Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. Calculate the speed of each particle

Explanation

The net gravitational force = Centripetal force
Force `F_(1)` between A and B , ` F_(1) = (GMM)/((sqrt(2)R)^(2))`
Force `F_(2)` between A and D , `F_(2) = (GMM)/((sqrt(2)R)^(2))`
Force `F_(3)` between A and C ( or B and D)
`F_(3) = (GMM)/((2R)^(2))`
The components F1 and F2 along the radius
`F_(1)` cos 45 and `F_(2)` cos 45 `F_(1) = F_(2) = F`
` :. ` Net force = `2F cos 45 +F_(3)`
`= 2 (GM)/(sqrt(2)R)^(2) xx (1/2) + (GM^(2))/(4R^(2))`
`(Mv_(0)^(2))/R = (GM^(2))/(2R^(2)) (2sqrt(2) +1)`
`v_(0)^(2) = (GM)/(4R) [ 1+2sqrt(2)] `
` v_(0) = 1/2 sqrt((GM)/R (1+2sqrt(2)))`