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Theory Question - Uniform Circular Motion

  1. Nov 30, 2011 #1
    -Question Removed-
     
    Last edited: Nov 30, 2011
  2. jcsd
  3. Nov 30, 2011 #2

    Doc Al

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    What was your reasoning? Show how you determined that answer.
     
  4. Nov 30, 2011 #3
    You teacher was right, the correct answer is false.

    If the radius of an object in orbit increases, the speed of said object must decrease.
     
  5. Nov 30, 2011 #4
    v=2*pi*radius/T

    As the radius increases, the speed also increases.
     
  6. Nov 30, 2011 #5

    Vanadium 50

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    And what about T?
     
  7. Nov 30, 2011 #6
    It stays constant in order to maintain uniform circular motion.
     
  8. Nov 30, 2011 #7

    Doc Al

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    You assume that T is constant. Rethink that.
     
  9. Nov 30, 2011 #8
    Shouldn't angular velocity always be constant no matter what the radius is?
     
  10. Nov 30, 2011 #9

    Doc Al

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    No it doesn't.

    Try getting back to basics. What force holds the satellite in orbit? Apply Newton's 2nd law.
     
  11. Nov 30, 2011 #10

    Doc Al

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    Your formula was for linear velocity, not angular velocity. And no, the angular velocity is not constant as the radius changes.
     
  12. Nov 30, 2011 #11
    (mv^2)/r = (Gmm)/r^2

    v^2 = Gm/r

    I'm confused. What do I do here?

    Yes, but 2PI/T is the formula for angular velocity, correct? In order to maintain uniform circular motion, how can the angular velocity not be constant?
     
    Last edited: Nov 30, 2011
  13. Nov 30, 2011 #12

    Doc Al

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    Excellent. Now realize that Gm = (v^2)r must be constant. That's what you need.

    Yes.

    For a given radius the angular velocity will be constant, but not if you change the radius.
     
  14. Nov 30, 2011 #13

    Doc Al

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    Sure it does. The formulas are derived for the case of uniform circular motion.
     
  15. Nov 30, 2011 #14
    Even if you change the radius, the angular velocity will stay the same.

    angular velocity=2PI/T

    You can change the radius as much as you'd like, but the angular velocity will stay the same.

    I can see why v^2 = Gm/r would work with large bodies that involve large gravitational forces, but what if we assumed gravity is negligible? Then, I guess we can use:

    v=2*PI*radius/T

    And from that we can see, since angular velcoity is constant (2PI/T), as you increase the radius, the velocity also increases.

    edit: Actually, it's impossible for large bodies of mass to maintain uniform circular motion after the radius is changed. If we plug in the v=2*PI*radius/T into the equation, then we can see that:

    v^2 = Gm/r
    4(PI^2)(r^2)/T^2 = Gm/r
    4(PI^2)(r^3)/T^2 = Gm

    Gm has to stay constant. 4PI^2 has to stay constant. In order to maintain uniform circular motion, T has to stay constant, therefore it is impossible to change the radius AND maintain uniform circular motion at the same time.
     
    Last edited: Nov 30, 2011
  16. Nov 30, 2011 #15

    Doc Al

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    Why do you think this? What's your reasoning?

    This is a question about satellites in orbit. You can't very well ignore gravity.

    You can always use that for uniform circular motion. But it doesn't tell you much.

    You merely assume that angular velocity is constant. Why?
     
  17. Nov 30, 2011 #16
    angular velocity = 2PI/T

    Because angular velocity comes from multiplying 2 times PI divided by T, the radius has no effect on the angular velocity.

    Assume the radius is 5 m and the period is 10 seconds.

    angular velocity 1 = 2PI/10

    Assume the radius is 10 m and the period is 10 seconds.

    angular velocity 2 = 2PI/10

    angular velocity 1 = angular velocity 2
     
  18. Nov 30, 2011 #17
    You can also think of it logically.

    To maintain the same angular velocity, if you increase the radius of anything, it will need to cover the larger amount of arc length over the same amount of time. Therefore, the velocity must increase, to maintain the same amount of time taken.
     
  19. Nov 30, 2011 #18

    Doc Al

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    For a given orbital radius, the angular velocity is constant. That's what uniform circular motion means. But when you change the radius, the angular velocity changes to a new value. So you cannot assume that angular velocity remains constant for different radii.
     
  20. Nov 30, 2011 #19
    Alright, so you agree that when there is uniform circular motion, the angular velocity is constant?
     
  21. Nov 30, 2011 #20

    Doc Al

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    Sure, if angular velocity remains the same. But it doesn't!

    (By the way, for uniform circular motion both angular and linear velocity are constant.)
     
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