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Math Help - Central force problem

  1. #1
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    Central force problem

    Central force F=-ar/(r^3) & Central potential energy,U=-(a/r)
    (not U_eff)
    Find the nature of orbits if (i)a>0 and (ii)a<0

    If we remember the attractive central force E=E(r) diagram,i.e.the one showing the graph of U_eff,we only need to know E_total=K+U.
    Where only PE is given.
    We see,

    U= -integration[F.dr]=integration[dW]=-integration[dK]=-K

    Then K=a/r and U=-(a/r)

    So,the E=K+U=0

    Then in positive and negative both caes we get a parabolic orbit.

    Please check if i am correct.
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  2. #2
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    Quote Originally Posted by kolahalb View Post
    Central force F=-ar/(r^3) & Central potential energy,U=-(a/r)
    (not U_eff)
    Find the nature of orbits if (i)a>0 and (ii)a<0

    If we remember the attractive central force E=E(r) diagram,i.e.the one showing the graph of U_eff,we only need to know E_total=K+U.
    Where only PE is given.
    We see,

    U= -integration[F.dr]=integration[dW]=-integration[dK]=-K

    Then K=a/r and U=-(a/r)

    So,the E=K+U=0

    Then in positive and negative both caes we get a parabolic orbit.

    Please check if i am correct.
    That's one possibility...

    Doesn't the form of the force law look familiar? That should lead you to what the correct answer should be. Now you have to prove it.

    -Dan
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  3. #3
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    There is the problem.The -ve force obviously corresponds to elliptical path...
    But I am getting total energy 0.
    But I annot understand where did I go wrong?
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  4. #4
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    OK,thank you.I got it.

    Consider a>0.
    Then, -(m(v^2)/r)=-(a/r^2)
    or,m(v^2)=(a/r)
    Hence total energy E=K+U= -(1/2)(a/r)

    =>elliptic orbit.

    Consider a<0.

    Then,F=+b/r where b=-a>0
    and U_p=b/r

    K is itrinsically positive.So,total energy positive.

    From e=sqrt[1+{(2L^2*E)/(m*b^2)}]
    e>1.
    hence hyperbolic trajectory.
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