A binary star system consists of a star P and a star Q, of mass 4.0 x 10^10 kg and 2.0 x 10^10 kg respectively, separated 6.3 x 10^9 m apart. Star P and Star Q move in circular orbits with their centers at the center of mass which remains at rest.
Find the position of the center of mass of the binary star system.
g = GM/r^2 (I used this and equated the field strengths from both)
But I have some worked solution to this from my teacher, and he used moments to solve this so I am just adding that equation here as well.
Moments = F x Perpendicular distance (This is just what my teacher used, but in many similar questions I have done, I never used this moments for such a question)
Or is it because this is a binary system that moments have to be used?
And in space normally, with no binary system or anything, we must use the g-field?
Also, I don't use calculus or vectors, so please don't use that for this question.
The Attempt at a Solution
Let x be the distance of the Center of Mass from Star P
At that point,
Gravitational Field Strength of P = Gravitational Field Strength of Q
g (p) = g (q)
(G(4 x 10^10))/(x^2) = (G(2 x 10^10))/(6.3 x 10^9 - x)^2
2 = (x^2)/(6.3 x 10^9 - x)^2
2(6.3 x 10^9 - x)^2 = x^2
solving the quadratic equation, I get x = (3.7 x 10^9) m
I would appreciate if someone could help me with this. And also, if someone could tell me if this is the way I am supposed to work it out for binary star systems and normal masses in space at a distance, or does it differ for this type of binary star system questions?