How Do Two Stars Orbit Each Other with Equal Masses?

[Masjo]

Their are 2 stars orbiting one another. Their masses are equal. The speed at which they rotate is 220 km/h, and their orbital period is 14.4 days. Find the mass of the Star.

Homework Equations

220km/h = 61.1m/s
14.4 days = 51840s

V= 2pi(r)/T
F=ma
F=mv^2/R
F= m4pi^2r/T^2

Fg= Gm(star1)m(star2)/r^2


These are all the equations I've been using. So far, using V= 2pi(r)/T I can deduce that the radius of the star is 504111.2m and using V=d/t I THINK I can say that the space between the stars is 3167424m, but I am not sure.
I've also tried equating F=mv^2/r with Fg = Gm(star1)^2/r^2, and then found the mass to be 3.616x10^13. I don't think this is right, as my teacher said hint: The mass of the sun is 1.99*10^30 kg.
I know I often make stupid math mistakes, but I can't seem to find out where I went wrong... Can anyone help me?

 

[jbsteele]

The mass of the star can be calculated using the equation Fg = Gm(star1)m(star2)/r^2, where G is the gravitational constant (6.67x10^-11 N m^2/kg^2). m(star1) = (Fg x r^2)/(G x m(star2))m(star1) = (6.67x10^-11 x 3167424^2)/(G x 1.99x10^30) m(star1) = 3.59x10^30 kg

 

[nlightner]

I would first like to commend you on your use of relevant equations to solve this problem. It shows a strong understanding of the concepts involved. However, your calculation for the mass of the star seems to be incorrect. Let's break down the problem and see where the mistake might have occurred.

First, we know that the two stars have equal masses, so let's call their mass m. Using the equation F=mv^2/r, we can equate the force of gravity between the two stars to the centripetal force keeping them in orbit. This gives us:

Fg = mv^2/r = Gm^2/r^2

We can rearrange this equation to solve for m:

m = v^2r/G

Plugging in the given values for v (61.1 m/s) and r (5.0411x10^8 m), and the universal gravitational constant G (6.67x10^-11 N*m^2/kg^2), we get a mass of 3.01x10^30 kg. This is very close to the mass of the sun (1.99x10^30 kg), which makes sense since the given hint was that the mass of the sun is involved in the solution.

So, it seems that the mistake was in your calculation for the radius of the star. It should be 5.0411x10^8 m, not 5.0411x10^5 m. This is because the units for velocity are m/s, not km/h, so your conversion to m/s should have given you a larger value.

In summary, the mass of each star in this system is approximately 3.01x10^30 kg, which is equal to the mass of the sun. This makes sense given that the two stars have equal masses and their orbital period is relatively short. I hope this helps clarify the problem for you. Keep up the good work!