The physics between stars and planets homework question?

[joeyjane]

Homework Statement

A binary star system consists of two equal mass stars that revolve in circular orbits
about their center of mass. The period of the motion, T = 25.5 days, and the
orbital speed v = 220 km/s of the stars can be measured from telescopic observations.
What is the mass of each star?

Homework Equations

I'm not sure at all what I could use for solving this...

The Attempt at a Solution

I was thinking I could solve it using Kepler's third law, but I think since the orbits are circular I can't use that. I'm also struggling with the concepts involved in this question, so if anyone has any tidbits of help I would greatly appreciate that :)

 

[gneill]

Circular motion implies a centripetal force (or acceleration). Gravity is supplying the force. Note that the stars are circling the center of gravity, so that the radius of their circular orbits is half the distance between the two stars.

For one star, write the equations for the gravitational force and the centripetal force for circular motion. An expression for the radius of the orbit can be found from the relationship between the circumference of the orbit, the velocity, and the period.

 

[leonne]

no you can use kelpers 3rd law, I am pretty sure.
I think you can use the velocity to find semi major axis( if i remember correctly) and than just solve for M (total mass) than divide by 2 to find each mass

 

[joeyjane]

Gneill, I'm a little confused on how you get the formula for the period without using kepler's 3rd. I just found in my homework instructions that it says to not use kepler's 3rd law. Any advice?

 

[gneill]

You're given the period and the velocity. You know how to compute the length of the path (it's a circle of some radius r). So you can find an expression for the radius from that information. That's the radius about the center of gravity.

Next write the expression for the centripetal force felt by each star according to the usual circular motion equations.

Next write the expression for the force due to gravity that is felt by each star. Note that the distance between the stars is twice the radius that you found above.

What's supplying the centripetal force? Equate things accordingly and substitute for the radius as required.

 

[SammyS]

The magnitude of the gravitational force between two objects of mass m₁ and m₂, separated by a distance of d:

$$F_{\text{Gravity}}=G\frac{m_1 m_2}{d^2}$$

 

[joeyjane]

Alright, so far I have F_g=(G(2m₁))/R²

(2m₁ because the masses are equal)

and I'm trying to use C=2$$\pi$$R

and A_c=v²/R

This is where I start to get a little lost when I'm trying to apply the period somehow to get R. I don't really think I need the centripetal acceleration formula, but it was the only one I could think of that had both V and R in it. (sorry I'm not as quick at grasping these concepts, this is a completely different shift from what we were just learning and I'm struggling to understand it)

 

[gneill]

Alright, so far I have F_g=(G(2m₁))/R²

(2m₁ because the masses are equal)

and I'm trying to use C=2$$\pi$$R

and A_c=v²/R

This is where I start to get a little lost when I'm trying to apply the period somehow to get R. I don't really think I need the centripetal acceleration formula, but it was the only one I could think of that had both V and R in it. (sorry I'm not as quick at grasping these concepts, this is a completely different shift from what we were just learning and I'm struggling to understand it)

In the gravitational force equation, note that 2m is not the same as m². The gravitational force is

$$F_g = G \frac{m_1 m_2}{d^2} = G \frac{m^2}{d^2}$$

when m₁ = m₂ = m, and d is the distance between the masses. NOTE THAT THE DISTANCE IS NOT R IN THIS CASE. It's 2R.

If you multiply the centripetal acceleration by the mass of the star, you get the centripetal force (remember, F = MA). So F_c=mv²/R