Find the position of the center of mass of the binary star system

[csharsha]

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?

 

[ehild]

The position of the center of mass is independent of all forces that act, it depends only on the masses and their positions. Your teacher used momentums to find a point with respect to the sum of the momentums is zero:

m₁v₁+m₂v₂=0.

Both stars orbit along concentric circles so as the straight line connecting them goes through the common center of the circles (the center of mass, CM) and their angular velocities are the same, ω. The total momentum of the binary star system is zero. See the figure: when one star moves up, the other down, so the direction of their momentums are opposite. If m₁ is at distance r₁ from the center and m₂ is at distance r₂, the velocities are

v₁=m₁r₁ω and

v₂=-m₁r₁ω .

The total momentum is

m₁v₁+m₂v₂=0, that is

m₁r₁ω -m₁r₁ω =0

m₁r₁=m₂r₂. *

You are given the distance between the stars,

d=r1+r2. **

_{Find r1 and r2} from equations * and **.


See also :

http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html


ehild