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Gravity due to a uniform ring of mass

  1. Apr 21, 2016 #1
    1. The problem statement, all variables and given/known data
    Several planets (Jupiter, Saturn, Uranus) are encircled by rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ring-like structures. Consider a homogeneous thin ring of mass 2.1 x 1022 kg and outer radius 4.3 x 108 m (the figure). (a) What gravitational attraction does it exert on a particle of mass 76 kg located on the ring's central axis a distance 4.5 x 108 m from the ring center? (b) Suppose that, starting at that point, the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring?

    2. Relevant equations
    Kepler's 3rd Law: T^2=(4*π^2/GM)*R^3
    F=GmM/R^2

    3. The attempt at a solution
    I haven't really been able to attempt this question, I have absolutely no idea how to establish a relation between the mass on the axis and the ring. Please enlighten me with your genius ideas!
     

    Attached Files:

  2. jcsd
  3. Apr 22, 2016 #2

    gneill

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    The problem should remind you of the similar one involving electric force: a ring of charge acting on a point charge along the axis of the ring.
     
  4. Apr 22, 2016 #3
    I haven't previously studied electric force, I'm not sure what you meant.
     
  5. Apr 22, 2016 #4

    haruspex

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    1. In what direction will the force from the whole ring be?
    2. Consider a small segment of the ring, length rdθ, say. What force does it exert at the given point on the axis? What component of that acts in the direction in (1)?
     
  6. Apr 22, 2016 #5

    gneill

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    Mentor's note:
    Please note that the the thread title has been changed to: Gravity due to a uniform ring of mass.

    The original title, "Challenging planetary problem", was too vague and non-descriptive of the actual problem.
     
  7. Apr 22, 2016 #6
    I'm guessing for 1) the force would be towards the centre, and 2) I don't see how r*dθ would exert a force, pls instruct me on how to analyse this system, thanks
     
  8. Apr 22, 2016 #7
    Okay I'll be more specific next time
     
  9. Apr 22, 2016 #8

    haruspex

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    The ring as a whole has a given mass M, so it has a mass per unit length density. If the ring has radius r, and a small piece of it subtends angle dθ at the centre of the ring then it has a mass (rdθ)M/(2πr). That will exert a gravitational force on the particle.
     
    Last edited: Apr 22, 2016
  10. Apr 22, 2016 #9
    I am starting to understand. Is this related to Kepler's Second Law? I will need to go thru the textbook again if it is
     
  11. Apr 22, 2016 #10

    haruspex

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    No, it's Newton's law of gravitation, which you quoted up front. What is the gravitational attraction between the particle and the small element of the ring?
     
  12. Apr 22, 2016 #11
    Right, F=GMm/R^2
    where m=(rdθ)M/(πr2)
     
  13. Apr 22, 2016 #12

    haruspex

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    I made an error in my post #8, which you have blindly copied. Trust no-one!
    Also, be careful with the M's - you have them mixed up. I suggest fixing on M for the mass of the ring and m for the mass of the particle.
     
  14. Apr 23, 2016 #13
    I have managed to got the correct answer for this question, thank you very much for the help!!
     
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