Question

Four stars, each having a mass M, are situated in a square pattern (each star is at the corner of the square) and are rotating about the system's center of mass. At what speed must they move for the system to maintain its square shape, such that the system does not collapse due to mutual gravitational attraction of the stars? Assume a side length of a for the square, and express your answer in terms of M, G, and a.

Answer 1

Net force acting on each star towards the center of the square,

Fnet = sqrt(2)*G*M^2/a^2 + G*M^2/(2*a^2)

= (G*M^2/a^2)*(sqrt(2) + 1/2)

= 1.9142*G*M^2/a^2

radius of the circle, r = distance from each star to center

= sqrt(2)*a/2

= 0.707*a

let v is the speed of each star.

now using Newtons second law, Fnet = M*a_radial

1.9142*G*M^2/a^2 = M*v^2/r

1.9142*G*M^2/a^2 = M*v^2/(0.707*a)

1.9142*G*M/a = v^2/0.707

==> v^2 = 1.35*G*M/a

v = sqrt(1.35*G*M/a) <<<<<<------------Answer