Circular motion and gravitation question

[chazgurl4life]

Q:Suppose that a binary star system consists of two stars of equal mass. They are observed to be separated by 340 million kilometers and take 5.0 Earth years to orbit about a point midway between them. What is the mass of each?

 

[Hootenanny]

Have you done any working? This question is similar to you previous one, just adding in a bit of circular motion(assuming the orbits are circular).

 

[kyle8921]

The mass of each star in the binary star system can be calculated using the formula for the period of circular motion, which is T = 2π√(r^3/GM), where T is the orbital period, r is the separation distance, G is the gravitational constant, and M is the combined mass of the two stars.

Substituting the given values into the formula, we get:

5.0 Earth years = 2π√(340 million kilometers)^3/(G x M)

Simplifying, we get:

5.0 Earth years = 2π√(340 x 10^9 meters)^3/(6.67 x 10^-11 m^3/kg/s^2 x M)

Squaring both sides and rearranging, we get:

M = (4π^2 x (340 x 10^9)^3)/(6.67 x 10^-11) x (5.0 x 365.25 x 24 x 3600)^2

M = (5.86 x 10^29)/5.77 x 10^15

M = 1.01 x 10^14 kg

Therefore, the mass of each star in the binary star system is approximately 1.01 x 10^14 kg. This is a very large mass, but it is not surprising given that the stars are separated by a distance of 340 million kilometers and have an orbital period of 5.0 Earth years.