What Determines the Mass Ratio in a Binary Star System?

[mkphysics]

1. The two components of a double star are observed to move in circles of radii r_1 and r_2. What is the ratio of their masses? (Hint: Write down their accelerations in terms of the angular velocity of rotation, w)

The answer is m1/m2 = r2/r1.

How does one cancel the velocity from the problem to get this result?

Homework Equations

For an isolated two body system

m1.a1=-m2.a2
where m are the body masses and a are the accelerations

w=v/r
angular velocity equals velocity divided by radius

a=dw/dt
angular acceleration

The Attempt at a Solution

m1.dw1/dt = m2.dw2/dt

therefore m1.w1=-m2.w2

m1/m2 = -w2/w1

m1/m2 = -(v2/r2)/(v1/r1)

m1/m2 = -v2r1/v1r2

 

[kuruman]

Do the stars rotate about any which point or about a specific point? What are this point's properties?

 

[mkphysics]

Unfortunately all information is contained in the question. This question is taken directly from chapter 1 problem 2 of Classical Mechanics by Kibble and Berkshire. I'm not sure if this is a good book yet or not. From your reply it seems the question does not contain enough information to find the solution given. Can one infer the conditions of the system from the result given (m1/m2 = r2/r1)?

 

[kuruman]

I am not familiar with the textbook, but it does not matter. Does "center of mass" ring a bell?

 

[iva]

I know this is an old post but for anyone looking up this problem, here's how I got to that answer:

Using Newtons law for 2 isolated bodies

m₁a₁ = -m₂a₂

The bodies are rotating ( here i guessed at the same velocity).
The motion is rotational so you can use tangential acceleration

a_T = w²r and substitute this into above to get

m₁ w²r₁ = -m₂w²r2

cancelling out w² on both sides, move m's to left and r's to right to get

m₁/m₂ = -r₂/r₁