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Is this force a central field ?

  1. Sep 26, 2004 #1

    cj

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    Is this force a "central field" ?

    A partical moves in a spiral orbit given by:

    [tex]r=a\theta[/tex]

    if [tex]\theta[/tex] increases linearly with time, is the force a central field? If not, how would [tex]\theta[/tex] have to vary with time for a central force?

    I believe that a central force is a function only of the scalar distance, r, to the force center, and its direction is along the radius vector.

    I also believe that the angular momentum of a particle is constant when it is moving under the action of a central force.

    Even though I seem to remember the above, I'm at a loss see whether or not the above is a central force -- nor how to modify it to make it one??
     
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  3. Sep 26, 2004 #2

    Gokul43201

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    What is the angular momentum, in terms of r and \theta ?
     
  4. Sep 26, 2004 #3

    cj

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    [tex]L = r \times mv = mr^2\dot{\theta}[/tex]

    I think ...

    So, this means

    [tex]L = r \times mv = mr^2\dot{\theta} = ma^2\theta^2\dot{\theta}[/tex]

    [tex]\dot{\theta}[/tex] is constant (since it varies linearly with t), but [tex]\theta^2[/tex] is not constant.

    So -- this does not correspond to a central field? Or am I still missing something?
     
  5. Sep 26, 2004 #4

    robphy

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    From the trajectory, you can calculate the velocity vector and the acceleration vector. [Using Newton's Law, you can find the force vector.] You can express this as a vector field. Check if its curl is zero... (I believe this is necessary but not sufficient for a central force).

    Of course, you can probably just check if the acceleration vector is radial.
     
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