# Period of tethered satellites

1. Aug 22, 2014

### jhoge

1. The problem statement, all variables and given/known data

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.

2. Relevant 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)

3. 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!

2. Aug 22, 2014

### jhoge

forgot to include photo

here's a picture of the problem

#### Attached Files:

• ###### tethered_satellites.jpg
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3. Aug 22, 2014

### Orodruin

Staff Emeritus
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...

4. Aug 22, 2014

### 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?

Last edited: Aug 22, 2014
5. Aug 22, 2014

### jhoge

also thank you for the prompt response

6. Aug 22, 2014

### Orodruin

Staff Emeritus
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?

7. Aug 22, 2014

### 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

8. Aug 22, 2014

### jhoge

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

9. Aug 22, 2014

### Orodruin

Staff Emeritus
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 ...)

10. Aug 22, 2014

### 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

11. Aug 22, 2014

### jhoge

but how can i use the period?

12. Aug 22, 2014

### Orodruin

Staff Emeritus
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 ...

13. Aug 22, 2014

### 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 :/

14. Aug 22, 2014

### Orodruin

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

15. Aug 22, 2014

### 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.

16. Aug 22, 2014

### Orodruin

Staff Emeritus
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 ...

17. Aug 22, 2014

### 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.

18. Aug 22, 2014

### Orodruin

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

19. Aug 22, 2014

### jhoge

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

20. Aug 22, 2014

### 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.