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Rods with Masses orbiting Earth

  1. Jan 19, 2015 #1
    1. The problem statement, all variables and given/known data
    Two small, equal masses are attached by a lightweight rod. This object orbits a planet; the length of the rod is
    smaller than the radius of the orbit, but not negligible. The rod rotates about its axis in such a way that it remains
    vertical with respect to the planet.
    • Is there a force in the rod? If so, is it tension or compression?
    • Is the equilibrium stable, unstable, or neutral with respect to a small perturbation in the angle of the
    rod? (Assume this perturbation maintains the rate of rotation, so that in the co-rotating frame the rod
    is still stationary but at an angle to the vertical.)

    (A) There is no force in the rod; the equilibrium is neutral.
    (B) The rod is in tension; the equilibrium is stable.
    (C) The rod is in compression; the equilibrium is stable.
    (D) The rod is in tension; the equilibrium is unstable.
    (E) The rod is in compression; the equilibrium is unstable.

    A picture may be found here http://www.aapt.org/physicsteam/2014/upload/exam1-2013-1-6-unlocked.pdf
    page 6, problem 12

    2. Relevant equations


    3. The attempt at a solution
    I know that there is a force of tension because the Earth is pulling the black mass (according to diagram) down and so the rod must being pushing the white mass up to keep it in equilibrium as it rotates. I'm just having trouble about whether the equilibrium is stable or not. Any thoughts?
     
  2. jcsd
  3. Jan 19, 2015 #2

    Nathanael

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    The force of gravity (Earth) is pulling on both masses...

    What would happen if you slightly displaced the rod so that it wasn't perfectly vertical with respect to the planet?
     
  4. Jan 19, 2015 #3
    OK, even though the Earth is pulling on both masses, isn't the rod still pulling up on the white mass to keep it taut as it revolves?
     
  5. Jan 19, 2015 #4

    Nathanael

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    I'm not sure what you mean by this.
     
  6. Jan 19, 2015 #5
    Like the system isn't collapsing as it revolves.
     
  7. Jan 19, 2015 #6

    Nathanael

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    But why would it collapse?
     
  8. Jan 19, 2015 #7
    Because the Earth is pulling the black mass with more force than the white mass. You need tension to balance it.
     
  9. Jan 19, 2015 #8

    Nathanael

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    There we go :)
    But if there was no tension then the system would not collapse, it would stretch apart! That is why I thought you were misunderstanding.

    About the stability of the object. I'm not sure if you already figured it out, but what would happen if you slightly displaced the object so it wasn't vertical? Would it start rotating more and more away from being vertical, (unstable) or would it go back to being vertical (stable)?
     
  10. Jan 19, 2015 #9
    I think it would go back to vertical, but I can't explain why.
     
  11. Jan 19, 2015 #10

    Nathanael

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    Is there any torque on the system?
     
  12. Jan 19, 2015 #11
    Yes because there is a force (tension) and it acts over the length of a rod, giving it both a force and a distance.
     
  13. Jan 19, 2015 #12

    Nathanael

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    Tension acts in the direction through the center of mass, therefore it produces no torque. Just like when the object is rotating vertically, gravity acts through the center of mass, so it provides no torque. But what about when we displace the object slightly; does gravity still act through the center of mass?
     
  14. Jan 19, 2015 #13
    Yes, gravity does acts through the center of mass.
     
  15. Jan 19, 2015 #14

    Nathanael

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    The center of mass is located in the center of rod. But the force of gravity acts on the two masses on the edges of the rod (because it said the rod's weight was negligible). Perhaps draw a free body diagram of the object slightly tilted from it's vertical position.
     
  16. Jan 19, 2015 #15
    Ok ok ,doing the FBD I see now that gravity does not act through the center of mass.
     
  17. Jan 19, 2015 #16

    Nathanael

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    So is there a net torque? If so, does it cause the rod to return to a vertical position or to rotate away from being vertical?
     
  18. Jan 19, 2015 #17
    Yes because the force is acting some distance away perpendicular to the center of mass.
     
  19. Jan 19, 2015 #18
    And it returns it to a vertical position.
     
  20. Jan 19, 2015 #19

    Nathanael

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    Your explanation is a bit unclear again. The torque from gravity on the white mass acts to rotate the object away from the vertical, but the torque on the black mass acts to rotate the object back to the vertical. Does one of these torques win? (Is one of them stronger?)
     
  21. Jan 19, 2015 #20
    I think that the torques are the same because even though the black mass is experiencing more force ,the white mass has more distance, hence balancing the torques.
     
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