrt(3))/2=r/(sqrt(3))`
When each body is describing a circular orbit with cenre of orbit at `O`, the force `F` provides the required centripetal force. The radius of the circular orbit is `OC=r//sqrt(3)`. If `v` is the speed of the body in circular orbit, then
`(Mv^(2))/(r//sqrt(3))=(sqrt(3)Gm^(2))/(r^(2))` or `v=sqrt((Gm)/r)`
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