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2D SHM Question with Ellipse

  1. Sep 18, 2010 #1
    The problem statement, all variables and given/known data

    A particle undergoes simple harmonic motion in both the x and y directions simultaneously. Its x and y coordinates are given by: x=asin(wt) y=bcos(wt) . Show that the quantity x(dy/dt) - y(dx/dt) is constant around ellipse, and what is the physical meaning of this quantity?

    The attempt at a solution

    Ok so I got the first part of the question:
    x(dy/dt) - y(dx/dt) = -abw

    Now I have no clue what the meaning of this quantity is? the units I see are m^2/s so is this angular area??? someone help please
     
  2. jcsd
  3. Sep 19, 2010 #2

    ehild

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    Think of planetary motion, Kepler's second law.

    ehild
     
  4. Sep 19, 2010 #3
    Ok great than you ehild! So it's just saying that the same area is covered in equal time all around ellipse.
     
  5. Sep 19, 2010 #4

    D H

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    Bad example.

    That's why this is a bad example. This is not true in this case.
     
  6. Sep 19, 2010 #5
    So what's the meaning of x(dy/dt) - y(dx/dt) then? i'm really confused now
     
  7. Sep 19, 2010 #6

    D H

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    What do you think it might mean? The rules of this forum preclude us from telling you directly. You need to show some work.
     
  8. Sep 19, 2010 #7
    Well the units of the constant are m^2/s so if that value is conserved around the ellipse this means the area per unit time is constant when traveling around ellipse which makes sense since ellipse is symmetric. I just don't know why you said Kepler's 2nd Law is a bad example it made perfect sense to me. If I'm wrong, can you point me in a better direction about how to think of this problem.
     
  9. Sep 19, 2010 #8

    ehild

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    That quantity is constant during the motion, so it is conserved. What conservation laws do you know?

    ehild
     
  10. Sep 19, 2010 #9
    Oh I just read ahead, gotcha it's conservation of angular momentum! Thanks
     
  11. Sep 19, 2010 #10

    ehild

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    Your solution is almost quite correct. Well, the quantity in question is the magnitude of the angular momentum divided by the mass. Anyway, the areal velocity is

    dA/dt=1/2 [rxv]=1/2(yvx-xvy)ez,

    half the vector product of the position vector with the velocity. This is constant both here and for the orbits of planets. The angular momentum is L=m [r x v]. It is conserved when a body moves in a central force field. Gravity is a central force. The force in your problem is also central, as the acceleration is anti-parallel with the position vector.

    ehild
     
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