Four particles each of mass 'm' are located at the four corners of a sqaure of side a. If the system is revolving in a circle due to thei mutual force of attraction then the angular velocity and time period of each particle is

Four particles each of mass 'm' are located at the four corners of a s

1.	Four particles each of mass 'm' are located at the four corners of a sqaure of side a. If the system is revolving in a circle due to thei mutual force of attraction then the angular velocity and time period of each particle is
Answer» Solution : From the FIG. `r = a//sqrt(2)` The resultant gravitational force on ONE of the particles `F_(R) = sqrt(2) (Gm^(2))/(a^(2)) + (Gm^(2))/(2a^(2))` This gravitational force provides the NECESSARY CENTRIPETAL force `mr omega^(2) = (Gm)/(a^(2))(sqrt(2)+(1)/(2)) rArr` `(a)/(sqrt(2)) omega^(2) = (Gm)/(a^(2)) ((2sqrt(2)+1)/(2))rArr omega = sqrt((Gm)/(a^(3))(2+(1)/(sqrt(2))))` and `T = (2pi)/(omega) = 2pisqrt((a^(3))/(Gm)((sqrt(2))/(2sqrt(2)+1)))`