What is the force of tension in a string connecting two revolving satellites?

[jhoge]

Homework Statement

Two satellites connected by a string revolve in concentric circular orbits of radius r and 2r
Two satellites of equal masses m revolve around the planet of mass M. The satellites have extremely small mass compared to the planet, m≪M. The radii of the orbits of the satellites are r and 2r. The satellites are connected by a light string, directed along the radius of the orbit, that keeps their periods of revolution equal. Find the force of tension Fτ in the string.

Homework Equations

I first drew an FBD and wrote the Newton's Second Law equations for both satellites, keeping in mind circular motion: (equation 1 corresponds to the inner satellite, and 2 to the outer)

(1) ∑ F = m(a_c1) = F_g1 - Fτ

(2) ∑ F = m(a_c2) = F_g2 + Fτ

Also, since I am given that the periods of revolution are constant, I used the following formula's defining the period:

(Kepler's 3rd law) T^2 = 4*pi^2*α^3/(G*M)
[α is the semi-major axis of an elliptical orbit... in this case either of the two given radii]
[ G is the universal gravitational constant, but can be expressed simply as G for the sake of the problem.]

and T = 2*pi*r/v_orbital, which can also be written as 2*pi/√(a_c/r)

The Attempt at a Solution

I used the two equations for the period to determine a value for a_c1 (i chose to work with the inner sattellite, so α = r..)

Thus:

T^2 = 4*pi^2*α^3/(G*M) = 4*pi^2*r/a_c1, given that α = r

we should get 4*pi^2*r^3/(G*M) = 4*pi^2*r/a_c1

--> G*M/r^2 = a_c

The issue with this is that if this is indeed the centripetal acceleration, then Fτ = zero when I plug the centripetal acceleration into equation (1)..

I'm stumped. Any help would be appreciated. Thank you in advance!

 

[jhoge]

forgot to include photo

here's a picture of the problem

 

[Orodruin]

Kepler's laws hold for an orbiting object which is only subject to the gravitational force from the object it orbits. Is the only force on either object gravity?
Hint: The law would also give different periods for the two satellites, and you have already stated they should be the same...

 

[jhoge]

There are two forces acting upon each mass: the gravitational force due to the planet and the tension force of the tether connecting them. I guess this would mean that I cannot use kepler's equation for period then... In that case, how can I endeavor to find the centripetal acceleration?

 

[jhoge]

also thank you for the prompt response

 

[Orodruin]

What is the acceleration needed for an object to stay in a circular orbit? How can you relate this to the forces acting on the satellites?

 

[jhoge]

the acceleration needed for an object to stay in circular orbit is a_c = v^2/r. I related this to the forces acting upon the satellite through my Newton's second law equations, considering the sum of the forces acting upon the satellite to be equal to m*a_c

 

[jhoge]

however this leaves me with a velocity that I cannot find..

 

[Orodruin]

What is the velocity of an object in circular orbit at radius r with period T? (Or angular velocity ω if you prefer - I know I do ...)

 

[jhoge]

The velocity of an object in circular orbit at radius r with period T is v = 2*r/T, or for angular ω = 2*pi/T

 

[jhoge]

but how can i use the period?

 

[Orodruin]

The velocity of an object in circular orbit at radius r with period T is v = 2*r/T, or for angular ω = 2*pi/T

Well, circumference is 2 pi r, so you are missing a pi but it does not matter much. What can you say about the relation between the angular velocities/periods of the two satellites?

Hint: You have already said it ...

 

[jhoge]

oh! thanks for pointing out the mistake. The fact that the satellites have the same period indicates that they have the same angular velocity. I feel like there is something that is blatantly obvious that I am missing :/

 

[Orodruin]

So what is the relation between the centripetal acceleration of each satellite and the angular velocity?

 

[jhoge]

the centripetal acceleration of each satellite is going to be equal to ω^2*r, however I am lacking a value for ω, and I'm not sure how to find it.

 

[Orodruin]

That is the centripetal acceleration for the inner satellite. That of the outer is two times that because it is at a larger r. Now if only you had two separate formulas that contain these and one other unknown quantity ...

 

[jhoge]

so i solved for ω^2*r using one of my equations and plugged that into the other to solve for tension. My result was F_T= 3*m*M*G/8/r^2, which is wrong according to the grader.

 

[Orodruin]

Can you show your equations and how you got there? It is not the same result as I obtained.

 

[jhoge]

never mind, it was an algebra error! Thank you friend! I really appreciate the help :)

 

[jhoge]

i reworked it, having left out a 2. This threw my answer off by a lot, because I didn't distribute it and so on.

 

[ClaudioCosta]

Did anybody get the correct answer ?