Problem related to the center of mass of a binary star.

[TheSodesa]

So here's my problem:

The distance between two stars is constant(d = 4,3 * 10¹⁰m), and they have a common center of mass. M_s = mass of our star, m_a = 0,82 * M_s and m_b = 2,2 * M_s.

What I'm supposed to do is calculate the period of orbit of both stars, which is the same for them both, since the distance between them stays constant. I knew to set the centripetal force equal to the gravitational force for star A as follows: (Gm_am_b)/(r_a + r_b)² = m_a * ((2pi*r_a)/T)²/r_a.

I know how to solve this equation for T. However, I'm missing some required information. Looking at the answer sheet, it says that because the stars have a common center of mass, m_ar_a = m_br_b => m_ar_a = m_b(d-r_b). I do not understand where this relationship comes from. At all. It's the only missing information I need to solve the problem.

Is there a simple explanation for this relationship? The thing that bugs me is that we didn't go over this information in class. Is it really that simple, that I'm just supposed to know this? Even our book says nothing about this, at least not the chapter that is related to this problem.

EDIT: There are chapters later on in the book related to torque and things along those lines, but this problem can't seriously expect me to know stuff we haven't covered yet. Because if it does, the authors of this book messed up when they were writing it.

 

[A.T.]

m_ar_a = m_br_b ... I do not understand where this relationship comes from.

Do you understand what a common center of mass is, and how it is computed?
http://en.wikipedia.org/wiki/Center_of_mass#A_system_of_particles

 

[TheSodesa]

Do you understand what a common center of mass is, and how it is computed?
http://en.wikipedia.org/wiki/Center_of_mass#A_system_of_particles

Annoyingly, no. Looking at the link you provided, it mentions torque, which I know will be covered later on in the course I'm taking. The sum formula alone on that wikipedia page isn't intuitive enough for me to figure it out. I need some solid, well explained theory to back it up.

 

[SteamKing]

Do you know the principle of a see-saw, or how to balance a beam resting on a fulcrum?

It's the same principle for finding the location of the center of gravity of two blocks or two stars.

You know the total distance between the stars, 4.3*10¹⁰ m (we will assume that this also represents the distance between the centers of the stars.

Our unknown center of gravity (or fulcrum point, to use the see-saw analogy) lies at some unknown distance between the two stars, say r_a to the first star and r_b to the second star. When the see-saw is balanced, the product of one mass and its distance from the fulcrum point is exactly balanced by the product of the second mass (on the opposite side of the fulcrum) and its distance from the fulcrum.

In math terms,

m_a*r_a = m_b*r_b, or

m_a*r_a - m_b*r_b = 0

We also have a second relationship which says that the distance between stars is equal to the sum of each star's distance from the fulcrum, or

r_a + r_b = d

We now have two equations in two unknowns, r_a and r_b, which can be solved very easily.

The formula in the OP uses the substitution r_b = d - r_a

[Note: The expression in the OP contains an error when it says m_ar_a = m_b(d-r_b).

It should read m_ar_a= m_b(d-r_a) instead.]

 

[TheSodesa]

Do you know the principle of a see-saw, or how to balance a beam resting on a fulcrum?

It's the same principle for finding the location of the center of gravity of two blocks or two stars.

You know the total distance between the stars, 4.3*10¹⁰ m (we will assume that this also represents the distance between the centers of the stars.

Our unknown center of gravity (or fulcrum point, to use the see-saw analogy) lies at some unknown distance between the two stars, say r_a to the first star and r_b to the second star. When the see-saw is balanced, the product of one mass and its distance from the fulcrum point is exactly balanced by the product of the second mass (on the opposite side of the fulcrum) and its distance from the fulcrum.

In math terms,

m_a*r_a = m_b*r_b, or

m_a*r_a - m_b*r_b = 0

We also have a second relationship which says that the distance between stars is equal to the sum of each star's distance from the fulcrum, or

r_a + r_b = d

We now have two equations in two unknowns, r_a and r_b, which can be solved very easily.

The formula in the OP uses the substitution r_b = d - r_a

[Note: The expression in the OP contains an error when it says m_ar_a = m_b(d-r_b).

It should read m_ar_a= m_b(d-r_a) instead.]

Yeah, that last mistake was entirely my doing. I blame frustration and having to deal with a text editor to write equations.

I any case I'm probably going to leave this problem for later, once we have actually gone through statics, torque and the like, that I believe are used to derive those two relationships. I don't want to just start plugging numbers into an equation that hasn't been justified to me in the slightest. I appreciate your input, but you didn't go into why the product of mass and distance from the center of gravity (or fulcrum) is constant. Something is definitely inversely proportional to something else, just like in Boyle's law p₀V₀ = p₁V₁. Here Temperature T is constant, and p is inversely proportional to V, and vice versa.

I'm just frustrated with the fact that the teacher would give us a problem that requires us to know something we haven't yet covered in class. :(

 

[dean barry]

Your initial equation works, transpose for T

Take at look also at the attached sheet. (for information only)

 

[SteamKing]

I appreciate your input, but you didn't go into why the product of mass and distance from the center of gravity (or fulcrum) is constant.

The moment about the fulcrum (which is what the product of the mass and its distance from the fulcrum is called) is not just constant, but from the reference point of the fulcrum, the mass of each star times its distance from the fulcrum must be equal. This is the principle of the lever, if you haven't guessed it:

http://en.wikipedia.org/wiki/Lever

Something is definitely inversely proportional to something else, just like in Boyle's law p₀V₀ = p₁V₁. Here Temperature T is constant, and p is inversely proportional to V, and vice versa.

Not necessarily. Not all mathematical relationships can be modeled on Boyle's Law or similar.

I'm just frustrated with the fact that the teacher would give us a problem that requires us to know something we haven't yet covered in class. :(

This is unusual, but not unheard of. That's why it always helps to be able to do a little research, even if you haven't covered the material previously. Think of it as preparation for when you encounter something unfamiliar after you finish school: not every problem gets solved in a classroom.

 

[TheSodesa]

The moment about the fulcrum (which is what the product of the mass and its distance from the fulcrum is called) is not just constant, but from the reference point of the fulcrum, the mass of each star times its distance from the fulcrum must be equal. This is the principle of the lever, if you haven't guessed it:

http://en.wikipedia.org/wiki/Lever
Not necessarily. Not all mathematical relationships can be modeled on Boyle's Law or similar.
This is unusual, but not unheard of. That's why it always helps to be able to do a little research, even if you haven't covered the material previously. Think of it as preparation for when you encounter something unfamiliar after you finish school: not every problem gets solved in a classroom.

Fair enough. Although, the teacher admitted last night after class, that he made a little blunder when he gave us that assignment, because the book we are using is the newest edition of it, and the order of topics has been switched around. Damn the book publishers and their greed.

We'll be covering the center of mass tomorrow, apparently.