Question about gravitational force(Binary star)

[Clara Chung]

Homework Statement

In a binary star system, the centre of rotation of the system coincides with its centre of mass. If the two centres do not coincide, there arises a net moment and the system will collapse.

Homework Equations

The Attempt at a Solution

But refer to the photo, isn't
Fh = Fh , so there is no net moment?

 

[haruspex]

If the two centres do not coincide, there arises a net moment and the system will collapse.

That's a very odd statement. First, "centre of rotation" implies circular orbits, whereas in general each will follow an elliptical orbit, the common mass centre being at a focus of each. Secondly, if circular, I do not understand how the centres might not coincide.

Fh = Fh

I think the author is referring to angular momentum rather than moments of forces, but I could be wrong.

 

[Clara Chung]

That's a very odd statement. First, "centre of rotation" implies circular orbits, whereas in general each will follow an elliptical orbit, the common mass centre being at a focus of each. Secondly, if circular, I do not understand how the centres might not coincide.

I think the author is referring to angular momentum rather than moments of forces, but I could be wrong.

Yes. It is a circular orbit. The question originally ask for the mass of each star if the radius of orbit P is twice that of Q. P and Q are the stars in a binary system. My approach is listing out an equation with equal force and angular velocity. However my book approach is :
Sum of moments about center of mass is zero.
Mp r = Mq 2r
So the mass of p is half of that q.
I don't understand the statement of sum of moments about cg is zero

 

[haruspex]

I don't understand the statement of sum of moments about cg is zero

Mp r = Mq 2r

Ok, I see. The book is using "moment" to refer to the first moment of mass, i.e product of mass and displacement from axis. When the chosen axis is the common mass centre, the sum of the first moments of mass is zero.