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## Homework Statement

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.

## Homework Equations

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?